Evolution of Navier-Stokes and Euler vortex filaments

Programa de doctorado matemáticas PhD Dissertation Evolution of Navier-Stokes and Euler vortex filaments Author Antonio Hidalgo Torné Under the supervision of Francisco Gancedo García May 2024 A mi mujer, Maite Agradecimientos Aunque un proyecto de estas magnitudes tiene inevitablemente algunos baches, en la meta solo queda la alegría de un buen final. Y si bien me siento satisfecho con el resultado, soy aún más afortunado por la compañía que he tenido en el camino. Paco, muchas gracias por dirigirme la tesis. Por tu tiempo, cercanía, paciencia, confianza... Trabajando contigo, los obstáculos no son más que motivación para mejorar. Eres, sin duda, un modelo a seguir. Una parte fundamental de este trabajo ha sido la colaboración con Fran. He tenido la suerte de disfrutar de la compañía de todas las personas con las que he trabajado, pero hacerlo con un amigo es otro nivel. Muchas gracias por toda tu ayuda en la etapa final de mi tesis. I would like to thank Luis Vega, Daniel Faraco, Christian Seis, Huy Nguyen and Martina Magliocca for being part of my committee and taking the time to read my thesis. También a Paco Torres, Eduardo García y Claudia García. Que estéis de suplentes es para mí todo un lujo y tranquilidad. I am looking forward to starting my new position at the Max Planck Institute in Leipzig, which has treated me so well during my research stays. I particularly appreciate the warmth of the Applied Analysis Research Group. I am deeply grateful to László Székelyhidi for giving me the opportunity to continue my research there in the upcoming years, and for sharing a challenging project with me. Also to Jan Burczak, for hosting me during my stays and patiently teaching me convex integration for another very interesting project. Elena, Fran, Víctor: Con vosotros allí, es más fácil despegarse de Sevilla. ¡Nos vemos pronto! También estoy enormemente agradecido a Ángel Castro y Eduardo García, por tomaros el tiempo en conocer mi trabajo, y recomendarme para los postdocs que he solicitado. En general, estoy muy agradecido con el grupo de investigación que iii Agradecimientos iv tenemos en Sevilla. De una forma u otra, todos me habéis ayudado en la tesis sin dudarlo. Echaré de menos el IMUS, que después de tantos años es mi segunda casa. Quiero dar las gracias a todo el personal, especialmente a Teresa, que siempre tiene una sonrisa y solución para cualquier problema. A mis compañeros de doctorado, que hacéis que ir al trabajo sea siempre agradable. No os dedico el tiempo que merecéis. Elena, eres la mejor hermana académica mayor que podía tener. Paula, ¡mucha suerte con tu tesis! Más allá de mi entorno científico, agradezco mucho a mi familia la fe ciega y el apoyo que siempre me han dado, sin necesidad de entender qué estoy haciendo. Especialmente a mis hermanos, Manuel y Carmen, que tienen gran parte de culpa en que hoy sea quien soy. Quiero agradecer también a mi familia política, en la que me siento uno más y a la que aprecio muchísimo. Gracias a mis amigos, porque siempre están ahí. Especialmente a Ale, Monte y Eugenio, que me acompañan desde tiempos inmemoriales. También a José Ignacio, que aunque duramos poco en la misma ciudad, cada vez que nos vemos no ha pasado el tiempo. A mis amigos del Aula de Cultura, a los amigos que he conocido a través de Maite... los momentos con vosotros son una recarga de pilas. A veces, por tenerte siempre presente, te dejo para el final. Infinitas gracias a mi mujer, Maite, porque más que un apoyo es un motor para mí. Gracias por tu compañía en todo momento, por darme ilusión y motivación cada día, y por hacerme vivir en una nube permanente. This thesis has been supported by the grant FPU19/02748, funded by Ministerio de Ciencia, Innovación y Universidades. The works in this thesis have been partially supported by the ERC through the Starting Grant H2020-EU.1.1.-639227, by the Fundación de Investigación de la Universidad de Sevilla through the grant FIUS23/0207, by MICINN through the grants EUR2020-112271 and PID2020- 114703GB-I00 and by Junta de Andalucía through the grant P20-00566. Resumen La comprensión de las ecuaciones de Euler y Navier-Stokes es esencial para el desarrollo matemático de la mecánica de fluidos, ya que sirven como pilares de diversos modelos dentro de este campo. Por otra parte, el estudio de filamentos de vorticidad viene motivado por su semejanza con algunos fenómenos en la naturaleza, pero nuestro conocimiento actual sobre ellos es limitado. Parte de la dificultad del estudio matemático de este tipo de estructuras es su singularidad. En particular, la energía alrededor del filamento y la velocidad efectiva sobre el mismo son infinitas, haciendo que plantear su evolución sea un problema no trivial. Los resultados que se encuentran al respecto en la literatura requieren pequeñez en el dato o tiempo de existencia, algún tipo de simetría, modificaciones en las ecuaciones, o regularización del dato inicial. Es por ello que el estudio de filamentos de vorticidad en las ecuaciones de Euler y Navier-Stokes es un tema candente en la literatura. Esta tesis se centra en el desarrollo de dos nuevos resultados de dinámica de filamentos de vorticidad. En el capítulo 1, se introducen las ecuaciones de Euler y Navier-Stokes. A continuación, se abordan consideraciones sobre los filamentos de vorticidad, al tiempo que se revisa la literatura existente sobre el tema. El capítulo 2se basa en el artículo [52]. En él, se estudia el problema de Cauchy de un filamento de vorticidad helicoidal en las ecuaciones de Navier-Stokes. Concretamente, se demuestra la existencia global en tiempo, unicidad y regularización de soluciones para este problema. Dado que las ecuaciones de Navier-Stokes no mantienen el swirl helicoidal nulo, este es el primer resultado de existencia global en tiempo de filamentos de vorticidad sin restricción de talla en presencia de vortex stretching. Para obtener dicho resultado, en primer lugar se demuestra que la evolución de filamentos de vorticidad periódicos en una dirección está bien vii Incompressible flows and vortex filaments 4 the velocity field, and the scalar variable pis the pressure. The constant ν > 0 denotes the viscosity of the fluid. If we set ν= 0 in the previous system, we obtain the Euler equations. Regarding differential operators, the term ∇·v= d X i=1 ∂vi ∂xi is the divergence of v, which is identically zero for incompressible fluids. Imposing zero divergence in a fluid implies that each subdivision of the domain retains its volume as it evolves with the fluid. The gradient operator is ∇= ∂ ∂x1 ,..., ∂ ∂xd!, and ∆ = d X i=1 ∂2 ∂x2 i is the Laplace operator. Lastly, v·∇v= d X i=1 vi ∂v ∂xi is a convective term. An essential variable in the study of fluids is the vorticity. In three dimensions, it is defined as ω= curlv=∇∧v, where ∧denotes the cross product. If we apply the curl operator to the system (1.1.1), we obtain the Navier-Stokes (or Euler if ν= 0) system in vorticity form (∂tω(t, x) + v(t, x)·∇ω(t, x)−ω(t, x)·∇v(t, x) = ν∆ω(t, x), t ∈R+, x ∈Rd, v(t, x) = ∇∧(−∆)−1ω(t, x). (1.1.2) It is important to note that one can recover the velocity field from the vorticity field through the so called Biot-Savart law, which in R3reads as v=∇∧(−∆)−1ω=ZR3K3D(x−y)∧ω(y)dy, (1.1.3) where K3Dis the three-dimensional Biot-Savart kernel K3D(x) = −1 4π x |x|3. 5 1.2. Introduction to vortex filaments The vorticity is related to the rotational motion of a fluid and therefore is also associated with turbulence, which is the chaotic motion of fluid particles. In many interesting problems, the vorticity of a fluid is space-localized, or at least, there is a zone with much higher vorticity than in the rest of the space. In such problems, we are usually interested in understanding what happens to the turbulent zone as time evolves. 1.2 Introduction to vortex filaments In this thesis, we will focus on the study of vortex filaments, a subject that was originally addressed by classical physicists such as Kelvin [93] and Helmhotz [65]. Definition 1.2.1. A velocity field vis a vortex filament if the associated vorticity is supported on a curve, and the direction of the vorticity is tangent to the curve. An intuitive way to visualize a vortex filament is to imagine a tornado, where particles move violently around the curve that defines the shape of the tornado. The rigorous study of the evolution of a vortex filament is difficult to model because the velocity and kinetic energy around the filament are infinite. We can easily see this in the particular case of a vortex filament supported on a straight line. Due to the analogy between the Biot-Savart law in fluids and electromagnetism, the velocity associated to a straight vortex filament is equivalent to the electric field created by an infinite conducting wire. In this case, it is well-known that the magnitude of the electric field behaves as d−1, where dis the distance to the wire. From this, we see that uis not bounded around the filament and it is easy to check that |u|2is not integrable in any neighbourhood intersecting the filament. Despite this fact, the direction of the velocity defines rotations around the axis, so they do not affect its motion. Therefore, the vortex column is a stationary solution of the Euler equations. The evolution of the vortex column can also be calculated in Navier-Stokes (1.1.2) and it is known as the Lamb-Oseen vortex. Due to its symmetry, it is easier to express it in cylindrical coordinates (r, θ, z). We have ωLO r(t, x) = 0, ωLO θ(t, x) = 0, ωLO z(t, x) = Γ 4πνt e−r2 4νt , vLO r(t, x) = 0, vLO θ(t, x) = Γ 2πr1−e−r2 4νt , vLO z(t, x)=0,(1.2.4) where Γis the circulation of the vortex core and it is a measure of the strength of the vortex. Scenarios get complex when one consider a curved vortex filament. In this case, a particular part of the filament is affected by the velocity field generated by other Incompressible flows and vortex filaments 6 parts of the filament. One possible approach is to study the evolution of a thin tube of vorticity and see what happens when the radius cof the tube vanishes. With this method, we will obtain a rotation component around the filament, which does not affect its motion. Ignoring this rotation, in [31] was derived that the velocity over the filament should be, to higher order, log cin the binormal direction. See [3, Section 5.2] for a reference in English. Therefore, the binormal component should lead to the evolution of the filament, but it also becomes infinite when c= 0. Thus, to study the evolution of the filament, additional considerations are needed. A common approach in Euler equations is to consider a vorticity tube around the curve and study the limit when the width cvanishes if simultaneously the time is rescaled as (−log c)−1. This leads to the question of whether, with this rescale, there are Euler solutions whose vorticity is concentrated near a curve that evolves as the binormal flow equation. That is, the evolution of a curve α(t, s), with sthe unit speed parametrization, given by the equation ∂tα(t, s) = ∂sα(t, s)∧∂2 sα(t, s).