Uniqueness for SQG patch solutions
Abstract
This paper is about the evolution of a temperature front governed by the surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by the third author (2008). Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 5, Pages 1–31 (January 19, 2018) http://dx.doi.org/10.1090/btran/20 UNIQUENESS FOR SQG PATCH SOLUTIONS ANTONIO C´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO Abstract. This paper is about the evolution of a temperature front governed by the surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by the third author (2008). Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result. 1. Introduction Among the more important partial differential equations of fluid dynamics we have the three dimensional Euler equation, modelling the evolution of an incompressible inviscid fluid, and the surface quasi-geostrophic (SQG) which describes the dynamics of atmospheric temperature [19]. SQG also has the extra mathematical interest of capturing the complexity of the 3D Euler equation but in a two dimensional scenario, as was described in the classical work [8]. This model reads θt+u·∇θ=0, u=(−R2θ,R1θ), where θ(x, t) is the temperature of the 2D fluid with (x, t)∈R2×[0,+∞). The velocity uis related to the temperature through the Riezs transforms Rjgiven by Rj(θ)(x)= 1 πR2 yj |y|3θ(x−y)dy. Within the equation there is an underlying particle dynamic which preserves the value of θ, implying that the norms θLp(t), 1 ≤p≤∞, remain constants under the evolution. In this paper we consider the patch problem, on which the temperature takes two constant values in two complementary domains and the solution of SQG has to be understood in a weak sense, namely: (1) ∞ 0R2 θ(x, t)(ϕt(x, t)+u(x, t)·∇ϕ(x, t))dxdt =R2 θ0(x)ϕ(x, 0)dx, u=(−R2θ,R1θ), for every ϕ∈C∞ c([0,∞)×R2). That is, the temperature reads (2) θ(x, t)=θ1,x∈D1(t), θ2,x∈D2(t)=R2\D1(t), Received by the editors May 21, 2016 and, in revised form, February 7, 2017, May 30, 2017, July 14, 2017, and August 2, 2017. 2010 Mathematics Subject Classification. Primary 35Q35. c 2018 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0) 1
2 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO where D1(t) is a simply connected domain. It gives rise to a contour equation for the free boundary (3) ∂Dj(t)={x(γ,t)=(x1(γ,t),x 2(γ,t)) : γ∈T}, which is moving with the fluid and whose exact formulation can be found in [10]. It is then clear that the evolution of the patch is equivalent to that of its free boundary ∂Dj(t). Therefore an important question for this problem is the propagation in time of the regularity of the interface ∂Dj(t) or to the contrary the existence of finite time blow-up phenomena. This problem was first considered by Resnick in his thesis [20]. Local-in-time existence and uniqueness was proven by Rodrigo [21] for C∞initial data using the Nash-Moser inverse function theorem. In [10] the third author proves local-in-time existence for the problem in Sobolev spaces, using energy estimates and properties of a particular parameterization of the contour. Namely, one such that the modulus of the tangent vector to the curve does not depend on the space variable, depending only on time [16] and giving us extra cancellations which allows to integrate the system. In the distributional sense, the gradient of the temperature is given by ∇θ(x, t)=(θ2−θ1)∂⊥ γx(γ,t)δ(x(γ,t)=x) for x(γ,t) a given parameterization of the contour and ∂⊥ γx(γ,t)=(−∂γx2(γ,t),∂ γx1(γ,t)). Then the Biot-Savart formula helps us to get the velocity field, outside the boundary, in terms of the geometry of the contour, that is, u(x, t)=I1(∇⊥θ)(x, t)=−θ2−θ1 2ππ −π ∂γx(γ,t) |x−x(γ,t)|dγ, where I1is the Riesz potential of order 1, which on the Fourier side is multiplication by |ξ|−1. The above integral diverges when xapproaches the boundary but only on its tangential component, while its normal component is well defined. This fact is crucial to assign a normal velocity field to the boundary governing its evolution. Since the contribution of the tangential component amounts to a reparameterization of the boundary curve, we are free to add such a component satisfying both purposes: to be bounded and having a tangent vector with constant length. For a given parameterization x(γ,t), approaching the boundary in both domains we obtain u(x(γ,t),t)·∂⊥ γx(γ,t)=−θ2−θ1 2ππ −π ∂γx(η,t)·∂⊥ γx(γ,t) |x(γ,t)−x(η,t)|dη. And we get the task of finding a good parameterization x(γ,t) and a function λso that u(x(γ,t),t)·∂⊥ γx(γ,t) =θ2−θ1 2ππ −π ∂γx(γ,t)−∂γx(η,t) |x(γ,t)−x(η,t)|dη +λ∂γx(γ,t)·∂⊥ γx(γ,t), and the two purposes mentioned above are achieved. Having the length of the vector ∂γx(γ,t) as a function in the variable tonly provides the following two identities: (4) ∂2 γx(γ,t)·∂γx(γ,t)=0 and ∂3 γx(γ,t)·∂γx(γ,t)=−|∂2 γx(γ,t)|2.