(1.2.5) This question is known in the literature as the vortex filament conjecture. This approach to study vortex filaments is known as the local induction approximation, and equation (1.2.5) is also called the vortex filament equation. Some sufficient conditions on Euler solutions to satisfy this conjecture are given in [69]. This conjecture is only known to be true for a straight line, a circle, and a helix. Part of the difficulty is that the binormal flow itself is a challenging equation that exhibits complex behavior (see e.g. [70,9,5] and the references therein). The existence of helical vortex tubes with small cross-section solving Euler equations was proven in [1]. In [33], this conjecture was proved when the filament is a helix by constructing a family of solutions without swirl that tend asymptotically to the binormal flow. In [38,63], this result is extended to more general initial data. See [23] for a result with helical symmetry on bounded domains. In the axisymmetric case without swirl, solutions concentrated close to a curve moving in its binormal direction were constructed in [47]. In [14], this result is generalized to more general initial data. See [22] and the references therein for more results on the existence, uniqueness, and stability of vortex rings. In [34], the leapfrogging phenomenon in vortex rings has been rigorously studied. The case of column vortices can be reduced to two dimensions and is equivalent to the point-vortex model. See [78,49,37] and the references therein for results in the point-vortex setting. See also [64] for a recent result on leapfrogging in this setting. In Navier-Stokes equations, the smoothing effect of the Laplacian is sufficient for the problem to make sense. The drawback of considering the Navier-Stokes equations is that the vorticity support instantaneously becomes the whole space. However, it is expected that the vorticity remains concentrated mainly near a 7 1.2. Introduction to vortex filaments curve for times t≪ν−1. More specifically, due to the scaling of the equation, the vorticity should have a behavior similar to a vortex tube with core radius √νt. See [88] for a derivation of the velocity of a viscous vortex ring. Hence, it is also possible to study the evolution of the filament in this setting. Despite having a general heuristic, the state of the art is still far from having such generality. The global well posedness for small initial vorticity belonging to the Morrey space M3/2, which includes vortex filaments, was demonstrated in [59]. For large data, the local existence and uniqueness under some restrictions of solutions for an arbitrary smooth and closed curve have been demonstrated in [12]. The global existence of solutions without size restriction is known only under symmetry conditions. The translational symmetry case can be reduced to a 2D problem, where there is abundant literature on the subject. See [60] for an existence proof, [48] for a uniqueness result, and [13] for a 3D approach. For axisymmetric fluids without swirl, the global existence of axisymmetric solutions was shown in [43], and the uniqueness in the symmetry class in [51]. The first main result of this thesis consists in proving the global existence of solutions when the filament is a helix. This is developed in Chapter 2, which is based on the submitted paper [52]. Regarding control on how the filament evolves, it has been shown in [51] that the vorticity of the unique axisymmetric solution for a circular filament remains close to a circle moving in its binormal direction. This result has been sharpened in [50] for vorticities sufficiently large compared to viscosity. For a small initial vorticity supported on a closed curve, it was proven recently in [45] that there is a solution with the vorticity remaining close to the evolution of the curve in its binormal direction. The stability of vortex filaments is a complex issue. On the one hand, there are natural phenomena that resemble vortex filaments and seem to evolve in a stable manner. We can think, for example, of tornadoes, smoke rings created by volcanoes, or bubble rings created in water. Of course, even if the area where turbulence is concentrated is visually small, it is difficult to bound it rigorously. Experimentally, it is observed that very thin vortex filaments are unstable [95,94]. Mathematically, the stability of a vortex column is known only as long as the perturbation amplitude is much smaller than the size of the vortex core [3, Chapter 5]. See [13] and the references therein for further discussion on why stability of vortex filaments is not expected. In this regard, it is worth noting that the nonuniqueness of Leray solutions to the forced Navier-Stokes equation has been proved recently for a specific unstable vortex ring [2]. See [6,7,8,10] for stability results on the binormal flow. Motivated by the instability of vortex filaments, it is natural to look for solutions with turbulence. The second main result of this thesis consists in constructing, by means of convex integration, the first weak solution for Euler equations Incompressible flows and vortex filaments 8 with vortex filament initial data. This is developed in Chapter 3, which is based on the submitted paper [53]. 1.3 General notation In Section 1.2 we have introduced the notation for several differential operators. In this section, we set basic notation we will use in the following chapters of this thesis. More specific notation will be set in each chapter. •Given two comparable quantities a, b ∈R, we denote a≲bif there exists a constant C > 0such that a≤Cb, and also a∼bif additionally b≲a. •We denote as a⊗bthe matrix whose component (i, j)is aibj. •We denote X([0, T];Y)by the set of functions fsuch that for each t∈[0, T] we have f(t)∈Yand ||f||Y(t)∈X. If Xis a Lebesgue space, we only ask that f(t)∈Yfor almost every t∈[0, T]. •We reserve the letter Kto denote the Biot Savart kernel, and Nfor the Newtonian potential, i.e., the inverse of the Laplacian. We denote et∆by the Heat operator, and also HDif we need to specify the domain D. CHAPTER 2 Navier-Stokes helical vortex filaments 2.1 Introduction ............................... 10 2.1.1 Introduction to helical symmetry ............... 10 2.1.2 Main result and sketch of the proof .............. 14 2.1.3 Notation specific to the chapter ................ 16 2.2 Construction of fixed-point scheme .................. 17 2.2.1 Geometrical setting ....................... 17 2.2.2 Approximate solution ...................... 20 2.2.3 Unknowns and fixed-point scheme ............... 20 2.3 Preliminar analysis ........................... 23 2.3.1 General analysis in R2×T................... 23 2.3.2 Analysis close to the curve γ.................. 25 2.3.3 Morrey spaces .......................... 26 2.4 Fixed-point Estimates, Short Time Existence ............. 27 2.4.1 Preliminary estimates ...................... 27 2.4.2 Estimates on ηc1........................ 35 2.4.3 Estimates on ωc2........................ 36 2.4.4 Short time existence ...................... 39 2.5 Preservation of helical symmetry ................... 40 2.5.1 Helicity of ωg.......................... 40 2.5.2 Helicity of ω........................... 41 2.6 Global in time existence for helical initial data ............ 42 This chapter is based on [52], a joint work with Francisco Gancedo. 9 Chapter 2. Navier-Stokes helical vortex filaments 10 2.1 Introduction The aim of this chapter is to prove the global-in-time existence and smoothing of solutions to the 3D incompressible Navier-Stokes equations for any helical vortex filament initial data. To this end, we consider the Cauchy problem for Navier- Stokes equations with one periodic coordinate        ∂tu+u·∇u+∇p=ν∆u, ∇·u= 0, u|t=0 =u◦, (2.1.1) posed on [0, T]×R2×T, where u◦is the prescribed initial data. We denote R2×T as the three dimensional space periodic in the third variable. For simplicity, in this chapter we will set ν= 1. We can also write the Navier-Stokes equations in vorticity formulation        ∂tω+u·∇ω−ω·∇u= ∆ω, u=∇∧(−∆)−1ω, ω|t=0 =ω◦=∇∧u◦. Using the Duhamel’s formula, we rewrite the equations in integral form as            ω(t) = et∆ω◦−Zt 0e(t−s)∆ div(u(s)⊗ω(s)−ω(s)⊗u(s))ds, u=∇∧(−∆)−1ω, ω◦=∇∧u◦. (2.1.2) The advantage of using formulation (2.1.2) is that it allows to set a fixed point scheme where the heat kernel plays a main role. In this chapter, we choose as initial data ω◦= Γδγ,(2.1.3) where Γ∈R,γ⊂R2×Tis a smooth curve and the measure δγsatisfies ⟨δγ, φ⟩=Zγφ·dl. 2.1.1 Introduction to helical symmetry Helical symmetry implies periodicity in x3with period κ. Each value of κ∈R+ defines a different helical symmetry. Without loss of generality, we fix κ= 2πfor simplicity. 11 2.1. Introduction Definition 2.1.1. Denote as Rθthe rotation matrix around the x3axis with angle θ Rθ=   cosθsinθ0 −sinθcosθ0 0 0 1  , and Sθthe superposition of Rθand a vertical translation of size θ Sθ(x) = Rθ(x) +    0 0 θ  .(2.1.4) We say that a scalar function f(x) : R37→ Ris helical or has helical symmetry if f(x) = f(Sθx)∀x∈R3, θ ∈R. We say that a vector field v(x) : R37→ R3is helical or has helical symmetry if Rθ(v(x)) = v(Sθx)∀x∈R3, θ ∈R. We say that a non-smooth function is helical if it is limit of helical smooth functions in the sense of distributions. The same applies to vector functions. We say that Ωis a helical domain if it satisfies SθΩ = Ω for all θ∈R. We say that a helical vector field v(x)has vanishing helical swirl if v(x)·(x2,−x1,1) = 0.(2.1.5) Remark 2.1.2. Using cylindrical coordinates (ρ, θ, z)and making the change θ→ θ+z=ζ, the previous definition is equivalent to saying that a scalar function fis helical when f(ρ, ζ, z)does not depend on the value of z. Geometrically, fixing ρ and ζdetermines the helix in which the point is located. Having vanishing helical swirl is an important property because it simplifies the vortex-stretching term to a rotation [17, Lemma 2.4]. Lemma 2.1.3. Smooth solutions of the Navier-Stokes equations preserve helical symmetry. Smooth solutions of the Euler equations preserve helical symmetry and vanishing helical swirl. Proof. First, we need to show that if ωhas helical symmetry, then ∂tωtoo. Let a(x), b(x)be helical vector fields. Then, with the notation of Definition 2.1.1, direct computations show that ∆(a(Sθx)) =Rθ∆a(x), (a(Sθx)·∇)(b(Sθx)) =Rθ((a·∇)b). Chapter 2. Navier-Stokes helical vortex filaments 12 It is also easy to check that if uand ωare related through the Biot-Savart law, uis helical if and only if ωis helical. In this case, u·∇uis helical and it can be seen that the Leray projector preserves helical symmetry, so pis helical too. This proves that Euler and Navier-Stokes preserve helical symmetry. Regarding the vanishing helical swirl, we need to check that in Euler equations, (2.1.5) implies that ∂tualso satisfies (2.1.5). We multiply Euler equations by ζ= (x2,−x1,1). Then, using that uis helical we check that ∂ζp= 0 and ζ·div(u⊗u) = 0to finish the proof. Being a symmetry conserved by the Euler and Navier-Stokes equations, it is natural to study helical data. In the inviscid case, imposing a vanishing helical swirl v(x)·(x2,−x1,1) = 0 is a standard assumption in the literature that rules out vortex stretching. In this setting, [39] showed global-in-time existence of helical solutions for u◦∈Cs, for s > 1. In [40] global-in-time well-posedness is shown for ω◦∈L∞. See [62] for another global well-posedness result. Global-in-time existence was shown in [17] for vorticity in L1∩Lpwith p > 4/3and in [71] for p > 1. As we mentioned in Chapter 1, there are some results involving vortex filaments and helical symmetry. We will also mention them now for the reader’s convenience. The existence of helical vortex tubes with small cross-section solving Euler equations was proven in [1]. In [33], the vortex filament conjecture was proved when the filament is a helix by constructing a family of solutions without swirl that tend asymptotically to the binormal flow. In [38,63], this result is extended to more general initial data. See [23] for a result with helical symmetry on bounded domains. The vanishing of the helical swirl is not preserved in the Navier-Stokes equations, which makes it harder to take advantage of the helical symmetry. The cases u◦∈L2and H1have been studied in [77], see also [72]. They obtain global-in- time well-posedness of weak solutions for u◦∈L2and strong solutions for u◦∈H1. Using the helical symmetry, one can reduce the system (2.1.1) to two dimensions [75], but the resulting system is also complex. As we will see in Section 2.3, in the domain R2×Tthe Biot-Savart kernel decays as in two dimensions, implying that in general u∈ L2(R2×T)for ω∈ C∞ c(R2×T), as occurs in R2. Therefore, in Navier-Stokes equations with helical symmetry it is natural to consider local energy weak solutions. Definition 2.1.4. For any 1≤p≤ ∞, Lp uloc(R2×T) = {u∈Lp loc(R2×T) : ||u||Lp uloc = sup x0∈R2×T||u||Lp(B(x0,1)) <∞}. We define also ˚ Epas the closure of divergence free compactly supported smooth functions with respect to the norm Lp uloc. 13 2.1. Introduction It is well known that ˚ Epcoincides with the functions in Lp uloc such that ||u||Lp(B(x,1)) →0when |x|→∞[73]. We will consider initial data which does not have finite energy, so Leray-Hopf solutions does not make sense and we need another definition. We follow [73] and [16]. See Section 2.1.3 for clarification on notation. Definition 2.1.5 (Local energy weak solutions).Fix 0< T < ∞. A vector field u∈L2 loc([0, T]×R2×T)is a local energy weak solution to (2.1.1)with divergence free initial data u◦∈L2 uloc(R2×T)if: 1. u∈L∞([0, T];L2 uloc),∇u∈L2 uloc([0, T]×R2×T). 