UNIQUENESS FOR SQG PATCH SOLUTIONS 3 The first one gives extra cancellations while the second allows us to perform convenient integration by parts. Although we cannot give justice to the many interesting contributions due to the different authors quoted in our references, let us say that, at the beginning, there was a conjecture about the formation of singularities in the evolution of a vortex patch for Euler equations in dimension two [2]. It was disproved by Chemin in a remarkable work [7] using paradifferential calculus, and later Bertozzi-Constantin [1] obtained a different proof taking advantage of an extra cancellation satisfied by singular integrals having even kernels. Between the patch problem for 2D Euler and SQG there is a continuous set of interpolated equations given by θt+u·∇θ=0, u=(−R2,R 1)(I1−αθ),0<α<1. (5) The case α= 0 is the most regular, 2D Euler, while for α= 1 one gets SQG. The patch problem for those equations was first studied in [9], where C´ordoba, Fontelos, Mancho, and Rodrigo introduced a very interesting scenario for which they could show numerical evidence of singularity formation: two patches with different temperature approach each other in such a way that they collide at a point where the curvature blows-up. Let us mention that recently it has been shown analytically [11] that if the curvature is controlled, then pointwise collisions cannot happen in the patch problem for SQG. In [22,23] a different finite time singularity scenario is shown where numerics point at a self-similar blow-up behaviour for SQG patches. The system above can also be considered in more singular cases than SQG, replacing the last identity by the following one: u=(−R2,R 1)(Λβθ),0<β<1, where here Λ = (−Δ)1/2, whose Fourier symbol is |ξ|. See [6] for results on this equation with patch solutions. A classical result in fluid dynamics is the existence for all time of vortex patches for the Euler equation which are rotating ellipses [2]. The patch problem for the system (5) and SQG present a more complex dynamic, as ellipses are not rotational solutions and some convex interfaces lose this property in finite time [5]. See [12] for a study of the growth of the patch support. Recently, in a remarkable series of papers and with an ingenious use of the Crandall-Rabinowitz mountain pass lemma, the authors have extended those global-in-time existence results to a more general class of geometrical shapes for the vortex patch problem [14,15], the α-system (5) [13] and also to the SQG equation [3,4]. There are two articles [17,18] where the patch problem for the α-system is considered in a half plane with Dirichlet’s condition. The system is proved to be well-posed for 0 <α< 1 12 in the more singular scenario where the patch intersects the fixed boundary. In this framework, singularity formation is shown when two patches of different temperature approach each other.
4 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO In this paper we will take advantage of a special parameterization of the boundary in the following terms: We say that a bounded simply connected domain D⊂R2is C2,δ(T)for0<δ<1 if there exists a parameterization of the boundary ∂D ={x(γ)∈R2:γ∈T,2π-periodic} such that x(γ)∈C2,δ(T). Specifically, a domain Ω ∈C2,δ(T)givenby ∂Ω={y(ξ)∈R2:ξ∈T,2π-periodic} is said to be equal to Dif there exists a change of variable ϕ:T→T,biyective,ϕ (γ)>0,ϕ(γ)−γ2π-periodic,ϕ∈C2,δ(T), such that x(γ)=y(ϕ(γ)). Furthermore, a time dependent simply connected domain D(t) belongs to C([0,T]; C2,δ(T))∩C1([0,T]; C1(T)) if there exist parameterizations of the boundaries ∂D(t)={x(γ,t)∈R2:γ∈T,t∈[0,T],2π-periodic in γ} such that x(γ,t)∈C([0,T]; C2,δ(T)) ∩C1([0,T]; C1(T)). Throughout the paper we shall also deal with time dependent simply connected domains in the space C([0,T]; Hk(T)), with HkSobolev spaces for k∈N, meaning that its evolving boundary x(γ,t) belongs to that time dependent space. Another main character of this play is the so-called arc-chord condition which help to control the absence of self-intersections of the boundary curve. This is done through the following quantity: F(x)(γ,η,t)= |η| |x(γ,t)−x(γ−η,t)|∀γ,η ∈[−π,π], with F(x)(γ,0,t)= 1 |∂γx(γ,t)|, whose L∞norm has to be controlled in the evolution. As was mentioned before, patch solutions for the SQG equation are understood in a weak sense. Any such solution with a free boundary given by a smooth parameterization x(γ,t) has to satisfy the equation below (6) xt(γ,t)·∂⊥ γx(γ,t)=−T ∂⊥ γx(γ,t)·∂γx(γ−η,t) |x(γ,t)−x(γ−η,t)|dη, wherewehavetakenθ2−θ1=πfor the sake of simplicity. On the other hand, any smooth parameterization x(γ,t) satisfying (6) provides a weak SQG solution with the temperature given by (2,3) (see [10] for more details). It is easy to check that the equation above is a reparameterization invariance object, and that the following formula, introduced in [20] and [21], has a well-defined tangential velocity and identical normal component: (7) xt(γ,t)=T ∂γx(γ,t)−∂γx(γ−η,t) |x(γ,t)−x(γ−η,t)|dη. The local-in-time existence result was given in [10] for initial data satisfying (4) and evolving by xt(γ,t)=T ∂γx(γ,t)−∂γx(γ−η,t) |x(γ,t)−x(γ−η,t)|dη +λ(x)(γ,t)∂γx(γ,t), (8)
UNIQUENESS FOR SQG PATCH SOLUTIONS 5 λ(x)(γ,t)=γ+π 2πT ∂ηx(η,t) |∂ηx(η,t)|2·∂ηT ∂ηx(η,t)−∂ηx(η−ξ,t) |x(η,t)−x(η−ξ,t)|dξdη −γ −π ∂ηx(η,t) |∂ηx(η,t)|2·∂ηT ∂ηx(η,t)−∂ηx(η−ξ,t) |x(η,t)−x(η−ξ,t)|dξdη. (9) We state the result here for completeness. Theorem 1.1. Let x0(γ)∈Hk(T)for k≥3with F(x0)(γ,η)<∞and ∂γx0(γ)· ∂2 γx0(γ)=0. Then there exists a time T>0so that there is a solution to (8) in C([0,T]; Hk(T)) with x(γ,0) = x0(γ)and λ(γ,t)given by (9). The main purpose of this paper is to show uniqueness for the patch problem for SQG which was until now an open problem. The following theorem provides this result: Theorem 1.2. Consider a solution of (1) with θ(x, t)given by a patch (2) and Dj(t)time dependent simply connected domains whose moving boundary satisfies the arc-chord condition for any t∈[0,T]and C([0,T]; C2,δ(T)) ∩C1([0,T]; C1(T)) regularity. Furthermore, assume that the function ¯ θ(x, t)given by ¯ θ(x, t)=θ1,x∈¯ D1(t), θ2,x∈¯ D2(t)=R2\¯ D1(t), satisfies (1) with ∂¯ Dj(t)∈C([0,T]; C2,δ(T)) ∩C1([0,T]; C1(T)) and θ(x, 0) = ¯ θ(x, 0).Thenθ(x, t)=¯ θ(x, t)for any t∈[0,T]. This is an important part of the paper and it is proved in its section 2. In particular we show that any weak solutions of (1) identified by a patch (2), for a given parameterization (3) with a certain regularity, can be reparameterized satisfying (4). This property is preserved in time and, together with a new reparameterized curve, help us to fix the tangential velocity for a contour that evolves by (8,9) giving the patch solution. Then, one just needs to get uniqueness for the system (8,9). Next we check the evolution of the H1Sobolev norm of the difference among two different curves evolving by (8,9). We close the estimate revisiting the previous existence results and introducing new cancellation and tools to find uniqueness by Gronwall’s lemma. However, in this process several different points of view with respect to the previous literature are introduced. In the following we are going to show how it is possible to go from (8,9) to equation (7) through a convenient change of variable. This procedure is also valid to go from (8,9) to an SQG patch contour equation with a different and more convenient tangential term. We denote by x(γ,t)∈C([0,T]; H3) a solution of (8,9) and let ˜x(ξ,t)begiven by ˜x(ξ,t)=x(φ−1(ξ,t),t),γ=φ−1(ξ,t), or equivalently x(γ,t)=˜x(φ(γ,t),t),ξ=φ(γ,t), where (10) φ(γ,t):R×R+→R,∂ γφ(γ,t)>0,φ(γ,t)−γ2π-periodic,
6 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO is a reparameterization in γfor any positive time. Here φis a solution of the linear system (11) φt(γ,t)=T ∂γφ(γ,t)−∂γφ(η,t) |x(γ,t)−x(η,t)|dη +λ(x)(γ,t)∂γφ(γ,t). The existence and uniqueness for that system is given in the following proposition, for whose formulation we introduce the space: Hk log ≡{f∈L2(T): n∈Z |n|2k log2(|n|+e)|ˆ f(n)|2=f2 Hk log <∞}. Proposition 1.3. Let φ0(γ)−γ∈Hk log for k≥3and x(γ,t)∈C([0,T]; Hk)be a solution of (8,9) with F(x)(γ,η,t)∈L∞and ∂γx(γ,0) ·∂2 γx(γ,0) = 0.Then there exists a unique solution to (11) with φ(γ,t)−γ∈C([0,T]; Hk log )such that φ(γ,0) = φ0(γ).In particular, if ∂γφ0(γ)>0,then∂γφ(γ,t)>0holds for any t∈(0,t p]with tp∈(0,T]. The proof of the proposition is given in section 3. The space Hk log is needed because we can only assume that λ(x)∈C([0,T]; Hk log )forx∈C([0,T]; Hk)(see the proof of Proposition 1.3). Observe that the logarithmic modification of Sobolev norms is not a problem in the proof of the existence theorem given in [10], because only control of the Hk−1norm of λ(x) is needed, which is far from the Hk log norm. In the energy estimates which provide local existence, one needs to consider the integral ∂k γx(γ,t)·∂k γxt(γ,t)dγ, whose most singular term coming form λ(x)isgivenby I=∂k γλ(x)(γ,t)∂k γx(γ,t)·∂γx(γ,t)dγ. Integration by parts yields I=−∂k−1 γλ(x)(γ,t)∂γ(∂k γx·∂γx)(γ,t)dγ, and using identity (4) one gets the bound I≤∂k−1 γλ(x)L2∂γ(∂k γx·∂γx)L2≤λ(x)Hk−1x2 Hk≤Cxp Hk with pand Cconstants depending on k≥3 (it is easy to observe that this extra cancellation cannot be used in the φequation). Next we shall show that ˜x(ξ,t) is a solution of (7). Here we consider φregular enough (φ(γ,t)−γ∈C([0,T]; Hk log ) with k≥3) so that it is a bona fide reparameterization satisfying (10). The chain rule implies (12) xt(γ,t)=˜xt(φ(γ,t),t)+φt(γ,t)∂ξ˜x(φ(γ,t),t).