2. For some p∈L3/2 loc ([0, T]×R2×T), the pair (u, p)satisfies ZT 0ZR2×T −u∂tφ+ (u·∇)u·φ−pdiv φ+u·∆φdxdt = 0, for all φ∈C∞ c((0, T)×R2×T). 3. For any compact K⊂R2×T,u(t)→u◦in L2(K)as t→0+. 4. The function t7→ RR2×Tu(x, t)·h(x)dx is continuous in t∈[0, T)for any compactly supported h∈L2(R2×T). 5. For almost any 0< t < T and all nonnegative φ∈C∞ c((0, T)×R2×T), ZR2×T|u(t)|2φ(t)dx + 2Zt 0ZR2×T|∇u|2φdxds ≤Zt 0ZR2×T|u|2(∂sφ+ ∆φ)+(|u|2+ 2p)(u·∇)φdxds. 6. For any x0∈R2×Tthere exist a function cx0∈L3/2(0, T)such that for any x∈C(x0,3/2) and 0< t < T, p(t, x)−cx0=−1 3|u(t, x)|2+ p.v.ZC(x0,2) ∂i∂jN(x−y)(ui(t, y)uj(t, y))dy +Z(R2×T)\C(x0,2)(∂i∂jN(x−y)−∂i∂jN(x0−y))(ui(t, y)uj(t, y))dy, (2.1.6) where Nis the Newtonian potential. Remark 2.1.6. In the last property of the previous definition we use cylinders instead of balls because we work in R2×T, but this change is not relevant. The radius of the cylinder in the pressure decomposition is not important and can be taken arbitrarily big, we fix it for ease of reading. The pressure decomposition formula can be deduced from the other conditions; see [73] or [16] for a more general discussion. Chapter 2. Navier-Stokes helical vortex filaments 20 2.2.2 Approximate solution First of all, since we are going to work with vorticities and velocities defined on the straightened and curved frame, we fix more notation. We will denote with ηvorticities defined on the straightened frame, and ωvorticities on the curved frame. For velocities, we will use vfor the ones defined on the straightened frame, and ufor the ones on the curved frame. To make the fixed-point argument work, we need to construct a good initial approximation so the remaining terms are small. Our choice is a suitable mapping of the Lamb-Oseen vortex (1.2.4). In the notation of this chapter, the Lamb-Oseen vortex and its correspondent velocity are ηg(t, x) = "0 0 Γ 4πte−|ex|2 4t#t , vg(t, x) = "Γe x⊥ 2π|e x|21−e−|ex|2 4t0#t , (2.2.8) where (x1, x2)⊥:= (−x2, x1). The corresponding magnitudes on the curved frame are denoted as ωg=QΦ(χ2rηg), ug=PΦ(χ4rvg).(2.2.9) We must check which equation satisfies ωgto know which equation must satisfy the remaining terms. Note that ηgsolves the heat equation since the bilinear term vanishes due to symmetry. Mollification commutes with derivatives, and again by symmetry the bilinear term vanishes for ηg, so we can write ∂tηg+B[χrvg, ηg] = ∆ηg, and now apply QΦ(χ2r·)and Lemma 2.2.1 to obtain ∂tωg+B[e χrug, ωg]=∆ωg+QΦ(χ2r∆ηg)−∆ωg | {z } Eg .(2.2.10) 2.2.3 Unknowns and fixed-point scheme By construction, the vorticity ωgattains as initial data Γδγ, so we can expect the remaining vorticity ωcto be small for short times provided that equation (2.2.10) is a good approximation to Navier-Stokes. It is convenient to split the remaining vorticity into two terms: ωc1to track the evolution close to the curve, and ωc2to do the rest. We will keep the same superindexes for the associated vorticity in the straightened frame and velocities. We leave without superindex the total vorticity 21 2.2. Construction of fixed-point scheme ω=ωg+ωc=ωg+ωc1+ωc2. Regarding velocities, uwill always be related to the corresponding ωby the Biot-Savart law u∗=∇∧(−∆)−1ω∗, and except for vgand ug,vand uwill always be related by v∗:=P−1 Φ(e χru∗).(2.2.11) With respect to vorticities, we choose the following relations ωg:=QΦ(χ2rηg), ωc1:=QΦ(χ2rηc1), ηc2:=Q−1 Φ(e χ2rωc2). We define ηc1through the equation ∂tηc1+B[v, ηc] + B[v−χrvg, ηg] = ∆ηc1,(2.2.12) with ηc1(0) = 0. This implies that (applying QΦ(χ2r·)) ∂tωc1+B[e χru, ωc] + B[e χr(u−ug), ωg]=∆ωc1+QΦ(χ2r∆ηc1)−∆ωc1 | {z } Ec .(2.2.13) The remaining ωc2must close Navier-Stokes on the curved frame, which is the physical one, so ∂tωc2+B[(1 −e χr)u, ω] = ∆ωc2−Eg−Ec,(2.2.14) with ωc2(0) = 0. To set the fixed-point scheme, the next step is to define the norm for each variable. Norms We will use the same norms as [12], with the difference that for us y∈R2×T instead of R3. We will have one norm ||·||Nβdefined on the straightened space, to be used with terms close to γ, and another norm ||·||Ffor the physical space. Their definitions are ||η∗||Nβ:= sup 0<t≤T √t*e x √t+m ⟨√t∇⟩βη∗Bx3L2 x , ||ω∗||F:= sup 0<t≤T √t(1 −e χ6r)*d √t+m ω∗L3 y , Chapter 2. Navier-Stokes helical vortex filaments 22 where β∈(0,1 4)is a real number, m≥2is a fixed integer, the function d(y):= dist(y, γ),and the norm ||f||Bx3Lp x:=X ξ∈Zˆ f(x1, x2, ξ)Lp x1,x2 . We are making a slight abuse of notation, the above norm should be written as Bx3Lp exor Bx3Lp x1,x2, but there is no confusion since we are also applying a norm in x3. For fractional β, we understand ∇βor ⟨√t∇⟩βas a Fourier multiplier. The norm that will be used at the fixed-point argument is ||ω||XT≡ ||(ηc1, ωc2)||XT:=||ηc1||Nβ+MQ−1 Φ[e χ8rωc2]N0+M||ωc2||F,(2.2.15) where Mis a constant that we will choose later. We will prove that Banach’s fixed-point theorem works in the ball Bε,T,r,M :=n(ηc1, ωc2) : ||(ηc1, ωc2)||XT≤εo, if constants are chosen appropiately. Mapping from fixed-point Theorem Now, we derive the fixed-point mapping from equations (2.2.12), (2.2.14). We will use Duhamel’s formula. For that, we choose an evolution operator for each variable, then rewrite equations (2.2.12), (2.2.14) coherently. Let S(t, s)be the solution operator associated to the Navier-Stokes linearized equation around the Lamb-Oseen vortex η(t) = S(t, s)η(s),(∂tη+B[vg, η] + B[v, ηg]=∆η, v=−∇∧(∆)−1η. We have used vinstead of vin order to avoid confusion, note that v∗=−∇ ∧ (∆)−1η∗, Biot-Savart law only holds for u∗and ω∗. We can rewrite (2.2.12) as ∂tηc1+B[vg, ηc1] + B[(−∆)−1∇∧ηc1, ηg]=∆ηc1−B[v−vg, ηc1] −B[v, ηc2]−B[vg−χRvg, ηg]−B[v−vg−(−∆)−1∇∧ηc1, ηg]. The first part of our mapping will be a1=−Zt 0S(t, s)fc1(s)ds, (2.2.16) 23 2.3. Preliminar analysis where fc1=B[v−vg, ηc1] + B[v, ηc2] + B[vg−χrvg, ηg] + B[v−vg−(−∆)−1∇∧ηc1, ηg]. Denoting by et∆the heat operator, we set from (2.2.14) the second part of our mapping as a2=−Zt 0e(t−s)∆fc2(s)ds, (2.2.17) where fc2=B[(1 −e χr)u, ω] + Eg+Ec. By construction, a fixed-point of the mapping Q(ηc1, ωc2):= (a1, a2)will solve (2.2.12) and (2.2.14). 2.3 Preliminar analysis 2.3.1 General analysis in R2×T Lemma 2.3.1. If 1≤p1, p2, q ≤ ∞ and 1 q=1 p1+1 p2, then ||fg||Bx3Lq x≲||f||Bx3Lp1 x||g||Bx3Lp2 x. Proof. Write the expression of the left-hand side. Apply Minkowsky’s inequality in x3and Hölder’s inequality in e x. Lemma 2.3.2. ∇∧(−∆)−1fBx3L4 x ≲||f||Bx3L4/3 x. Proof. Write the expression of the left-hand side norm, then apply the change xξ =e x. It reduces to proving standard estimates for 2Dkernels. Lemma 2.3.3. If 1<p<∞, for a constant depending on p (∆)−1∇2fBx3Lp x ≲||f||Bx3Lp x. Proof. Fix a frequency ξ. For each ξthe inequality is true, so it also works for Bx3. Lemma 2.3.4. For 1< p ≤2, we have ||f||Bx3Lp x≲||f||1−1 p Lp x||∂x3f|| 1 p Lp x+||f||Lp x. Chapter 2. Navier-Stokes helical vortex filaments 24 Proof. We need the last term because fmight not have 0mean. We take C∈N, p′−1+p−1= 1.We apply Hölder inequality in ξfollowed by Hausdorff-Young inequality to obtain C X |ξ|=0 ˆ f(e x, ξ)Lp x ≲C1 pˆ f(e x, ξ)lp′ ξLp x ≲C1 pˆ f(e x, ξ)Lp xlp′ ξ ≲C1 p||f(x)||Lp x. Above, we can change the order of the norms because p′≥p. We bound the high frequencies similarly by X |ξ|>C ˆ f(e x, ξ)Lp x ≲C1 p−1ξˆ f(e x, ξ)lp′ ξLp x ≲C1 p−1||∂x3f(x)||Lp x. Taking C= max (1,⌊||∂x3f||Lp x ||f||Lp x,z ⌋), we obtain the result. Lemma 2.3.5. Let K(y)be the Biot-Savart kernel on R2×T. Then, for any multiindex a∈N3, |∂aK(y)|≲|e y|−(1+a),1≤ |y|, |∂aK(y)|≲|y|−(2+a),|y| ≤ 1. Proof. A different proof can be seen in [17, Lemma 3.3]. We show that the kernel in R2×Tis obtained summing in the third component the 3D kernel, i.e., K(e y, y3) = 1 4πX k∈Z (y1, y2, y3+ 2kπ) (|e y|2+|y3+ 2πk|2)3/2.(2.3.18) This sum is y3-periodic. Passing the vector norm inside the sum, we bound |K(e y, y3)|≲X k∈Z 1 |e y|2+|y3+ 2πk|2. Since y3∈[−π, π), we have that |y3+ 2πk|is always equal or greater than |y3|. Then we can bound X k∈Z 1 |e y|2+|y3+ 2πk|2≲∞ X k=0 1 |e y|2+|y3+ 2πk|2≲1 |e y|2+|y3|2+Z∞ 1 dy3 |e y|2+|y3|2 ≲1 |e y|2+|y3|2+1 |e y|+ 1. This shows that the sum (2.3.18) is absolutely convergent away from 0, so the definition makes sense. We need to show that such Ksatisfies curlZK(x−y)∧ω(y)dy =ω(x). 25 2.3. Preliminar analysis From the 3D case, we can easily check that this is true for the term k= 0 of the sum, while for the other terms the curl is identically 0. Due to absolute convergence, we can switch sum and derivatives. The bound for derivatives of K using this sum is analogue. Remark 2.3.6. The Newtonian kernel can also be pointwise bounded in R2×T\ {0},obtaining the same heuristic. Near the origin it behaves like the kernel in R3, and far from zero like the kernel in R2. See [17, Section 3.1] for more details. Lemma 2.3.7. Let HDbe the heat kernel on the space D,e y∈R2,y3∈T. Then HR2×T(t, e y, y3) = X k∈Z HR3(t, e y, y3+ 2πk) = HR2(t, e y)X k∈Z HR(t, y3+ 2πk) = HR2(t, e y)HT(t, y3). Proof. To prove the first equality we can proceed as in the previous Lemma. We can separate the variables due to properties of the exponential function. 2.3.2 Analysis close to the curve γ In this section we proof some technical results involving the mappings defined in (2.2.7). Proposition 2.3.8. Let ηbe supported in {|e x| ≤ 16r}. If 0< r ≪1is small enough, then the following holds: χ8rP−1 Φ(−∆)−1∇∧QΦηBx3L4 x +∇(χrP−1 Φ(−∆)−1∇∧QΦη)Bx3L4/3 x ≲||η||Bx3L4/3 x, (2.3.19) (1 −χr)χ8rP−1 Φ(−∆)−1∇∧QΦ(χr 4η)Bx3L4 x ≲r−1 2||η||Bx3L1 x.(2.3.20) Proposition 2.3.9. Let m > 1,0< r ≪1be small enough, ηsupported in {|e x| ≤ 16r}. Assuming 0<√t≲r≪1and v=χrP−1 Φ∇∧(−∆)−1QΦη, we have t1 4v−∇∧(−∆)−1ηBx3L4 x +t1 4∇(v−∇∧(−∆)−1η)Bx3L4/3 x ≲(t1 4r−1 2+rlnr)||η||N0. (2.3.21) Lemma 2.3.10. Let Kbe the Biot-Savart kernel in R2×T.Then the corresponding kernel f Kof the operator FzQ−1 Φe χ16rKQΦχ16rFzsatisfies |f K(e x, ξ, e x′, ξ′)|≲|e x−e x′|−1⟨ξ−ξ′⟩−2. Chapter 2. Navier-Stokes helical vortex filaments 26 Proof. The three previous results are proven in [12, Appendix A] for x∈R2×T, y∈R3. We have instead y∈R2×T, so the Biot-Savart and Newtonian kernels are different. We can still use the same proof because we can restrict |y|< C due to the presence of χcr,e χcr through the computations, so essentially the kernels behave identically. Lemma 2.3.11. It holds 1. For f1with support in {|e x| ≤ 16r}, Q−1 Φe χ8rZt 0e(t−s)∆(QΦdiv f1(s) + div f2(s)dsN0 ≲sup 0<s≤ts3 4⟨s−1 2e x⟩mf1(s)Bx3L4/3 x +s3 4⟨s−1 2d⟩mf2(s)L2 y. 2. For fwith support in {|x| ≤ 4r}and any multi-index α∈N3with |α| ≤ 2, Q−1 Φe χ8rZt 0e(t−s)∆QΦ∇α x,zf(s)dsN0 ≲t1−|α| 2||f||Nβ. Proof. See [12, Lemma 7.12]. In this case f2can be nonzero far from γ, but the result is still valid since only Lpnorms of the heat kernel are used, and in both R3 and R2×Twe have ∇ket∆Lp≲t−k 2−3(p−1) 2p. 2.3.3 Morrey spaces Here we proof some results for the spaces defined in Definition 2.1.7. Lemma 2.3.12. If 1 p=1 p1+1 p2and 1 q=1 q1+1 q2 ||fg||Mp q≤ ||f||Mp1 q1||g||Mp2 q2. Proof. Use the usual Hölder inequality. Lemma 2.3.13. Lp⊂Mp q∀1≤q≤p, Bx3Lq x⊂M 3q 2 q. Proof. The first embedding is a consequence of the Hölder inequality. For the second one, apply Hölder in the x3variable and use Bx3⊂L∞. 27 2.4. Fixed-point Estimates, Short Time Existence Lemma 2.3.14. For any k∈N3. ∇ket∆fM3 2≲t−|k| 2||f||M3 2. Understanding ∇as a Fourier multiplier, for any κ≥0, ∇κet∆fM3 2≲t−κ 2||f||M3 2. Proof. For the first statement, Use that ∇ket∆L1≲t−|k| 2. For the second statement, see [91, Theorem 3.8]. 2.4 Fixed-point Estimates, Short Time Existence In this section we first prove the estimates needed to make the fixed-point argument work. We will control several quantities with the norm ||·||XTand constants r, T, M. We assume that r < 1, M > 1. Any other constant will be implicitly included in the notation ≲. Then, we will conclude the local-in-time existence of solutions. We adapt [12, Section 7]. Since we work on a different domain, the kernels behave differently and some estimates change. For ease of reading, we try to include at least a short proof of each estimate. 