UNIQUENESS FOR SQG PATCH SOLUTIONS 7 On the other hand, the equation for the evolution provides xt(γ,t)=∂γx(γ,t)−∂γx(η,t) |x(γ,t)−x(η,t)|dη +λ(x)(γ,t)∂γx(γ,t) =∂ξ˜x(φ(γ,t),t)∂γφ(γ,t)−∂γx(η,t) |x(γ,t)−x(η,t)|dη +λ(x)(γ,t)∂ξ˜x(φ(γ,t),t)∂γφ(γ,t), and therefore xt(γ,t)=∂ξ˜x(φ(γ,t),t)∂γφ(γ,t)−∂γφ(η,t) |x(γ,t)−x(η,t)|dη +λ(x)(γ,t)∂ξ˜x(φ(γ,t),t)∂γφ(γ,t) +∂ξ˜x(φ(γ,t),t)∂γφ(η,t)−∂γx(η,t) |x(γ,t)−x(η,t)|dη. (13) The fact that φis a solution of (11) together with identities (12,13) allow us to get ˜xt(φ(γ,t),t)=∂ξ˜x(φ(γ,t),t)−∂ξ˜x(φ(η,t),t) |˜x(φ(γ,t),t)−˜x(φ(η,t),t)|∂γφ(η,t)dη. Introducing the change of variable φ(η,t)=ζin the integral above and taking γ=φ−1(ξ,t)weobtain˜x(ξ,t) as a solution of (7) replacing xby ˜x,γby ξand ηby ζ. Therefore, ˜x∈C([0,T]; Hk log ) as a consequence of the Leibniz rule for derivatives of composite functions. An interesting feature in this process is the logarithm loss of derivative which affects the solutions of (7); nevertheless, we will show later how to take care of that. Once at this point one can see clearly how this reparameterization process helps to solve the following system: (14) ˜xt(ξ,t)=∂ξ˜x(ξ,t)−∂ξ˜x(ζ,t) |˜x(ξ,t)−˜x(ζ,t)|dζ +˜μ(ξ,t)∂ξ˜x(ξ,t) for any ˜μ(ξ,t) having the same regularity as ˜x(ξ,t). We just have to repeat the argument but with the equation φt(γ,t)=T ∂γφ(γ,t)−∂γφ(η,t) |x(γ,t)−x(η,t)|dη +λ(x)(γ,t)∂γφ(γ,t)−μ(γ,t), where the function μacts as a source term, and as long as φand μhave the same regularity, the argument works. We then arrive at (14) with ˜μ(ξ,t)=μ(φ−1(ξ,t),t). This shows that the systems (14) or (7) come from the system (8,9) by a change of variable. Theorem 1.1 together with Proposition 1.3 yield the existence of solutions for the system (7). Then Theorem 1.2 implies uniqueness: Theorem 1.4. Let x0(γ)∈Hk(T)for k≥3with F(x0)(γ,η)∈L∞. Then there exists a time T>0so that there exists a unique solution to (7) in C([0,T]; Hk log (T)) with x(γ,0) = x0(γ). The uniqueness part of this theorem will be discussed in section 4. Its proof will not assume property (4) and it will be done controlling the evolution of the L2 norm of the difference between any two given solutions.
8 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO An important linear operator in the study of patch solutions for SQG is given by (15) L(f)(γ)=π −π f(γ)−f(γ−η) |η|dη for f2π-periodic. Since Lis a translations invariance (where we have extended |η|−1periodically), the operator is a Fourier multiplier given by (16) L(f)(k)=O(log(2|k|)) f(k)fork∈Z{0}, L(f)(0) = 0. Uniqueness for the 2D Euler vortex patch problem was obtained in the classical Yudovich work [24]. The results presented in that paper hold in a more general setting but it is also valid for any 2D Euler weak solution with vorticity in L∞(0,T;L∞∩L1). For the α-system, weak solutions given by patches have been shown to be unique in [18]. The uniqueness result in the present paper corresponds to the more singular and physically relevant case: α= 1, but the arguments can be extended for 0 <α<1. In those cases the equations for the reparameterization are more regular than (11) and there is no logarithm derivative loss in the change of variable process. Solutions for one of the contour evolution equations were shown to be unique in [10] for 0 <α<1. 2. Uniqueness for the SQG patch problem This section is devoted to showing the proof of uniqueness of SQG weak solutions given by patches: Theorem 1.2. As a consequence of its proof, the solutions found in [10] are unique: Corollary 2.1. Consider a solution of the system (8,9) given by Theorem 1.1 with x(γ,t)∈C([0,T]; H3(T)).Thenx(γ,t)is unique as a solution of (8,9) with initial data x(γ,0). Furthermore, it provides the unique weak solution of (1,2,3) with Dj(t) a time dependent simply connected domain in C([0,T]; C2,δ(T))∩C1([0,T]; C1(T)), 0<δ≤1/2. Proof of Theorem 1.2.We consider a solution θ(x, t) satisfying the hypothesis in Theorem 1.2. Then, it is shown in [10], the parameterization of the free boundary has to fulfill equation (6) where, without loss of generality, we can assume that θ2−θ1=π. The length of the curve is l(t)=T|∂γx(γ,t)|dγ, and we shall consider the following change of variable: φ(·,t):T→T,φ(γ,t)=−π+2π l(t)γ −π|∂γx(η,t)|dη. Consequently, we get the reparameterization ˜x(ξ,t)=x(φ−1(ξ,t),t),x(γ,t)=˜x(φ(γ,t),t),ξ=φ(γ,t), satisfying property (4) (|∂ξ˜x(ξ,t)|=(2π)−1l(t)) and having the same regularity (˜x(ξ,t)∈C([0,T],C2,δ)∩C1([0,T]; C1(T))). As we pointed out before, the curve ˜x(ξ,t) is a solution of (6) with the tilde notation. We mean by this that ˜x(ξ,t)is a solution of (6) replacing xby ˜xand γby ξ.
UNIQUENESS FOR SQG PATCH SOLUTIONS 9 For this new evolving curve ˜x,theidentity ˜xt(ξ,t)=˜xt(ξ,t)·∂ξ˜x(ξ,t) |∂ξ˜x(ξ,t)| ∂ξ˜x(ξ,t) |∂ξ˜x(ξ,t)|+˜xt(ξ,t)·∂⊥ ξ˜x(ξ,t) |∂ξ˜x(ξ,t)| ∂⊥ ξ˜x(ξ,t) |∂ξ˜x(ξ,t)| together with (6) provides ˜xt(ξ,t)=˜μ(ξ,t)∂ξ˜x(ξ,t) |∂ξ˜x(ξ,t)|+T (∂ξ˜x(ξ,t)−∂ξ˜x(ξ−ζ,t)) |˜x(ξ,t)−˜x(ξ−ζ,t)|dζ ·∂⊥ ξx(ξ,t) |∂ξ˜x(ξ,t)| ∂⊥ ξ˜x(ξ,t) |∂ξ˜x(ξ,t)|, wherewehavedefined˜μ(ξ,t)=˜xt(ξ,t)·∂ξ˜x(ξ,t)/|∂ξ˜x(ξ,t)|. Taking ˜μ(ξ,t)=T (∂ξ˜x(ξ,t)−∂ξ˜x(ξ−ζ,t)) |˜x(ξ,t)−˜x(ξ−ζ,t)|dζ ·∂ξ˜x(ξ,t) |∂ξ˜x(ξ,t)|+˜ λ(˜x)(ξ,t)|∂ξ˜x(ξ,t)|, it is easy to find that ˜xsatisfies (8) with the tilde notation: ˜xt(ξ,t)=T ∂ξ˜x(ξ,t)−∂ξ˜x(ξ−ζ,t) |˜x(ξ,t)−˜x(ξ−ζ,t)|dζ +˜ λ(˜x)(ξ,t)∂ξ˜x(ξ,t), where ˜ λ(˜x)(ξ,t)=˜xt(ξ,t)−T ∂ξ˜x(ξ,t)−∂ξ˜x(ξ−ζ,t) |˜x(ξ,t)−˜x(ξ−ζ,t)|dζ·∂ξ˜x(ξ,t) |∂ξ˜x(ξ,t)|2. The regularity of ˜x(ξ,t) yields ˜ λ(˜x)(ξ,t)∈C([0,T]; C1(T)). Then we can find a function a∈C1([0,˜ T]; R) as a unique solution of the o.d.e. a(t)=˜ λ(˜x)(−π−a(t),t),a(0) = 0, where 0 <˜ Tby the Picard-Lindel¨of theorem. Since sup[0,T ]˜ λ(˜x)L∞(t)≤Cm(x), for Cm(x) depending on sup[0,T ]F(x)L∞(t)andsup [0,T ](xC2,δ (t)+xtL∞(t)), the function a(t) can be extended to [0,T] satisfying that |a(t)|≤TCm(x) for any t∈[0,T]. The new curve given by ˜x(ξ,t)=¯x(ξ+a(t),t) satisfies ¯xt(α, t)=T ∂α¯x(α, t)−∂α¯x(α−β,t) |¯x(α, t)−¯x(α−β,t)|dβ +¯ λ(¯x)(α, t)∂α¯x(α, t) for α=ξ−a(t)and¯ λ(¯x)(α, t)=˜ λ(˜x)(α−a(t),t)−˜ λ(˜x)(−π−a(t),t). Since ∂α|∂α¯x(α, t)|=0and¯ λ(¯x)(−π,t) = 0, we proceed as in [10] (see pg. 2585) to find that ¯xevolves according to equations (8,9) replacing xby ¯xand γby α.In particular it is easy to check that ¯x(α, t) has the same regularity as ˜x(ξ,t)and ˜x(ξ,0) = ¯x(ξ,0). We consider next another solution ¯ θ(x, t), satisfying the hypothesis above with the free boundary parameterized by y(γ,t)∈C([0,T]; C2,δ(T)) ∩C1([0,T]; C1(T)). As θ(x, 0) = ¯ θ(x, 0), we use a function ϕ∈C2,δ(T) to define ˘y(γ,t)=y(ϕ(γ),t)in such a way that ˘y(γ,0) = x(γ,0). Therefore, it is easy to see that ˘yhas the same regularity as yand fulfills equation (6), providing the free boundary of the same patch solution ¯ θ(x, t). Next, we reparameterize ˘y(γ,t)aswedidforx(γ,t)toget ˜y(ξ,t) satisfying ∂ξ(|∂ξ˜y(ξ,t)|)=0and˜y(ξ,0) = ˜x(ξ,0). Then we obtain ¯y(α, t) similarly as before providing us a solution of equations (8,9) after replacing xby ¯y and γby α. In particular, all this reparameterization process provides ¯y(α, t) with the same kind of regularity and satisfying ¯x(α, 0) = ¯y(α, 0). From now on, we will drop the bars for simplicity, using the variables γand ηinstead of αand β. As before we shall write f=f(γ,t), f=f(γ−η,t), f−=f−f,and=T, when there is no danger of confusion in the writing