2.4.1 Preliminary estimates In this subsection we show estimates that are derived directly from definitions of norms and spatial estimates, i.e., we do not use the evolution equations. We do not intend to make them sharp, we just present what we will use later. Lemma 2.4.1 (Weight bounds).Let 0< c,1< p ≤ ∞. (1 −χcr)⟨t−1 2e x⟩−mBx3Lp x +(1 −e χcr)⟨t−1 2d⟩−mLp y ≲tm/2r2/p−m,(2.4.22) ⟨t−1 2e x⟩−mBx3Lp x ≲t1/p.(2.4.23) Proof. For (2.4.22) use that far from 0,⟨·⟩ behaves like |·|. For (2.4.23) use the change of variables e x′=e xt−1/2. Lemma 2.4.2 (Vorticity bounds in straightened frame). Chapter 2. Navier-Stokes helical vortex filaments 28 1. Full space bounds ||ηg||Nβ+t1 4||ηg||Bx3L4/3 x+⟨√t∇⟩βηgBx3L1 x ≲1,(2.4.24) ⟨√t∇⟩βηc1Bx3L1 x +t1 4||ηc1||Bx3L4/3 x≲||ω||XT,(2.4.25) Q−1 Φ(e χ8rωc2)Bx3L1 x +t1 4||ηc2||Bx3L4/3 x+t1 4Q−1 Φ(e χ8rωc2)Bx3L4/3 x ≲M−1||ω||XT, (2.4.26) t1 2⟨t−1 2e x⟩mηc1L2 x +M−1t1 2⟨t−1 2e x⟩mηc2L2 x ≲||ω||XT.(2.4.27) 2. Bounds far from filament For c > 0and 1< p ≤2, ||(1 −χcr)ηg||Bx3Lp x≲Tm−1 2r−m+2−p p,(2.4.28) ||(1 −χcr)ηc1||Bx3Lp x≲Tm−1 2r−m+2−p p||ω||XT,(2.4.29) (1 −χcr)Q−1 Φ(e χ8rωc2)Bx3Lp x ≲M−1Tm−1 2r−m+2−p p||ω||XT,(2.4.30) ||(1 −χcr)ηg||L2 x≲Tm−1 2r−m,(2.4.31) ||(1 −χcr)ηc1||L2 x≲Tm−1 2r−m||ω||XT.(2.4.32) Proof. 1. (2.4.24) follows from the explicit formula (2.2.8). (2.4.25) follow from (2.2.15), Hölder inequality (2.3.1) and weight estimate (2.4.23). (2.4.26) is similar, using that ||ηc2||Bx3Lp x≤ ||χ2r||Bx3L∞ xQ−1 Φ(e χ8rωc2)Bx3Lp x . For (2.4.27) use Bx3⊂L∞⊂Lpand (2.2.15). 2. For (2.4.28), (2.4.29) and (2.4.30) proceed as before, using now (2.4.22). (2.4.31) and (2.4.32) are identical, using again Bx3⊂L2. Lemma 2.4.3 (Vorticity bounds in physical frame, close to the filament). e χr 4ωgL1 y ≲1(2.4.33) e χr 4ωc1L1 y +M−1e χr 4ωc2L1 y ≲||ω||XT,(2.4.34) t1 2⟨t−1 2d⟩mωc1L2 y ≲||ω||XT,(2.4.35) t1 2⟨t−1 2d⟩me χ8rωc2L2 y +t1 2⟨t−1 2d⟩m(1 −e χ6r)ωc2L3 y ≲M−1||ω||XT,(2.4.36) 29 2.4. Fixed-point Estimates, Short Time Existence Proof. To prove (2.4.33) and (2.4.34), we apply Q−1 Φ. Then, we bound the change of variable (2.2.7) with Lemma 2.2.2 and use Lemma 2.4.2. (2.4.35), (2.4.37) and (2.4.38) follow similarly. Also the first term in (2.4.36), while the second is consequence of the definition (2.2.15). Lemma 2.4.4 (Vorticity bounds in physical frame, far from the filament).For 0< c ≤6, (1 −e χr 4)ωgL2 y ≲Tm−1 2r−m,(2.4.37) (1 −e χr 4)ωc1L2 y ≲Tm−1 2r−m||ω||XT,(2.4.38) ||(1 −e χcr)ωc2||L2 y≲M−1Tm−1 2r−m||ω||XT,(2.4.39) ||(1 −e χcr)ωc2||L4/3 y≲M−1r1 2−mTm−1 2||ω||XT.(2.4.40) Proof. To prove (2.4.37) and (2.4.38), we proceed as in the previous lemma. For (2.4.39) and (2.4.40), the strategy is the following. We split (1 −e χcr)ωc2= (1 −e χcr)e χ6rωc2+ (1 −e χcr)(1 −e χ6r)ωc2. To bound the first part, apply Q−1 Φand use (2.4.30). To bound the second part, use Hölder, (2.2.15), (2.4.22) and r < 1. Lemma 2.4.5 (Velocity bounds in straightened frame). 1. Full space bounds t1 4||vg||Bx3L4 x+t1 4||∇vg||Bx3L4/3 x≲1,(2.4.41) t1 4||vc1||Bx3L4 x+t1 4||∇vc1||Bx3L4/3 x≲||ω||XT,(2.4.42) t1 4||vc2||Bx3L4 x+t1 4||∇vc2||Bx3L4/3 x≲M−1(1 + Tm−1 2r−m−3 4)||ω||XT.(2.4.43) 2. Difference of velocities bounds t1 4||vg−χrvg||Bx3L4 x+t1 4||∇(vg−χrvg)||Bx3L4/3 x ≲T1 4r−1 2+rlnr−1+Tm 2−1 4r−m+1 2,(2.4.44) t1 4vc1−(−∆)−1∇∧ηc1Bx3L4 x ≲(T1 4r−1 2+rlnr−1+Tm 2−1 4r−m+1 2)||ω||XT. (2.4.45) Chapter 2. Navier-Stokes helical vortex filaments 36 and next we use (2.4.46), (2.4.44), (2.4.42), (2.4.43) and (2.2.15). For the second term, we split vand use (2.4.41), (2.4.42), (2.4.43) and (2.2.15). In the third term we apply (2.4.44) and (2.4.24). For the last term, we use (2.4.45),(2.4.43) and (2.4.24). To bound the difference of solutions, we first note that the terms having only the superindex gare explicitly constructed, and thus are equal in a1(ω)and a1(ω′), so they cancel. Then, we use that a1is linear and subtract similar terms. For example, B[v1, ηc2 1]−B[v2, ηc2 2] = B[v1−v2, ηc2 1]−B[v2, ηc2 2−ηc2 1]. Then we can proceed exactly as we did with the bound of ||a1||Nβ. 2.4.3 Estimates on ωc2 In this subsection we bound (2.2.17). We use two different norms for ωc2, so we need control on both. Proposition 2.4.9. Assume ε < 1, M > 1, T ≤r22. If ω∈Bε,T,r,M , then MQ−1 Φ[e χ8ra2(ω)]N0≲Mr + (Mr +r1 2)ε+Mε2. If ω, ω′have the same initial data, then MQ−1 Φ[e χ8r(a2(ω)−a2(ω′))]N0≲(Mr +r1 2+Mε)ε. Proof. Checking (2.2.17), the different contributions to a2come from B[(1 −e χr)u, ω] + Eg+Ec. From Definitions (2.2.10) and (2.2.13) together with Lemma 2.2.1, we note that both Q−1 Φ(Eg)and Q−1 Φ(Ec)are supported on {|e x| ≤ 4r}and can be written as Q−1 Φ(E∗) = χ2r∆(η∗)−∆Φ(χ2rη∗) = ∇α x,z(Cαη∗),(2.4.54) for some matrices Cα. By Lemma 2.2.2, the matrices Cαare smooth and satisfy |C2|≲r, |C1|≲r−1,|C0|≲r−2. By Lemma 2.3.11, (2.2.15) and (2.4.24), the term with Eg+Ecis bounded by (r+T1 2r−1+Tr−2)(1 + ||ω||XT). Now, we split B[(1 −e χr)u, ω] = B[(1 −e χr)ug, ω] + B[(1 −e χr)ur, ω]. 37 2.4. Fixed-point Estimates, Short Time Existence To control the contribution of B[(1 −e χr)ug, ωg+ωc1+e χ6rωc2], we use Lemmas 2.3.11 with f2≡0and 2.3.1 to get Q−1 Φe χ8rZt 0e(t−s)∆B[(1 −e χr)ug, ωg+ωc1+e χ6rωc2]dsN0 ≲t3 4P−1 Φ(e χ7r(1 −e χr)ug)Bx3L4 x⟨t−1 2x⟩m(ηg+ηc1+Q−1 Φ(e χ6rωc2))Bx3L2 x ≲T1 4r−1 2(1 + ||ω||XT+M−1||ω||XT). Above, we used for the last inequality (2.4.47), (2.4.24) and (2.2.15). For the term with B[(1 −e χr)ug,(1 −e χ6r)ωc2], we apply Lemma 2.3.11 with f1≡0followed by Hölder inequality, (2.4.51) and (2.2.15) to obtain Q−1 Φe χ8rZt 0e(t−s)∆B[(1 −e χr)ug,(1 −e χ6r)ωc2]dsN0 ≲T1 4r−3 2M−1||ω||XT. The bound for B[(1 −e χr)ur, ω]is obtained in a similar way. Instead of (2.4.47), we now use (2.4.48) and (2.4.49). Also, instead of (2.4.51), we use (2.4.52) and (2.4.53). The estimate for a2(ω)−a2(ω′)follows from a similar procedure. Proposition 2.4.10. Assume ε < 1, M > 1, T ≤r22. If ω∈Bε,T,r,M , then Ma2F≲Mr +Mrε +Mε2. If ω, ω′have the same initial data, then Ma2(ω)−a2(ω′)F≲(Mr +Mε)ε. We start with two auxiliary lemmas Lemma 2.4.11. With the notation of Lemma 2.3.7, 1 r≤|y|HR2×T(y)≲e−r2 16te−|y|2 16tt−3 2e−|ey|2 8tX k∈Z e−|y3+2kπ|2 8t. Proof. We use the triangular inequality and observe that e−|y3+2kπ|2 4t≤e−|y3|2 8te−|y3+2kπ|2 8t for all k∈Z. We do the same with e−|ey|2 4t, and then we use r≤ |y|. Lemma 2.4.12. For any y, y′∈R2×Tand 0< s ≤t, *d(y) √t+m e−|y−y′|2 16t≲*d(y′) √s+m . Chapter 2. Navier-Stokes helical vortex filaments 38 Proof. We split in the cases |y| ≥ 2|y′|and |y| ≤ 2|y′|. In the first case, the LHS is bounded because |d(y)| → ∞ ⇐⇒ |y|→∞and the exponential has faster decay. In the second case, we have d(y)≈d(y′). Now we proceed with the proof of Proposition 2.4.10. Proof. We begin with the bound for a2. For the term with Ec, use the formula (2.4.54). We can pass derivatives through QΦby adding lower-order terms, due to equation (2.2.7) and Lemma 2.2.2. Then, we pass the derivatives to the heat kernel. Then, we use the separation of supports of Cα,(1−e χ6r)to apply Lemma 2.4.11, and 2.4.12 to pass the weight to the vorticity term. Then, using the Young inequality, we obtain (1 −e χ6r)⟨t−1 2d⟩mZt 0e(t−s)∆QΦ(∇α xCαηc1)L3 y ≲Zt 0(t−s)−1+2|α| 4e−r2 16(t−s)s−1 2||Cα||L∞||ω||XTds. The exponential can control any power of (t−s), so at the end we get Zt 0e(t−s)∆EcF ≲Tr−2||ω||XT. The term with Egis bounded similarly. We now treat the term with B[(1 −e χr)ug, ωc2]. We begin by splitting in Zt 0e(t−s)∆B[(1 −e χr)ug,e χ6rωc2]F |{z } I +Zt 0e(t−s)∆B[(1 −e χr)ug,(1 −e χ6r)ωc2]F |{z } II . The strategy to bound both terms will be the same, but using different Lpnorms. In both cases we pass the divergence to the heat kernel and apply Young and Hölder inequalities to get I≤t1 2Zt 0⟨s−1 2d⟩mdiv e(t−s)∆L 4 3 y||(1 −e χr)ug||L12 y⟨s−1 2d⟩me χ6rωc2L2 y ds ≲t1 2Zt 0t−11 8r−7 4||ω||XT(1 −s/t)−7 8(s/t)−1 2ds ≲t1 8r−7 4||ω||XT≤r||ω||XT. For II, we get II ≤t1 2Zt 0⟨s−1 2d⟩mdiv e(t−s)∆L 6 5 y||(1 −e χr)ug||L6 y⟨s−1 2d⟩m(1 −e χ6r)ωc2L3 y ds ≲t1 2Zt 0t−5 4r−3 2||ω||XT(1 −s/t)−3 4(s/t)−1 2ds ≲t1 4r−3 2||ω||XT. 39 2.4. Fixed-point Estimates, Short Time Existence To bound the term with B[(1 −e χr)ug, ωc1], we can proceed as we did with I. We skip the details. We deal with the term B[(1 −e χr)ug, ωg]in the same way, but considering that now ||ω||XTdoes not appear. The remaining term, B[(1 − e χr)ur, ω], is treated identically using the corresponding inequalities from Lemma 2.4.6. The bound of a2(ω)−a2(ω′)is similar. 2.4.4 Short time existence Theorem 2.4.13. There exists T, M, r, ε such that Q:Bε,T,r,M →Bε,T,r,M and for ω1, ω2∈Bε,T,r,M there holds the contraction property ||Q(ω1)− Q(ω2)||Xt≤ 1 2||ω1−ω2||Xt. It follows that there exists a unique fixed-point Q(ω) = ω∈Bε,T,r,M . Proof. The Theorem is a consequence of Propositions 2.4.8,2.4.9 and 2.4.10. We assume that M > 1, T < r22, ε < 1. If ω∈Bε,T,r,M , we have shown that ||Q(ω)||XT≲rlnr−1+Mr +ε(rlnr−1+M−1+Mr +r1/2) + ε2M. To get a bound smaller than ε, we take M≫1, ε ≪M−1, r ≪ε. This choice also makes Qcontractive, which implies uniqueness of the fixed-point. Theorem 2.4.14. The solution constructed in Theorem 2.4.13 is a solution of the 3D Navier-Stokes equations in the sense of Definition 2.1.8. Therefore, Theorem 2.1.9 holds. Proof. The fact that for positive times ω∈L1∩L2follows from the vorticity decomposition and the norms used in (2.2.15). The integrability for the velocity ucan be deduced from the Biot-Savart law. The rest of the proof is analogue to [12, Proposition 7.18], so we skip it. See Section 2.3.3 to verify that the needed properties of Morrey spaces are analogue in R3and R2×T. Corollary 2.4.15 (Uniqueness).Let ω∈C2((0, T∗)×R3)be another mild solution such that for sufficiently small r > 0we have lim T→0+Q−1 Φ(e χ8r(ω−ωg))N0+||ω−ωg||F= 0.(2.4.55) Then ω=ω, where ωis the solution constructed in Theorem 2.4.13. Proof. In the proof of Theorem 2.4.13, the last constants we choose are rand T. We keep the choice of Mand ε. For any r > 0small enough, we can decompose ω in ω=ωg+ωc1+ωc2, where Q−1 Φ(e χ2rωc1)is forced to satisfy equation (2.2.12) and ωc2is the remainder. We note that this decomposition only depends on r. Then, use (2.4.55) to take a sufficiently small Tso that the fixed-point theorem can be applied on the ball (2.4.55), where we have uniqueness. For more details, see [12, Theorem 7.19]. Chapter 2. Navier-Stokes helical vortex filaments 40 2.5 Preservation of helical symmetry In this section we check the symmetry of the solution obtained in Theorem 2.4.13, provided the initial data have helical symmetry, which is defined in Definition 2.1.1. Due to rotation invariance of the Navier-Stokes equations, any single filament with helical symmetry can be reduced to α(x3)=(ρcos(x3), ρ sin(x3), κx3).Once fixed, the parameters ρ, κ do not play any role. To simplify the exposition, we will assume ρ=κ= 1, so that the initial vorticity is supported on the curve α(x3) = (cos(x3),sin(x3), x3). Navier-Stokes equations preserve helical symmetry [77], but we cannot apply this property directly because the initial data is not smooth. We could try to smooth the initial data and show convergence, but this approach does not work. More precisely, if we use time-independent mollifiers and denote the smoothed initial data in the straightened variables by ηε 0, one can check that ||η0−ηε 0||N0↛0as ε→0. We now check the conservation of symmetry in our construction. 2.5.1 Helicity of ωg Following subsection 2.2.1, we obtain that t(x3) = 1 √2(−sin(x3),cos(x3),1),n(x3) = (−cos(x3),−sin(x3),0), b(x3) = 1 √2(sin(x3),−cos(x3),1), so the mapping between the straightened and curved frame is Φ(x)=(cos(x3),sin(x3), x3)+x1(−cos(x3),−sin(x3),0)+ x2 √2(sin(x3),−cos(x3),1), (2.5.56) with jacobian J="n b 2−x1 √2t+−x2 √2n+x1 √2b!#,(2.5.57) and determinant at=2−x1 √2. We prove that ωgis helical in three steps. Lemma 2.5.1. The image by Φ : Σr→γrof vertical lines are helical lines. The preimage of helical lines are vertical lines. Proof. All we need is to focus on equation (2.5.56). If we fix x1, x2, we obtain vectors that rotate in the horizontal plane with angle x3and a vertical translation with jump x3. For the second statement, we find the preimage of a point, and then use that Φis bijective and the first statement of the Lemma. 41 2.5. Preservation of helical symmetry Corollary 2.5.2. By the previous lemma, the distance from Φ(x)to the initial vortex curve γis constant as a function of x3. Proof. The set of points Φ(x)with e xfixed is a helix with the same symmetry as γ. Let x3be a value of x3that minimizes the distance between Φ(x)and γ. Applying the operator Sθdefined in (2.1.4), we can easily see that the distance does not depend on x3. Proposition 2.5.3. If ηdoes not depend on x3,QΦ(η)has helical symmetry. Conversely, if ωhas helical symmetry, Q−1 Φ(ω)does not depend on x3. Proof. For the first statement we must check that, restricted to a helix, QΦ(η) rotates with vertical axis and angle x3. Due to the previous lemma, in this restriction x1, x2are fixed. QΦis defined in (2.2.7), where we see that everything on the right-hand side is constant except J(2.5.57), which rotates as needed. The second statement is completely analogue since our operators are invertible. Corollary 2.5.4. If γis helical, then ωgdefined in (2.2.9)has helical symmetry. Proof. Recall that ωgis defined from ηg, and we see in (2.2.8) that ηgdoes not depend on x3. Note that multiplying by χ2ror e χ2rdoes not affect symmetry. By Proposition 2.5.3, we conclude that ωgis helical. 2.5.2 Helicity of ω Proposition 2.5.5. Let ωbe the solution of Theorem 2.4.13, with helical initial data. Then ω(t)is helical for all 0≤t≤T. Therefore, Proposition 2.1.12 holds. Proof. We note that the Biot-Savart operator and the Laplacian applied to functions that are x3-independent keep the x3-independence. These operators also preserve helical symmetry. By Proposition 2.5.3,x3-independence of v∗or η∗and the helical symmetry of u∗or ω∗are equivalent. Also, the operator Sused in (2.2.16) preserves x3-independence when the variables involved are x3-independent. By the same reasoning, the E∗appearing in (2.2.17) are helical if the corresponding variables are x3-independent in the straightened frame, and helical in the physical frame. We recall that ω=ωg+QΦ(χ2rηc1) + ωc2. At time 0,ηc1and ωc2are 0, so initially they have the desired symmetry. By the previous reasoning, if fc1and fc2in (2.2.16) and (2.2.17) have the desired symmetry, then a1, a2will also have it. Since each iteration will preserve the symmetry, the fixed-point solution ωhas the desired symmetry. Chapter 2. Navier-Stokes helical vortex filaments 42 2.6 Global in time existence for helical initial data By the previous section, we know that the solution to the Navier-Stokes equations obtained in theorem 2.4.13 is helical if γis helical. Our purpose now is to use this fact to extend the solution local-in-time to a global-in-time smooth solution. We cannot apply the techniques of [77] because the solution obtained in Theorem 2.1.9 belongs only to L2 uloc. We apply results from local energy weak solutions instead. First, we show that smooth helical functions with compact support are dense in the space of helical functions belonging to ˚ E2. Omitting the symmetry restrictions, this result is well known. Lemma 2.6.1. For any u∈˚ E2with helical symmetry and for any ε > 0, there exist divergence free uε∈C∞ c(R2×T)helical such that ||u−uε||L2 uloc < ε. Proof. Following [73, Lemma 6.1] and using cylinders instead of balls, we can obtain a function vRsupported in C(0,2R)such that div vR=u·∇χR, vRL2 uloc ≲||u||L2 uloc R.(2.6.58) From the construction, it is not clear whether vRis helical or not. However, using that div vRis helical, we can construct a helical version of vR. Take cylindrical coordinates (r, θ, z)and make the change θ→ζ=θ+z, z →ζ=θ−z. These coordinates define ortonormal vectors ei,i=r, ζ, ζ. Let vR ibe the component of vRin the eidirection. Then, define vR i(r, ζ, ζ) = 1 2πZ2π 0vR i(r, ζ, ζ +s)ds. By construction, vRis helical and also satisfies (2.6.58). This can be checked by expressing the divergence operator in the new coordinates and using the fact that u·∇χRis helical. We conclude that uR=uχR−vRis divergence free and u−uRL2 uloc ≤ ||u−uχR||L2 uloc +vRL2 uloc < ε if R≫1. We mollify uRto get uε. Theorem 2.6.2. Let u0∈˚ E2. Then the Navier-Stokes equations (2.1.1)have at least one global-in-time solution in the sense of 2.1.5. Furthermore, if u0is helical, the solution obtained is helical. 43 2.6. Global in time existence for helical initial data Proof. Without the helical symmetry, this result is proven in R3in [74, Chapter 33] and also in [73]. We explain the steps of the proof and how it can be adapted to our case. The first step is to obtain the local-in-time existence by performing energy estimates on a regularized system. It can be checked that the regularized system preserves helical symmetry. Due to the change of domain it is needed to use Lemma 2.3.5 to estimate the pressure, but the proof is similar. For more details, see [73, Proposition 1.6]. It is important that this solution belongs to ˚ E3almost at all times. An intermediate step is to show that for any given T > 0, if ||u||˚ E3is small enough, there exists a solution to the Navier-Stokes equations in L∞((0, T); ˚ E3). This is shown in [74, Theorem 17.1] for R3. The proof is based on estimates of the heat kernel and a fixed-point argument. It can be adapted to our domain using Lemma 2.3.7, which shows that we will obtain essentially equivalent estimates. We can conclude the proof with a continuation argument. We fix T > 0and choose a time t1such that u(t1)∈˚ E3. As a conclusion of Lemma 2.6.1, we can split u=u1+u2in such a way that u1(t1)∈˚ E3is small, and u2(t1)∈L2. We solve the Navier-Stokes equations for u1 (∂tu1+u1·∇u1+∇p1=ν∆u1, x ∈R2×T, t ∈(t1, T), ∇·u1= 0,(2.6.59) leaving for u2the following system (∂tu2+u2·∇u2+u2·∇u1+u1·∇u2+∇p2=ν∆u2, x ∈R2×T, ∇·u2= 0. t ∈(t1, T),(2.6.60) Choosing u1(t1)∈˚ E3with ||u1||˚ E3≪1guarantees the existence of u1∈˚ E3 solving (2.6.59). To find u2solving (2.6.60), we can perform energy estimates to get a solution u2∈L∞((t1, T);L2). See [73, Section 5] for a proof in R3, which can be adapted to our domain, giving ||u2||L∞ t((t1,T);L2)≤ ||u2(t1)||L2. Now, we will perform local energy estimates to show that the helical solution constructed in Theorem 2.6.2 is unique. We will need to estimate terms that involve the pressure, for which we will use the following proposition. Proposition 2.6.3. For any x0∈R2×Tand (t, x)∈(0, T)×C(x0,3/2), the pressure passociated to the solution uof Theorem 2.6.2 can be written as px0(t, x) = px0,1(t, x) + px0,2(t, x), px0,1(t, x) = −1 3|u(t, x)|2+1 4πZC(x0,2) ∂i∂jN(x−y) : u(t, y)⊗u(t, y)dy, px0,2(t, x) = 1 4πZ(R2×T)\C(x0,2)(∂i∂jN(x−y)−∂i∂jN(x0, y)) : u(t, y)⊗u(t, y)dy. Chapter 2. Navier-Stokes helical vortex filaments 44 In particular, for any q > 1we have ||px0,1||Lq(C(x0,1)) ≲||u||2 L2q uloc , ||px0,2||L∞(C(x0,1)) ≲||u||2 L2 uloc . Proof. The decomposition comes from equation (2.1.6). The kernel ∂i∂jNbehaves as the analogue kernel in R3near the origin, and like the kernel in R2far from zero. See Lemma 2.3.5 or [17, Proposition 3.1] for a different proof. To prove the bound for px0,1, we use theory of singular integral operators. In order to obtain the estimate for px0,2, we first note that in the domain of the integral |∂i∂jN(x−y)−∂i∂jN(x0, y)|≲1 |x0−y|3. Then, we can bound |p(t, x)x0,2|≲∞ X i=1 ZC(x0,2i+1)\C(x0,2i) 1 |x0−y|3|u(t, y)|2dy ≲∞ X i=1 1 23iZC(x0,2i+1)|u(t, y)|2dy ≲∞ X i=1 1 23i22i||u||2 L2 uloc =||u||2 L2 uloc . So far, we have shown that the helical symmetry is preserved but we have not exploited it. We do that in the following lemma. Lemma 2.6.4. Let ube a smooth helical vector field, x= (x1, x2, x3)∈R2×T. Then ||u||4 L4(C(x,1)) ≲||u||2 L2(C(x,2)) ||∇u||2 L2(C(x,2)) +||u||4 L2(C(x,2)) .(2.6.61) Proof. The idea is to use the symmetry to reduce the estimate to a two dimensional estimate, and then recover the three dimensional domain. We use (ρ, θ, z) as cylindrical coordinates, ζ=θ+z, and x= (ρx, θx, zx). Observe that if (ρ1, θ1, z1),(ρ2, θ2, z2)∈C(x, 1), then |θ1−θ2|≲1 1+ρx. This implies that for fixed ρ, ζ; if (ρ, ζ, z1),(ρ, ζ, z2)∈C(x, 1), then |z1−z2|≲1 1+ρx. We will use cylindrical (ρ, θ, z)and helical (ρ, ζ, z)coordinates. Due to helical symmetry, in helical coordinates udoes not depend on z, so we can write u(ρ, θ, z) = u(ρ, ζ). From this expression, we deduce ∂θu(ρ, θ, z) = ∂ζu(ρ, ζ). We bound ||u||4 L4(C(x,1)) =Z{(ρ,θ,z)∈C(x,1)}|u(ρ, θ, z)|4ρdρdθdz ≲Z{(ρ,ζ,z)∈C(x,1)}|u(ρ, ζ)|4ρdρdζdz ≲1 1 + ρxZπ −πZ{|ρ−ρx|<1}|u(ρ, ζ)|4ρdρdζ, 45 2.6. Global in time existence for helical initial data The last integral can be seen as a two-dimensional norm in polar coordinates. By Gagliardo-Nirenberg inequality, it follows that ||u||4 L4(C(x,1)) ≲1 1 + ρxZπ −πZ{|ρ−ρx|<1}|u(ρ, ζ)|2ρdρdζ Zπ −πZ{|ρ−ρx|<1}(|∂ρu(ρ, ζ)|2+1 ρ2|∂ζu(ρ, ζ)|2)ρdρdζ +1 1 + ρx Zπ −πZ{|ρ−ρx|<1}|u(ρ, ζ)|2ρdρdζ!2 . Now we use that fixed ρ, ζ such that |ρ−ρx|<1, ζ ∈T, then µ{z: (ρ, ζ, z)∈ C(x, 2)}>1 ρx+1, where µis the one-dimensional Lebesgue measure. This is equivalent to saying that, in cylindrical coordinates, if we fix ρ, z such that |ρ−ρx|<1, then µ{θ: (ρ, θ, z)∈C(x, 2)}>1 ρx+1, which is more intuitive. Thus, we can recover the integral in zby adding a factor ρx+ 1. Therefore, ||u||4 L4(C(x,1)) ≲Z{(ρ,ζ,z)∈C(x,2)}|u(ρ, ζ)|2ρdρdζdz Z{(ρ,ζ,z)∈C(x,2)}(|∂ρu(ρ, ζ)|2+1 ρ2|∂ζu(ρ, ζ)|2)ρdρdζdz + Z{(ρ,ζ,z)∈C(x,2)}|u(ρ, ζ)|2ρdρdζdz!2 . Undoing the change of variables, ||u||4 L4(C(x,1)) ≲Z{(ρ,θ,z)∈C(x,2)}|u(ρ, θ, z)|2ρdρdθdz Z{(ρ,θ,z)∈C(x,2)}(|∂ρu(ρ, θ, z)|2+1 ρ2|∂θu(ρ, θ, z)|2)ρdρdθdz + Z{(ρ,θ,z)∈C(x,2)}|u(ρ, θ, z)|2ρdρdθdz!2 , which is the desired estimate (2.6.61). Remark 2.6.5. The estimate (2.6.61)is known in cylinders centered at 0[77]. In that case, the increase of domain for the norms on the right-hand side is not needed. The novelty of the previous lemma is how to take advantage of the helical symmetry restricted to a non-helical domain. Theorem 2.6.6. If u0∈˚ E2is helical, the helical solution obtained in Theorem 2.6.2 is unique. Therefore, Theorem 2.1.13 holds. Chapter 3. Euler solutions originating from a circular vortex filament 52 3.1 Introduction The aim of this chapter is to prove the existence of weak solutions to the Euler equations ∂tv+v·∇v+∇p= 0, divv= 0, v|t=0 =v◦, (3.1.1) posed on [0, T]×R3, where v(t, x)is the velocity field and p(t, x)is the scalar pressure. The initial velocity v◦is given by the Biot-Savart law (1.1.3) in terms of the vorticity ω◦, which we assume to be ω◦= ΓδC,(3.1.2) where Γ= 0 is the vorticity strength, and Cis an oriented circle with radius L > 0. Due to the invariances of the Euler equation, we can assume without loss of generality that Cis located in the plane x3= 0 and centered at the origin (x1, x2) = (0,0). 