16 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO Proof of Corollary 2.1.Let us consider x(γ,t)andy(γ,t)twosolutionsof(8,9) given by Theorem 1.1 in C([0,T]; H3) with y(γ,0) = x(γ,0) and satisfying the arc-chord condition in [0,T]. Proceeding as in the proof above we get the estimate d dtzH1≤CzH1 for z(γ,t)=x(γ,t)−y(γ,t), and then Gronwall’s inequality provides uniqueness. We are left with the task of proving that the patch weak solution given by x(γ,t) is unique. In order to obtain that result one just has to check that, for θ(x, t)given by a patch and ∂Dj(t) parameterized by x(γ,t), the regularity needed in Theorem 1.2 is achieved. The fact that H3⊂C2,δ,0<δ≤1/2, provides the appropriate regularity for x(γ,t). Next we will show that xt(γ,t)∈C([0,T]; H2 log ) and since H2 log ⊂H3 2+δ⊂C1,0<δ<1/2, the regularity for xt(γ,t) follows. At this point it is easy to check that xt(γ,t)∈C([0,T]; L2). Next we consider ∂2 γxt=G1+G2,where (21) G1=∂2 γ∂γx− |x−|dη,G 2=∂2 γλ∂γx, using the notation above. The term G1is decomposed further G1=G1,1+G1,2+G1,3+G1,4,where G1,1=∂3 γx− |x−|dη, G1,2=−2∂2 γx− |x−|3x−·∂γx−dη, G1,3=−∂γx− |x−|3x−·∂2 γx−dη, gathering in G1,4the terms in which only derivatives of order lower than 2 are involved. As before, the splitting easily yields sup [0,T ]G1,4L2≤√2πsup [0,T ]G1,4L∞≤C, and, furthermore, sup [0,T ] (G1,2L2+G1,3L2) ≤sup [0,T ] (3√2πF(x)2 L∞∂2 γxL∞∂2 γxCδ|η|1−δdη)≤C. The most singular term can be decomposed one more time as G1,1=G1,1,1+G1,1,2, where G1,1,1=1 |∂γx|L(∂3 γx),G 1,1,2=∂3 γx−1 |x−|−1 |∂γx||η|dη. It yields |G1,1,2|≤(F(x)2 L∞∂2 γxL∞))(2π|∂3 γx|+|∂3 γx|dη), and, therefore, sup [0,T ]G1,1,2L2≤Csup [0,T ]xH3.
UNIQUENESS FOR SQG PATCH SOLUTIONS 17 It remains to deal with G1,1,1, which cannot be placed in C([0,T],L 2). However, since less regularity is needed for xt(γ,t), we have sup [0,T ] n∈Z | G1,1,1(n)|2 log2(|n|+e)≤sup [0,T ]F(x)L∞ n∈Z | L(∂3 γx)(n)|2 log2(|n|+e)≤Csup [0,T ]xH3, wherewehaveused(16). HencewearedonewithG1,1,1and consequently with G1. Let us observe that we have obtained a better regularity for G2due to the fact that λ(x)∈C([0,T]; H3 log ) (see (37) and below in the next section). That is, G2∈C([0,T]; L2) and, therefore, xt∈C([0,T]; H2 log ), as desired. 3. Existence of an appropriate parameterization and commutator estimate First let us define the operators used within the proofs, namely ∂log and Ilog,a derivative and potential operator, respectively, as the following Fourier multipliers: (22) ∂logf(j)= j log(|j|+e)ˆ f(j), Ilogf(j)= 1 log(|j|+e)ˆ f(j) for f∈L2(T). Clearly we have that f∈L2(T) belongs to Hk log if ∂log∂k−1 γf∈L2or ∂k γIlogf∈L2. Next we show a commutator estimate needed in the existence and uniqueness proofs. Lemma 3.1. Let l1be the space of an absolutely convergence series. Then (23) ∂log∂γ(gf)−g∂log∂γfL2≤C( ∂γgl1∂logfL2+∂log∂γgL2 fl1), where Cis a universal constant. In particular, Sobolev’s embedding implies that for any >0there is a constant C>0such that (24) ∂log∂γ(gf)−g∂log∂γfL2≤Cε(gH3/2+∂logfL2+∂log∂γgL2fH1/2+). Proof. We have that |(∂log∂γ(gf)−g∂log∂γf)ˆ( j)|≤ lj2 log(|j|+e)−(j−l)2 log(|j−l|+e)|ˆ f(j−l)||ˆg(l)|, and the function h(j)=j2/log(|j|+e) satisfies h(j)−h(j−l)=1 0 d dr h((j−l)+rl)dr =l1 0 h(rl +(j−l))dr, and, therefore, |h(j)−h(j−l)|≤|l|1 0|h(rl +(j−l))|dr ≤3(|l|+|j−l|) log(|l|+|j−l|+e)|l|. It yields |h(j)−h(j−l)|≤ 3|l|2 log(|l|+e)+3|l||j−l| log(|j−l|+e), and finally |(∂log∂γ(gf)−g∂log∂γf)ˆ( j)| ≤ l 3|l|2 log(|l|+e)|ˆ f(j−l)||ˆg(l)|+ l 3|l||j−l| log(|j−l|+e)|ˆ f(j−l)||ˆg(l)|.