3.1.1 Main result Theorem 3.1.1 (Main result of [53]).There exists T > 0such that (3.1.1),(3.1.2) admits infinitely many weak solutions v∈C([0, T], L2−). For any 0< t ≤T, the energy becomes finite and decreasing with 1 2ZR3|v(t, x)|2dx=−LΓ2 2log c(t) + O(1),(3.1.3) as t→0. Furthermore, v(t)is analytic outside the “turbulence ring” Ωtur(t) = {x∈R3: dist(x, h(t)e3+C)< c(t)}, where c(t)and h(t)denote the thickness and height of the ring respectively c(t) = √νturt, h(t) = Γ 8πL(1 −2log c(t))t, and νtur >0represents the “turbulence viscosity”, a constant proportional to |Γ|. The proof of Theorem 3.1.1 relies on finding a suitable solution to the Euler- Reynolds equation, also known as a “subsolution”, and later applying the convex integration method [36]. As we will explain in more detail in Sections 3.2 and 3.3, we seek a subsolution whose vorticity concentrates around a filament evolving 53 3.1. Introduction in time. Heuristically, this evolution should be driven by the binormal flow with a displacement velocity amplified by a factor log c, where cis the radius of the vortex core. If the vorticity remained concentrated in a circle (c= 0), it would ascend to infinity instantly, making it impossible to solve the Euler equation. However, given the singularity of the initial data, it is conceivable that turbulence may be triggered by infinitesimal perturbations, thus expanding the support of the vorticity, slowing down the ascent of the turbulence ring, and decreasing the energy during the process. We will show that, by choosing c(t)>0, it is possible to construct Euler flows satisfying the aforementioned properties. The proper scaling c(t)∼√tappears naturally as we aim to attain a Reynolds stress comparable to the nonlinear term. This construction is inspired by prior work [89] on the Kelvin-Helmholtz instability for the 2D Euler equation. In this example, the vorticity is initially concentrated in the horizontal axis, that is, ω◦= Γδ{x2=0}. For positive times, the author proved the existence of dissipative Euler flows by devising a subsolution whose vorticity is concentrated inside a turbulence strip Ωtur(t) = {x∈R2:|x2|< νturt}. Remarkably, the presence of symmetries simplifies the derivation of the subsolution to a direct computation. Despite the initial data (3.1.2) also enjoys certain symmetries, the computation of the subsolution becomes intricate due to various factors. Firstly, in contrast to the previous works on modeling hydrodynamical instabilities through convex integration, this is the first one in three dimensions. Although the axisymmetric without swirl case reduces the problem to two dimensions, the corresponding Biot-Savart law involves integrals that cannot be explicitly computed, introducing a series of errors that need to be controlled. To address this, we draw upon previous work [43] on viscous vortex rings. Secondly, the turbulence zone involves three dynamics: vertical displacement, width expansion, and internal rotation. The first example with two dynamics was provided in [24] for the Saffman-Taylor instability in porous media. In this work, the authors proved the existence of mixing solutions by concentrating the vorticity inside a mixing zone Ωmix(t) = {x∈R2:|x2−f(t, x1)|< νturt}surrounding a dynamical interface f. The work [46] simplified the construction by introducing the idea of concentrating the vorticity on the boundary of the mixing zone. Later, this approach was implemented in [81] for the Kelvin-Helmholtz instability with dynamical vortex sheets. Motivated by the work [81], our first attempt consisted of trying to concentrate the vorticity at the boundary of the turbulence ring. Unfortunately, this approach did not fit suitably to the requirements of our subsolution. Based on the previous work on power-law vortices for the 2D Euler equation [80], we had hope in being able to concentrate the vorticity inside the turbulence ring. However, in contrast Chapter 3. Euler solutions originating from a circular vortex filament 54 to [80], the errors introduced by the Biot-Savart law prevents the explicit computation of the Reynolds stress. To overcome this difficulty, we made use of the antidivergence operator in [68]. This operator has the crucial property of maintaining the support of the turbulence zone, but at the cost of introducing several compatibility conditions. In this work, we ensure these conditions by choosing the vorticity density carefully and introducing a suitable corrector in the pressure. Thirdly, the initial data (3.1.2) is the first one satisfying v◦/∈L2 loc in the context of the Euler equation. Although the h-principle allows for the construction of solutions with finite energy [36], it does not immediately extend to our case: the energy blows up as time approaches zero. To address this, we adapt the convex integration method from [36] for an appropriate time-weighted space. Although the initial data cannot be reached in L2, we prove strong convergence in all Lp with 1<p<2, in short, L2−. 3.1.2 Introduction to convex integration In the seminal work [35], non-trivial Euler flows are constructed by adapting the convex integration method, which traces back to works in differential geometry [83] and calculus of variations [82], in hydrodynamics. Given that the solutions in [35] satisfy v◦= 0, their energy must necessarily increase. In their second paper [36], upon which the present work is based, they initiated the investigation on “admissible” Euler flows, those characterized by non-increasing energy. After a decade of refinements to this method, it allowed to construct Euler velocities with Hölder regularity strictly less than 1 /3exhibiting nonuniqueness [67] and dissipating the kinetic energy [19]. This milestone thereby resolves the dissipative part of the 3D Onsager conjecture. More recently, the L3-based Onsager conjecture was proved in [61]. The conservative part of the Onsager conjecture had been proven previously in [41,30]. See also [28] for critical regularity. Remarkably, this method allowed for proving nonuniqueness of solutions to the Navier-Stokes equation [20] and sharpness of the Ladyzhenskaya-Prodi-Serrin criteria [29]. In the context of turbulent transport, it was recently shown in [4] anomalous dissipation of scalar fields advected by deterministic vector fields, and in [21] by typical Euler flows below the Onsager threshold. In the aforementioned results, the emphasis is placed on upgrading the global regularity of the solutions, rather than addressing the Cauchy problem for specific data. According to the weak-strong uniqueness principle, initial data exhibiting nonuniqueness of admissible solutions, known as “wild” data, cannot be smooth. In [32] it is proved that wild data are dense in the space of divergence-free vector fields. The underlying idea is that a small but rough perturbation can be added to any (smooth) data, triggering turbulence. Despite this genericity, determining whether a given initial data is wild, that is, constructing admissible solutions emanating 55 3.1. Introduction from it, turns out to be quite challenging. Recently, [80] proved the existence of wild data in two dimensions satisfying ω◦∈L1∩Lpfor any given p < ∞. This showcases both sharpness of the weak-strong uniqueness principle and Yudovich’s proof of uniqueness [96]. The construction relies on devising a subsolution in the form of a self-similar vortex, with Ωtur(t) = {x∈R2:|x| ≤ (νturt)p /2}and νtur ∼ |Γ|p−2. As we will in Section 3.3, the cross section of our vortex ring shares similar properties (see (3.3.30)) to that of [80] when p→1. The difficulty in determining if a given data is wild arises from the fact that these are often associated with some form of hydrodynamical instability. As we mentioned at the beginning of the introduction, [89] provided the first example of wild data in the context of the Kelvin-Helmholtz instability (see also [81]). This initiated a promising program of modeling hydrodynamical instabilities through convex integration. Apart from the Kelvin-Helmholtz instability, this approach has been successfully applied to the Rayleigh-Taylor instability [57,56,55,54], and the Saffman-Taylor instability (see e.g. [90,24,46,84,25] and the references therein). In these works, the fluid is initially smooth outside an interface, where the hydrodynamical instability is concentrated. The general approach consists of opening a turbulent zone around a dynamical interface. Outside the turbulent zone, the corresponding Reynolds stress must vanish, meaning it must be a solution of the original system. Inside the turbulent zone, whenever the h-principle applies, it is sufficient to construct a subsolution. Despite the degree of freedom introduced by the Reynolds stress, the physical properties we seek in our solutions restrict the evolution of the turbulent zone. As an illustrative example, [27] showed that the thickness of the mixing zone is controlled by the inequality c(t, α)≤t(1−σ(t, α)), where σis the Saffman-Taylor function, which takes values in [−1,1]. This constraint prevents the fluids from mixing at asymptotically stable points (σ= 1), thus forcing the mixing zone to be localized around the instability. Although nonuniqueness appears inherent in these phenomena, there is hope for restoring determinism at the average level by selecting the Reynolds stress (equivalently the subsolution) as a function of the average flow. This challenge is known in the literature as the “closure problem”. For the Kelvin-Helmholtz instability, [89] and [81] studied a selection criterion based on maximal energy dissipation. For the Rayleigh-Taylor instability, [54] proposed a selection criterion based on the least action principle. Regarding the Saffman-Taylor instability, [85] and [90] obtained a selection criterion based on a Lagrangian and Eulerian relaxation approach, respectively (see also [26,79,66]). For the flat interface, this approach yields a unique entropy solution as the candidate for the macroscopic solution. Recently, in [25] these entropy solutions for dynamical interfaces have been constructed. Chapter 3. Euler solutions originating from a circular vortex filament 56 3.1.3 Notation specific to the chapter In this subsection we clarify some notation specific to this chapter. •We will use x= (x1, x2, x3)as cartesian coordinates, (r, θ, z)as cylindrical coordinates, and (ρ, α)as the polar coordinates of the cross section of the turbulence ring. •As is standard in the convex integration literature, in this chapter we will denote by Rthe Reynold stress, a positive semi-definite and symmetric tensor. •In axisymmetric coordinates, we will identify H:= {(r, z)∈R2:r > 0}, with the complex half-plane, and we will write compactly ζ=r+iz. Moreover, we will denote ζ∗=r−iz and ζ⊥=iζ =−z+ir. •We will denote Ωtur(t)by the turbulence ring at time t, and Ωtur ={(t, x) : x∈Ωtur(t)}by the space-time turbulence zone. 3.1.4 Organization of the chapter We start by providing some heuristics and outlining the proof of Theorem 3.1.1 in Sections 3.2 and 3.3 respectively. In Section 3.4 we derive the Euler-Reynolds equation for axisymmetric without swirl subsolutions, and the axisymmetric without swirl Biot-Savart law. In Section 3.5 we construct the subsolution, while the energy is computed in Section 3.6. In Section 3.7 we show that the h-principle can be adapted to a certain time-weighted space, concluding the existence of weak solutions. 3.2 Heuristics Theorem 3.1.1 gives the first example of weak solutions to the Euler equation with vortex filament data. For this reason, we would like to provide a heuristic explanation of its existence. Let us consider the Cauchy problem for the 3D Navier-Stokes equations ∂tvν+ div(vν⊗vν) + ∇pν=ν∆vν, divvν= 0, vν|t=0 =v◦, 57 3.2. Heuristics where ν > 0is the kinematic viscosity. In [51], it is proved that, within the axisymmetric without swirl class, there is a unique Navier-Stokes flow vνoriginating from the circular vortex filament (3.1.2), which becomes smooth for positive times. This solution behaves like a vortex ring of thickness proportional to cν(t) = √νt, ν =|Γ| Re, which is located, according to the Kelvin-Saffman formula [88], in the x3-axis with height hν(t) = Γ 8πL(C−2log cν(t))t, where Re >0is the (circulation) Reynolds number, and Cis a dimensionless constant that depends on the distribution of vorticity inside the cross section of the ring. Observe that, as the Reynolds number increases, this vortex ring ascends more and more rapidly, while vorticity dissipation decreases. Formally, in the inviscid limit ν→0, which corresponds to Re → ∞ when Γis fixed, one would observe the circle ascending towards to infinity instantaneously. For this reason, one might tend to think that there cannot exists Euler flows originating from the circular vortex filament (3.1.2), thus contradicting Theorem 3.1.1. Our goal in this section is to explain why this is not necessarily the case. At this point, we would like to caution the reader that, although it is tempting to draw parallels between the cand hin our Theorem 3.1.1 with cνand hνabove, our scaling arises from the (local) self-similarity of the initial data and turbulent dissipation, rather than from viscous dissipation. 