18 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO Then Parseval’s theorem gives ∂log∂γ(gf)−g∂log∂γfL2≤ j l 3|l|2 log(|l|+e)|ˆ f(j−l)||ˆg(l)|21/2 + j l 3|l||j−l| log(|j−l|+e)|ˆ f(j−l)||ˆg(l)|21/2. The Minkowski inequality provides (23). The proof ends by Sobolev’s embedding in dimension one. Proof of Proposition 1.3.Without loss of generality we may consider the case k= 3, because the extension to k>3 is just a straightforward exercise once we know how to handle k= 3. Also, in order to be concise we will show only the main part of the proof. That is, we will deal with the more dangerous terms in the needed estimates, leaving as an exercise to the reader the treatment to all the other more benevolent characters. In the main core of the proof are energy estimates; from them and with recent well-known mollifying arguments one can apply the classical Picard to conclude existence. The whole strategy can be found in [2, Chapter 3]. Often, in the following we will have to write double integrals in variables, say γand η, and differences f(γ)−f(γ−η). To simplify notation we shall write f=f(γ,t), f=f(γ−η,t), and f−f=f−when there is no danger of confusion. Furthermore, we shall write =Tand denote id as the identity, C(t) will be a polynomial function in F(x)L∞and xH3so that C(t)∈C([0,T]). As was mentioned before, most of the time we will show how to estimate the most singular terms: those in which the derivative of higher order is involved by the use of Leibnitz’s derivative rule. The rest of the terms are denoted by l.o.t. standing for lower order terms. Writing l.o.t. ∈Xmeansthatthelowerordertermsbelongto the space X. First we consider the evolution of the L2norm: 1 2 d dtφ−id2 L2=(φ−id)φtdγ =I1+I2, where I1=(φ−id)∂γφ− |x−|dηdγ, I2=λ(x)(φ−id)∂γφdγ. For I1we find I1=(φ(γ,t)−γ)∂γφ(γ,t)−∂ηφ(η,t) |x(γ,t)−x(η,t)|dηdγ =−(φ(η,t)−η)∂γφ(γ,t)−∂ηφ(η,t) |x(γ,t)−x(η,t)|dηdγ =1 2(φ(γ,t)−γ−(φ(η,t)−η))∂γφ(γ,t)−∂ηφ(η,t) |x(γ,t)−x(η,t)|dηdγ, hence I1=1 2(φ−id)− ∂γ((φ−id)−) |x−|dηdγ. Integration by parts yields I1=1 4 |(φ−id)−|2x−·∂γx− |x−|3dηdγ.
UNIQUENESS FOR SQG PATCH SOLUTIONS 19 Now we use (4) to rewrite x−·∂γx− |x−|3=x−·∂γx−−∂γx·∂2 γxη2 |x−|3, and obtain (25) x−·∂γx− |x−|3≤ 2x2 C2,1 2|η|2+ 1 2 |x−|3≤2F(x)3 L∞x2 C2,1 2|η|−1 2. This yields I1≤1 2F(x)3 L∞x2 C2,1 2|η|−1 2|(φ−id)−|2dγdη ≤F(x)3 L∞x2 C2,1 2|η|−1 2(|φ−id|2+|(φ−id)|2)dγdη ≤2F(x)3 L∞x2 C2,1 2φ−id2 L2(t)≤C(t)φ−id2 L2(t). The term I2can be rewritten as follows: I2=λ(x)(φ−id)(∂γφ−1)dγ +λ(x)(φ−id)dγ. The first term above can be handled by integration by parts. In the second the Cauchy-Schwarz inequality yields I2≤C(t)φ−id2 L2(t)+1 2λ(x)2 L2. The bounds for λ(x) (below we show that λ∈H3 log ) finally provide d dtφ−id2 L2(t)≤C(t)φ−id2 L2(t)+C(t).(26) Next, we consider the evolution of the higher order norm 1 2 d dt∂log∂2 γφ2 L2=∂log∂2 γφ∂log∂2 γφtdγ =∂log∂2 γφ∂log∂2 γ∂γφ− |x−|dηdγ +∂log∂2 γφ∂log∂2 γλ(x)∂γφdγ =J+K, (27) to bound the Jand Kterms. With Jwe split further J=J1+J2+J3,where J1=∂log∂2 γφ∂log∂3 γφ− |x−|dηdγ, J2=−2∂log∂2 γφ∂log∂2 γφ−x−·∂γx− |x−|3dηdγ, and J3=−∂log∂2 γφ∂log∂γφ−∂γx−·∂γx− |x−|3dηdγ. The fact that |∂γx|does not depend on γgives ∂log∂2 γφ∂log∂3 γφ− |∂γx||η|dηdγ =1 |∂γx|∂log∂2 γφ∂logL(∂3 γφ)dγ =0,
20 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO where Lwas defined in (15) and has properties (16). Therefore, one obtains J1=∂log∂2 γφ∂log∂3 γφ−1 |x−|−1 |∂γx||η|dηdγ. This extra cancellation suggests the further splitting J1=J1,1+J1,2,where J1,1=∂log∂2 γφ∂log∂3 γφ1 |x−|−1 |∂γx||η|dηdγ, J1,2=−∂log∂2 γφ∂log∂3 γφ1 |x−|−1 |∂γx||η|dηdγ, and J1,1=J1 1,1+J2 1,1+J3 1,1,where J1 1,1=∂log∂2 γφ[∂log∂γ(A∂2 γφ)−A∂log∂γ(∂2 γφ)]dγ, J2 1,1=∂log∂2 γφA∂log∂γ(∂2 γφ)dγ, J3 1,1=−∂log∂2 γφ∂log(∂γA∂2 γφ)dγ, with (28) A=1 |x−|−1 |∂γx||η|dη. In J1 1,1we use the commutator estimate (24) to find J1 1,1≤C∂log∂2 γφL2(AH2∂log∂2 γφL2+∂log∂γAL2∂2 γφH 1 log ) ≤CAH2∂log∂2 γφ2 L2. Furthermore, we have (29) ∂γA=−x−·∂γx−−∂γx·∂2 γxη2 |x−|3dη, and, therefore, ∂2 γA=−x−·∂2 γx−−∂γx·∂3 γxη2 |x−|3dη +l.o.t., where l.o.t.L2≤C(t). Identity (4) yields x−·∂2 γx−−∂γx·∂3 γxη2=(x−−∂γxη)·∂2 γx−+η∂γx·∂2 γx−−∂γx·∂3 γxη2 =(x−−∂γxη)·∂2 γx−−η∂γx−·∂2 γx+|∂2 γx|2η2, implying x−·∂2 γx−−∂γx·∂3 γxη2=(x−−∂γxη)·∂2 γx−−η(∂γx−−∂2 γxη)·∂2 γx+η2∂2 γx·∂2 γx−. The above configuration provides |x−·∂2 γx−−∂γx·∂3 γxη2|≤3xC2xC2,1 2|η|2+ 1 2, and, therefore, −(x−−∂γxη)·∂2 γx−−∂γx·∂3 γxη2 |x−|3dη≤3xC2xC2,1 2F(x)3 L∞, implying that ∂2 γA∈C([0,T],L 2) and the estimate J1 1,1≤C(t)∂log∂2 γφ2 L2.