3.2.1 Spontaneous stochasticity and anomalous dissipation As already mentioned in [51], a natural question is whether uniqueness remains true among all reasonable solutions that approach the initial vortex filament in a suitable sense. Both from an experimental and numerical point of view, it is indeed more realistic to consider potential deviations of the initial data. In this regard, we consider v◦ η=∇∧(−∆)−1ω◦ η, ω◦ η= Γδ(C+η),(3.2.5) for some perturbations ηpossibly breaking the symmetries of the circular filament (3.1.2). If preferred, one could consider smoother data by mollifying (3.2.5) or thickening the filament C+η. As we aim to examine small perturbations, we consider a family ενP, for some fixed class of perturbations P, and some εν>0 that vanishes as ν→0. Chapter 3. Euler solutions originating from a circular vortex filament 58 Whether what we observe later resembles the ideal case vνor not will depend on the stability of the system. As we have mentioned in Chapter 1, vortex filaments are expected to be unstable. In this sense, it is conceivable that these deviations could writhe the viscous ring, slowing down the ascent and disintegrating the vorticity into a thicker region of turbulence during the process. In the event that this turbulent region continues to be significant for a large class of perturbation η∈ ενPin the inviscid limit (ν→0), one might expect the Euler equations (3.1.1) for the singular initial data (3.1.2) to lose determinism (εν→0), a phenomenon known in the literature as “spontaneous stochasticity” (see e.g. [42,92]). This would align with the nonuniqueness and the turbulence ring described in our Theorem 3.1.1. In [12], it is proved that, in a suitable class, there is a local in time unique Navier-Stokes flow vη,ν originating from (3.2.5), whose energy becomes finite for positive times. These solutions can be extended globally in time by picking one of the (potentially non-unique) Leray solutions. It is well-known that these solutions satisfy the energy inequality 1 2ZR3|vη,ν(t, x)|2dx≤1 2ZR3|vη,ν(tν, x)|2dx−νZt tνZR3|Dvη,ν(s, x)|2dxds, for any tν>0. According to the zeroth law of turbulence, it is predicted that the last term will remain uniformly negative in the inviscid limit due to the loss of regularity of turbulent flows. This phenomenon is also known in the literature as “anomalous dissipation” (see e.g. [18]). We believe that, in our particular case, this phenomenon could compensate for the infinite initial energy and eventually lead to the production of dissipative Euler flows, as described in our Theorem 3.1.1. Figure 3.1: Cartoon of three different vortex filaments, originating from perturbations of the ideal circle, inside the turbulence ring. Left: The initial perturbations C+η. Right: The vortex filaments at a positive time t, included in a ring representing the turbulence zone Ωtur(t). 59 3.2. Heuristics 3.2.2 The Reynolds stress While nonuniqueness appears inherent to the phenomenon of spontaneous stochasticity, there is still hope to restore determinism by averaging these turbulent solutions. With this aim, we consider some probability measure µdefined on P(that extends trivially to ενP), with which we define the “averaged solution” vν=ZενPvη,ν dµ. (3.2.6) Given that averages do not commute with nonlinearities, this averaged solution satisfies the Navier-Stokes-Reynolds equation ∂tvν+ div(vν⊗vν+Rν) + ∇pν=ν∆vν, divvν= 0, vν|t=0 =v◦, where pνdenotes the averaged pressure, and Rνis the “Reynolds stress” Rν=vν⊗vν−vν⊗vν= (vν−vν)⊗(vν−vν). Notice that Rνis a positive semi-definite symmetric tensor. Since trRcan be absorbed in the pressure, the deviation from being a true Navier-Stokes flow is captured by the traceless Reynolds stress ˚ Rν=Rν−1 3(trRν)I3. In turn, this term contributes to the energy density by esub(vν,˚ Rν) = 1 2|vν|2+3 2λmax(˚ Rν), where λmax is the largest eigenvalue, which is nonnegative for traceless symmetric matrices. Notice that esub coincides with the usual energy density if and only if ˚ Rν= 0. As discovered in [36], the energy density esub characterizes the relaxation of the Euler equation, in particular it is convex within the space of subsolutions. By applying the Jensen inequality, we deduce that the energy of the averaged solution remains below the averaged energy ZR3esub(vν,˚ Rν)dx≤ZενP 1 2ZR3|vη,ν(t, x)|2dxdµ. In the inviscid limit, we obtain formally a solution to the Euler-Reynolds equation ∂tv+ div(v⊗v) + ∇p=−divR, (3.2.8a) divv= 0,(3.2.8b) v|t=0 =v◦. Chapter 3. Euler solutions originating from a circular vortex filament 60 Notice that the nonlinear term may not necessarily converge. However, since it is expected that the averages cancel out highly oscillating terms, let us assume, for simplicity, that this is not the case (otherwise, this contribution would be added to R). Finally, the Fatou lemma allows to bound the energy by ZR3esub(v, ˚ R)dx≤liminf ν→0ZR3esub(vν,˚ Rν)dx. We have seen that the existence of a subsolution (a solution to the Euler- Reynolds equation) with decreasing energy is a necessary condition for the occurrence of spontaneous stochasticity and anomalous dissipation. As mentioned in the previous section, it is natural to assume that the vorticity of this subsolution accumulates in a turbulence ring that expands around a dynamical filament and makes the energy finite for positive times. Figure 3.1 shows a cartoon illustrating these ideas. By virtue of the h-principle for the Euler equation [36], the existence of weak solutions as described in Theorem 3.1.1 is reduced to find a suitable subsolution. We remark that it remains an open question whether these solutions arise from an inviscid limit as described before. Notice that the Reynolds stress introduces several degrees of freedom in the system, making the problem overdetermined. Despite this freedom, we will see in the next section how the physical properties we seek end up roughly determining the evolution of the turbulent zone and the energy dissipation. 3.3 Sketch of the proof In this section we outline the main steps for the proof of Theorem 3.1.1. The precise computations and technical details will be rigorously addressed in the following sections. Our first task consists of constructing a subsolution (v, p, R), that is, a solution to the Euler-Reynolds equation (3.2.8). If we interpret the subsolution as the averaged solution (3.2.6), it makes sense for it to preserve the symmetries of the initial data. In our case, the initial data is axisymmetric without swirl. Specifically, ω◦= ΓδCis the vector-valued measure satisfying ⟨ω◦, φ⟩= ΓZCφ·dl= ΓLZ2π 0φ(Ler)·eθdθ, for any test function φ∈C(R3), where er= (cosθ, sin θ, 0), eθ= (−sinθ, cos θ, 0), ez= (0,0,1), 61 3.3. Sketch of the proof is the standard orthonormal basis in cylindrical coordinates (r, θ, z). Hence, we assume that the vorticity of the subsolution is of the form ω(t, x) = ωθ(t, r, z)eθ,(3.3.9) for some vorticity flux ωθ, which we assume to be bounded for positive times, to be determined. Notice that divω= 0. It is well-known (see Section 3.4) that the corresponding velocity field is of the form v(t, x) = −1 4πZR3 (x−x′)∧ω(t, x′) |x−x′|3dx′=vr(t, r, z)er+vz(t, r, z)ez.(3.3.10) As we aim to attain R(t, x)∼v⊗v, we assume that the Reynolds stress is of the form R(t, x) = Rrr(t, r, z)(er⊗er) + Rrz(t, r, z)(er⊗ez+ez⊗er) + Rzz(t, r, z)(ez⊗ez). (3.3.11) Similarly, we assume that p(t, x) = p(t, r, z).(3.3.12) Notice that all the coefficients in (3.3.9)-(3.3.12) do not depend on the azimuth θ. This makes possible to write the Euler-Reynolds equation (3.2.8) in the (complex) half-plane H:= {(r, z)∈R2:r > 0},(3.3.13) by identifying (with slightly abuse of notation) v="vr vz#, R ="Rrr Rrz Rrz Rzz #.(3.3.14) Under the axisymmetric without swirl assumption (3.3.9)-(3.3.12), we prove in Section 3.4 that the momentum equation (3.2.8a) is rewritten for (3.3.14) as ∂tv−ωθv⊥+∇q=−1 rdiv(rR),(3.3.15) on [0, T]×H, where q=p+1 2|v|2, is the Bernoulli pressure. We remark that, by the Biot-Savart law (3.3.10), the velocity field is divergence-free (3.2.8b) and satisfies the boundary condition vr= 0 on ∂H. As we were arguing in the previous sections, it is natural to assume that the vorticity is supported in a ring that moves in the vertical direction, and expands Chapter 3. Euler solutions originating from a circular vortex filament 68 Proposition 3.4.2. In the setting of Lemma 3.4.1, the axisymmetric without swirl Euler-Reynolds equation (3.4.33)(3.4.34)is equivalent to r(∂tv−ωθv⊥+∇q) = −div(rR),(3.4.35a) v|t=0 =v◦ where q=p+1 2|v|2 is the Bernoulli pressure. Proof. By the Biot-Savart law (3.3.9)(3.3.10), the conditions (3.4.33b) and (3.4.34) are automatically satisfied, while ω= curlxv= (∂zvr−∂rvz)eθ. Therefore, we have divζ(rv)=0,curlζv=−ωθ.(3.4.36) Finally, the equivalence between (3.4.33a) and (3.4.35a) follows from the identity divζ(rv ⊗v) = rv ·∇ζv=r1 2∇ζ|v|2−ωθv⊥, where we have applied (3.4.36). 3.4.1 Biot-Savart law In this subsection we give the velocity by means of the axisymmetric without swirl Biot-Savart law. Lemma 3.4.3 (Axisymmetric without swirl Biot-Savart law).It holds v(t, ζ) = ZH Kax(ζ, ζ′)ωθ(t, ζ′) dζ′, for ζ= (r, z)∈H, where Kax(ζ, ζ′) = i 2πr  sr′ rH(s)−ζG(s) , and H(s) = Zπ 0 1−cosφ (2(1 −cosφ) + s)3 /2dφ, G(s) = Zπ 0 cosφ (2(1 −cosφ) + s)3 /2dφ, for the variables ζ=ζ−ζ′ √rr′, s =|ζ|2. 69 3.4. Axisymmetric Euler-Reynolds equation Proof. the Biot-Savart law in R3is given by v(t, x) = ZR3K3D(x−x′)ω(t, x′)dx′=−1 4πZR3 (x−x′)∧ω(t, x′) |x−x′|3dx′. By writing x−x′= (reiθ −r′eiθ′, z −z′), and ω(t, x) = ωθ(t, r, z)eθ, we deduce that (x−x′)∧eθ′= e1e2e3 rcosθ−r′cosθ′rsinθ−r′sinθ′z−z′ −sinθ′cosθ′0 =−(z−z′)er′+ (rcos(θ−θ′)−r′)ez, and also (φ=θ′−θ) |x−x′|2=|rei(θ−θ′)−r′|2+|z−z′|2=rr′(2(1 −cosφ) + s), where ζ=ζ−ζ′ √rr′, s =|ζ|2. On the one hand, we have Z2π 0ZH ωθ (z−z′) |x−x′|3eiθ′r′d(r′, z′) dθ′ =eiθ ZH (z−z′)r′ (rr′)3 /2ωθZ2π 0 eiφ (2(1 −cosφ) + s)3/2dφd(r′, z′). Thus, vr=1 2πr ZH z−z′ √rr′Gωθdζ′, where G(s) = Zπ 0 cosφ (2(1 −cosφ) + s)3 /2dφ. (3.4.37) On the other hand, we have ZH r′ (rr′)3 /2ωθZ2π 0 rcosφ−r′ (2(1 −cosφ) + s)3 /2dφd(r′, z′) =ZH (r−r′)r′ (rr′)3 /2ωθZ2π 0 cosφ (2(1 −cosφ) + s)3 /2dφd(r′, z′) −ZH (r′)2 (rr′)3 /2ωθZ2π 0 1−cosφ (2(1 −cosφ) + s)3 /2dφd(r′, z′), Chapter 3. Euler solutions originating from a circular vortex filament 70 and thus vz=−1 2πr ZH r−r′ √rr′Gωθdζ′+1 2πr ZHsr′ rHωθdζ′, where H(s) = Zπ 0 1−cosφ (2(1 −cosφ) + s)3 /2dφ. (3.4.38) Therefore, we can write compactly v=−i 2πr ZH ζGωθdζ′+i 2πr ZHsr′ rHωθdζ′. It can be checked using elliptic integrals that this formula agrees with the one in [43]. Lemma 3.4.4. The function G(s)defined in (3.4.37)satisfies G(s) = 1 s+O(log s), as s→0. Proof. We make a Taylor expansion of the cosine in the numerator to obtain G(s) =Zπ 0 1 (2(1 −cosφ) + s)3 /2dφ−1 2Zπ 0 φ2 (2(1 −cosφ) + s)3 /2dφ +Zπ 0 O(φ4) (2(1 −cosφ) + s)3 /2dφ=G1+G2+G3. We further split G1in the following way G1(s) = Zπ 0 1 (φ2+s)3 /2dφ+Zπ 0 1 (2(1 −cosφ) + s)3 /2−1 (φ2+s)3 /2dφ. The first integral can be computed exactly and is equal to Zπ 0 1 (φ2+s)3 /2dφ=π s√s+π2=1 s+O(1). For the second integral is equal to Zπ 0 (φ2+s)3−(2(1 −cos(φ)) + s)3 (2(1 −cosφ) + s)3 /2(φ2+s)3 /2((2(1 −cos φ) + s)3 /2+ (φ2+s)3 /2)dφ. Note that there exist real constants C1, C2>0such that C1φ2≤1−cos(φ)≤C2φ2 for 0≤φ≤π/2. For π/2≤φ≤π, the integrand is bounded. Therefore, in the 71 3.5. The subsolution last integral we can cancel the s3terms and bound the absolute value of the integral by Zπ/2 0 O(φ4)O((φ2+s)2) (φ2+s)9 /2dφ +O(1) ≲Zπ/2 0 dφ (φ2+s)1 /2+O(1) = O(log s). The same idea let us bound |G2| ≤ O(log s),|G3| ≤ O(1). Lemma 3.4.5. The function H(s)defined in (3.4.38)satisfies H(s) = −1 4log s+O(1), as s→0. Proof. We follow the same strategy than in Lemma 3.4.4. We can split H(s) = 1 2Zπ 0 φ2 (2(1 −cosφ) + s)3 /2dφ +Zπ 0 O(φ4) (2(1 −cosφ) + s)3 /2dφ. We split the first integral into Zπ 0 φ2 (2(1 −cosφ) + s)3 /2dφ =Zπ 0 φ2 (φ2+s)3 /2dφ +Zπ 0 φ2 (2(1 −cosφ) + s)3 /2−φ2 (φ2+s)3 /2dφ. The first part can be computed as Zπ 0 φ2 (φ2+s)3 /2dφ = arcsinh π √s−π √s+π2=−1 2log s+O(1). The remaining terms are bounded similarly. 3.5 The subsolution In this section we construct the axisymmetric without swirl subsolution (3.3.9)- (3.3.12). Firstly, we assume that the vorticity flux (3.3.9) is given by (3.3.17) and (3.3.18): ωθis supported in the shifted ball (3.3.16), which is parametrized by D. The vorticity density must satisfy the compatibility conditions (3.3.19) and (3.3.30). In the next lemma we show that there exists ϖ(ρ)satisfying these conditions. Chapter 3. Euler solutions originating from a circular vortex filament 72 Lemma 3.5.1. It is possible to select a vorticity profile ϖ∈C(R+), supported on (0,1), and satisfying (3.3.19)and (3.3.30). Proof. For instance, we can make the ansatz for 0≤ρ≤1 ϖ(ρ)=Γc1ρ(1 −ρ)1−c2log(eρ), and adjust the parameters c1, c2∈R. Then, we extend ϖ(ρ) = 0 outside the interval [0,1]. Remark 3.5.2. The particular choice of ϖ(ρ)in the previous lemma is not important in this work. In fact, we will show in Section 3.6 that the leading term in the energy does not depend on ϖ(ρ). We will make all computations for any ϖ(ρ) that satisfies Lemma 3.5.1. 3.5.1 The velocity In this section we derive asymptotic formulas for the Biot-Savart operator evaluated on D v(t, D) = Z2π 0Z1 0Kax(D,D′)ϖ(ρ′)ρ′dρ′dα′, where we have abbreviated D = D(t, ρ, α)and D′= D(t, ρ′, α′). To obtain the leading order terms, we will assume that c≪1. Our c(t)will be continuous and null at t= 0, so this condition is satisfied for short times. Proposition 3.5.3. It holds v(t, D) = −iρΓρ 2πc ei(α+a)−iΓ 4πL log c+O(1), where Γρis defined in equation (3.3.21). Proof. We follow the notation of Lemma 3.4.3. We decompose v=v⟳+v↑, where v⟳=−i 2πr ZH ζG(s)ωθdζ′, v↑=i 2πr ZHsr′ rHωθdζ′. Firstly, we compute v⟳. Notice that we can write ζ=cρ−ρ′eiα √rr′ei(α+a), s =c2|ρ−ρ′eiα|2 rr′, 73 3.5. The subsolution where α(t, α, α′, ρ, ρ′) = α′+a(t, ρ′)−α−a(t, ρ). From Lemma 3.4.4, we know that G(s) = 1 s+O(log s), as s→0. Hence, v⟳=−i 2πr Z1 0Z2π 0ζG(s)dα′ϖ(ρ′)ρ′dρ′ =−i 2πr Z1 0Z2π 0ζ1 s+O(log s)dα′ϖ(ρ′)ρ′dρ′ =−i 2πr Z1 0Z2π 0 1 ζ∗dα′ϖ(ρ′)ρ′dρ′+i 2πr Z1 0Z2π 0ζO(log s)dα′ϖ(ρ′)ρ′dρ′. Since |ζ|=√s, the integrand in the second term is bounded, so the result is O(clog c) = O(1). The first term can be computed (ρ′=λρ) −i 2πr Z1 0Z2π 0 dα′ ζ∗ϖ(ρ′)ρ′dρ′=−i 2πcei(α+a)Z1 0Z2π 0 dα ρ−ρ′e−iα ϖ(ρ′)ρ′dρ′ −i 2πrcei(α+a)Z1 0Z2π 0 (√rr′−r)dα ρ−ρ′e−iα ϖ(ρ′)ρ′dρ′ =−i cei(α+a)Zρ 0ϖ(ρ′)ρ′ ρdρ′+O(1) =−i cei(α+a)Z1 0ϖ(ρλ)ρλ dλ+O(1), where we have applied 1 2πZ2π 0 dα ρ−ρ′e−iα =1 2πZ2π 0 eiα dα ρeiα −ρ′=1 2πρi Z∂Bρ dz z−ρ′=1 ρ1ρ>ρ′, and  1 rc Z1 0Z2π 0 (√rr′−r)dα ρ−ρ′e−iα ϖ(ρ′)ρ′dρ′ ≲ 1 √rZ1 0Z2π 0 ρcos(a+α)−ρ′cos(a+α′) (ρ−ρ′e−iα)(√r′+√r)ϖ(ρ′)ρ′dαdρ′ ≲Z1 0Z2π 0O(1)ϖ(ρ′)ρ′dαdρ′=O(1). Therefore, v⟳=−iρΓρ 2πc ei(α+a)+O(1). Chapter 3. Euler solutions originating from a circular vortex filament 74 Secondly, we compute v↑. From Lemma 3.4.5, we know that H(s) = −1 4log s+O(1), as s→0. Hence, v↑=i 2πr Z1 0Z2π 0sr′ rHdα′ϖ(ρ′)ρ′dρ′ =i 2πr Z1 0Z2π 0sr′ r−1 4log s+O(1)dα′ϖ(ρ′)ρ′dρ′ =−i 8πr Z1 0Z2π 0sr′ rlog sdα′ϖ(ρ′)ρ′dρ′+O(1). We further compute v↑=−i 8πr Z1 0Z2π 0sr′ r 2 log c+ log |ρ−ρ′eiα|2 rr′!!dα′ϖ(ρ′)ρ′dρ′+O(1) =−i 4πr Z1 0Z2π 0sr′ rlog cdα′ϖ(ρ′)ρ′dρ′+O(1) +i 2πr Z1 0Z2π 0sr′ rlog |ρ−ρ′eiα|dα′ϖ(ρ′)ρ′dρ′+O(1) =−iΓ 4πL log c+O(1)ZB(0,2) |log |x||dx +O(1) = −iΓ 4πL log c+O(1), where we have used that r=L+O(c). 3.5.2 The pressure In this section we give the pressure by means of the Bernoulli law. Lemma 3.5.4 (The Bernoulli law).There exists q0satisfying ∂tv+∇q0=ωθ∂tD⊥. Proof. Let us denote vax =v, and define v2D(t, ζ) = ZR2K2D(ζ−ζ′)ωθ(t, ζ′) dζ′, where K2D(ζ) = i 2πζ∗ 75 3.5. The subsolution is the 2dBiot-Savart kernel. Since curlζ(v2D) = ωθ=−curlζ(vax) in the simply-connected domain H, there exists e qsatisfying ∂t(vax +v2D) = ∇e q. We claim that q0,0(t, ζ) = −Z2π 0Z1 0K2D(ζ−D) ·∂tDϖρ dρdα, where D = D(t, ρ, α)satisfies ∂tv2D+ωθ∂tD⊥=∇q0,0.(3.5.39) Then, the statement holds for q0=q0,0−e q. Let ϕ∈C∞ c((0, T)×H)be a test function. On the one hand, ⟨∂tv2D, ϕ⟩=−⟨v2D, ∂tϕ⟩=−ZT 0ZH v2D∂tϕdζdt =−ZT 0ZHZ2π 0Z1 0K2D(ζ−D)ϖρ dρdα∂tϕdζdt =−Z2π 0Z1 0lim ε→0 ZT 0ZH\Bε(D) K2D(ζ−D)∂tϕdζdt!ϖρ dρdα. We parameterize the (space-time) cylinder ∂Bε(D) by X(t, β)=(t, D(t) + εeiβ). The normal vector to the boundary pointing into the interior of the cylinder is given by n=∂tX∧∂βX |∂tX∧∂βX|, ∂tX∧∂βX=ε(∂tD·(eiβ),−eiβ). Therefore, an integration by parts yields ZT 0ZH\Bε(D) K2D(ζ−D)∂tϕdζdt=ZT 0Z∂Bε(D) K2D(ζ−D)ϕntdζdt(3.5.40) −ZT 0ZH\Bε(D) ∂t(K2D(ζ−D))ϕdβdt.(3.5.41) Chapter 3. Euler solutions originating from a circular vortex filament 76 The first term equals (3.5.40)=ZT 0Z2π 0εK2D(εeiβ)ϕ(t, D+εeiβ)∂tD·(eiβ)dζdt. Since εK2D(εeiβ) = ieiβ 2π, we get lim ε→0(3.5.40)=1 2πZT 0ϕ(t, D)Z2π 0ieiβ∂tD·(eiβ) dβdt=1 2ZT 0ϕ(t, D)∂tD⊥dt. For the second term (3.5.41) we have ∂t(K2D(ζ−D)) = −DζK2D(ζ−D)∂tD. By plugging everything together we obtain ⟨∂tv2D, ϕ⟩=−Z2π 0Z1 0lim ε→0 ZT 0ZH\Bε(D) DζK2D(ζ−D)∂tDϕdζdt!ϖρ dρdα −1 2Z2π 0Z1 0 ZT 0ϕ(t, D)∂tD⊥dt!ϖρ dρdα. On the other hand, ⟨∇q0,0, ϕ⟩=−⟨q0,0,∇ϕ⟩=−ZT 0ZH q0,0∇ϕdζdt =ZT 0ZHZ2π 0Z1 0K2D(ζ−D) ·∂tDϖρ dρdα∇ϕdζdt =Z2π 0Z1 0lim ε→0 ZT 0ZH\Bε(D) K2D(ζ−D) ·∂tD∇ϕdζdt!ϖρ dρdα. An integration by parts yields ZT 0ZH\Bε(D) K2D(ζ−D) ·∂tD∇ϕdζdt=ZT 0Z∂Bε(D) K2D(ζ−D) ·∂tDϕnζdζdt (3.5.42) −ZT 0ZH\Bε(D) ∇(K2D(ζ−D) ·∂tD)ϕdζdt. (3.5.43) Since the first term equals (3.5.42)=ZT 0Z2π 0εK2D(εeiβ)·∂tDϕ(t, D+εeiβ)(−eiβ)dβdt, 77 3.5. The subsolution we get lim ε→0(3.5.42)=−1 2πZT 0ϕ(t, D)Z2π 0(ieiβ)·∂tDeiβ dβdt=1 2ZT 0ϕ(t, D)∂tD⊥dt. For the second term (3.5.43), we have ∇(K2D·∂tD) = (DζK2D)†∂tD = DζK2D∂tD+(curlζK2D)∂tD⊥. Since curlζK2D=δζ′, we indeed have ∇(K2D·∂tD) = DζK2D∂tD, for ζ∈H\Bε(D). By plugging everything together, we obtain ⟨∇q0,0, ϕ⟩=−Z2π 0Z1 0lim ε→0 ZT 0ZH\Bε(D) DζK2D(ζ−D)∂tDϕdζdt!ϖρ dρdα +1 2Z2π 0Z1 0 ZT 0ϕ(t, D)∂tD⊥!ϖρ dρdα. Finally, we get ⟨∂tv−∇q0,0, ϕ⟩=−ZT 0Z2π 0Z1 0ϕ(t, D)∂tD⊥ϖρ dρdαdt, that is, our claim (3.5.39). Finally, we take the Bernoulli pressure as q=q0+q1, where q1is a corrector introduced to guarantee several compatibility conditions that appear in the next section. We take q1smooth and with the same support as ω. Therefore, ∂tv+∇q=ωθ∂tD⊥+∇q1.(3.5.44) 3.5.3 The Reynolds stress In this section we give the Reynolds stress by means of the Isett-Oh antidivergence operator [68]. In view of (3.5.44), the Euler-Reynolds equation (3.4.35) reads as −div(rR) = F, (3.5.45) Chapter 3. Euler solutions originating from a circular vortex filament 84 We rewrite (3.3.21) as Γρ=1 ρ2Zρ 0ϖ(ρ′)ρ′dρ′=1 ρ2f(ρ), obtaining Z1 0ϖ(ρ)ρ3Γρdρ=Z1 0f′(ρ)f(ρ)dρ =1 2(f2(1) −f2(0)) = Γ2 2(2π)2, independently of ϖ. Thus, (3.6.52)=−LΓ2 2 ˙c c. We can do similar computations for the remaining terms to obtain ∂tEv=−LΓ2 2∂t(log c) + O(˙clog c). Finally, notice that the last term is integrable in time. Lemma 3.6.2. It holds, ER(t) = O(1), as t→0. Proof. Notice that λmax(˚ R)≲∥˚ R∥L∞≲∥R∥L∞≲∥rR∥L∞≲c∥F∥L∞, where we have applied Lemma 3.5.5 in the last inequality. From equations (3.3.27) and (3.5.50), we deduce that ||F||L∞≲c˙c+ 1 c3, concluding that ER=ZR3 3 2λmax(˚ R)≲|Bc|||rR||L∞=O(c˙c+ 1), which is O(1) by our choice of c. 85 3.7. Time-weighted h-principle 3.7 Time-weighted h-principle In this section, we show how the De Lellis-Székelyhidi h-principle for the Euler equation [36] can be adapted for constructing velocities in the space Clog tL2. We define this space, for any 1≤p < ∞, by the time-weighted norm ∥f∥Clog tLp:= sup t∈(0,T] ∥f(t)∥Lp max{1,|log t|}1 /p<∞. Firstly, it is convenient to rewrite the Euler-Reynolds equation (3.2.8) in linear form ∂tv+ divS+∇π= 0, divv= 0, v|t=0 =v◦, where we have made the change of variables π=p+1 3(|v|2+ trR), S =v⊚v+˚ R, with v⊚v=v⊗v−1 3|v|2I3,˚ R=R−1 3(trR)I3. Thus, a triple (v, π, S)∈Clog t(L2×L1 loc ×L1)(3.7.54) is a weak solution to the Euler-Reynolds equation (3.7.53) if vis a weakly divergencefree vector field, Sis a traceless symmetric tensor, and πis a scalar pressure, such that ZT 0ZR2(v·∂tφ+S:∇φ+πdivφ)(t, x) dxdt+ZR2v◦(x)φ(0, x)dx= 0,(3.7.55) for all test function φ∈C1 c([0, T)×R2). Notice that the integrability condition (3.7.54) is enough to make the expression (3.7.55) meaningful. Let (v, π, S)be the subsolution we constructed in the previous sections. The associated energy density is given by the convex function (see [36, Lemma 3]) esub(v, S) = 3 2λmax(v⊗v−S) = 1 2|v|2+3 2λmax(˚ R). The total energy Esub satisfies (3.6.51). We define the energy density of the (true) solutions by e=esub(v, S) + e e, Chapter 3. Euler solutions originating from a circular vortex filament 86 where e eis a smooth energy corrector. We take e evanishing outside Ωtur, strictly positive inside, and in such a way that the corresponding energy E(t) = ZR3e(t, x)dx=Esub(t) + O(1),(3.7.56) is strictly decreasing. We define the subspace X0of velocities v∈Clog tL2satisfying: (i) There exists Ssuch that (v, π, S)is a weak solution to the Euler-Reynolds equation. (ii) (v, S)=(v, v ⊚v)outside Ωtur. (iii) (v, S)is continuous and satisfy esub(v, S)< e inside Ωtur. By definition, we have v∈X0. For any v∈X0and t∈(0, T], since 1 2ZR3|v(t, x)|2dx≤E(t), we have V(t) = v(t) max{1,|log t|}1 /2∈B, for some closed ball in L2. By the Banach–Alaoglu theorem, we can take dBby the metric of Bwith respect to the weak convergence. Then, d(v1, v2) = sup t∈(0,T] dB(V1(t), V2(t)), defines a metric on Clog tL2 weak. Let Xbe the closure of X0with respect to the metric d. Then, Xis a complete metric subspace of Clog tL2 weak. Lemma 3.7.1. Any v∈Xsatisfies the conditions i,ii and iii with esub(v, S)≤e, (3.7.57) almost everywhere in Ωtur. Proof. Since X0=X, there exists a sequence (vj)⊂X0converging to v. Let Sjbe the corresponding traceless symmetric tensor from i. By applying the conditions ii, iii and e∼1/t, we deduce that the sequence (tSj)is bounded in L∞([0, T]×R2). By the Banach-Alaoglu theorem, we may assume (by taking a subsequence if necessary) that tSjconverges to some tS in L∞ weak∗([0, T]×R2). By applying the Mazur lemma, we deduce (3.7.57). Finally, it follows that (v, π, S)is a weak solution to the Euler-Reynolds equation. 87 3.7. Time-weighted h-principle Given a (space-time) cylinder I×Ω⊂⊂ Ωtur, we consider the functional J: X→[0,∞) J(v) = sup t∈IZΩe−1 2|v|2(t, x)dx. Lemma 3.7.2. The functional Jis well defined. Moreover, J(v)=0if and only if S=v⊚vin I×Ω. Proof. By (3.7.57), any v∈X⊂Clog tL2 weak satisfies 1 2|v(t, x)|2≤e(t, x), for almost every x∈Ωand all t∈I. Moreover, since eis continuous in I×Ω⊂⊂ Ωtur, we have J(v)≤sup t∈IZΩe(t, x)dx < ∞. Suppose now that J(v) = 0 and let Sbe the traceless symmetric tensor from Lemma 3.7.1. By (3.7.57), we have 1 2|v|2+3 2λmax(S−v⊚v)≤e=1 2|v|2, and thus S=v⊚vin I×Ω. Next, we check that any continuity point of the functional satisfies J(v)=0, which turns out to form a residual set in X. The proofs appear in [36, Lemma 4.3 and Proposition 4.5]. Since the time interval Iis away from t= 0, the same proofs hold in our space X. Lemma 3.7.3. The functional Jis upper-semicontinuous. Hence, XJ={v∈X:Jis continuous at v} is a countable intersection of open and dense sets. Proposition 3.7.4 (Perturbation property).For all α > 0there exists β(α, I × Ω) >0such that, whenever v∈X0satisfies J(v)≥α, there exists a sequence (vk)⊂X0converging to vand satisfying limsup k→∞ J(vk)≤J(v)−β. Chapter 3. Euler solutions originating from a circular vortex filament 88 Corollary 3.7.5. It holds XJ⊂J−1(0). Proof. We prove it by contradiction. Let us suppose that there exists v∈XJ with J(v)>0, and take α= 2J(v). Since X=X0, there exists a sequence (vj)⊂X0converging to v. Since Jis continuous at v, we may a assume (by taking a subsequence if necessary) that J(vj)≥1 2J(v) = α > 0for all indices j. By the perturbation property, for each vjthere exists a sequence (vj,k)⊂X0 converging to vjand satisfying limsup k→∞ J(vj,k)≤J(vj)−β(α, I ×Ω). Since Xis a complete metric space, we can construct a diagonal sequence vj,k(k)→ vsatisfying J(vj,k(k))↛J(v), which contradicts v∈XJ. 3.7.1 Proof of Theorem 3.1.1 Firstly, we consider a covering {Ij×Ωj}of Ωtur, and the corresponding functionals {Jj}. By Lemma 3.7.3 and Corollary 3.7.5, the set Xsol =\ j J−1 j(0) contains a countable intersection of open and dense sets in X. Since Xis a complete metric space, the Baire category theorem implies that Xsol is dense. Let v∈Xsol. On the one hand, recall that the conditions iand ii are already satisfied by Lemma 3.7.1. In particular, v=vis analytic outside Ωtur. On the other hand, we have 1 2|v|2=e. Hence, by (3.7.56), we have Ev(t) = 1 2ZR2|v(t, x)|2dx=ZR2e(t, x)dx=E(t). Since v∈Clog tL2 weak and |v|2= 2e∈Clog tL1, we also have v∈Clog tL2. Moreover, by Lemma 3.7.2, we have S=v⊚vin Ωtur. Hence, vis a weak solution to the Euler equation with pressure p=π−2 3e. We remark that the pressure is also analytic outside Ωtur. Moreover, since the vorticity is continuous by Lemma 3.5.1, it is not difficult to check that the pressure is indeed continuous in R3for positive times. 89 3.7. Time-weighted h-principle Finally, we check that v∈C([0, T], L2−). Let us fix 1<p<2. On the one hand, the subsolution already satisfies v∈C([0, T], Lp). Since v=voutside Ωtur ⊂[0, T]×BRfor some R > 0, we also have v∈C([0, T], Lp(R3\BR)). 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