UNIQUENESS FOR SQG PATCH SOLUTIONS 21 Then, integration by parts yields J2 1,1≤∂γAL∞∂log∂2 γφ2 L2≤AH2∂log∂2 γφ2 L2≤C(t)∂log∂2 γφ2 L2. In order to estimate J3 1,1we use the following inequalities: J3 1,1≤∂log∂2 γφL2∂log(∂γA∂2 γφ)L2 ≤C∂log(∂γA)L2∂log∂2 γφ2 L2≤CAH2∂log∂2 γφ2 L2 ≤C(t)∂log∂2 γφ2 L2. Hence J1,1≤C(t)∂log∂2 γφ2 L2. It remains to control J1,2.WerewriteJ1,2=J1 1,2+J2 1,2given by J1 1,2=∂log∂2 γφ∂log∂3 γφΩ1dηdγ, J2 1,2=∂log∂2 γφ∂log∂3 γφΩ2dηdγ, where |∂γx|−1|η|−1−|x−|−1=Ω 1+Ω 2with (30) Ω1=x− η−∂γx2 |∂γx||∂γx|+|x− η||x−|,Ω2=2 x− η−∂γx·∂γx |∂γx||∂γx|+|x− η||x−|. Next we will show how to deal with J2 1,2and since the kernel Ω2is more singular than Ω1, we leave to the reader the analogous details for J1 1,2. Identity (4) allows us to rewrite (31) Ω2=2 x− η−∂γx+1 2∂2 γxη·∂γx |∂γx||∂γx|+|x− η||x−| and the splitting J2 1,2=J2,1 1,2+J2,2 1,2,where (32) J2,1 1,2=∂log∂2 γφ∂log∂2 γφ∂ηΩ2dηdγ, J2,2 1,2=−∂log∂2 γφ∂log∂2 γφΩ2 η=π η=−πdγ. In the case of J2,2 1,2let us observe that the functions Ω2(γ,±π) are regular enough to obtain J2,2 1,2≤∂log∂2 γφL2∂log(∂2 γφΩ2 η=π η=−π)L2≤C∂log∂2 γφ2 L2 ±∂logΩ2|η=±πL2 ≤C∂log∂2 γφ2 L2 ±∂γΩ2|η=±πL2≤C(t)∂log∂2 γφ2 L2. Regarding J2,1 1,2, we proceed as follows: J2,1 1,2≤∂log∂2 γφL2∂log∂2 γφ∂ηΩ2dηL2 ≤∂log∂2 γφL2∂γ∂2 γφ∂ηΩ2dηL2 ≤∂log∂2 γφL2∂2 γφ∂γ∂ηΩ2dηL2+∂3 γφ∂ηΩ2dηL2, (33)
22 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO and two new terms appear that have to be controlled in L2: (34) B=∂2 γφ∂γ∂ηΩ2dη, D =∂3 γφ∂ηΩ2dη. In order to do that first we will prove the bound ∂γ∂ηΩ2L2≤C(t)toobtain (35) BL2≤∂2 γφL1∂γ∂ηΩ2L2≤C(t)∂log∂2 γφL2. With the help of formula (31) we split ∂ηΩ2=∂Ω2,1+∂Ω2,2+∂Ω2,3+∂Ω2,4,where ∂Ω2,1=−2 η2 (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η||x−|,∂Ω2,2=2 η (∂γx−∂γx+∂2 γxη)·∂γx |∂γx||∂γx|+|x− η||x−|, (36) ∂Ω2,3=−2 η (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η||x−|2 x−·∂γx |x−|, and ∂Ω2,4=−2 η (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η|2|x−| x− η·∂γxη−x− η2 x− η. Next we will show how to deal with ∂γ∂Ω2,1and since the other kernels are similar or even easier to handle we will skip the details. We have ∂γ∂Ω2,1=−2 η2 (∂γx−−∂2 γxη +1 2∂3 γxη2)·∂γx |∂γx||∂γx|+|x− η||x−|+l.o.t., where l.o.t.L2≤C(t). The identity ∂γx−−∂2 γxη +1 2∂3 γxη2=η21 0 r(∂3 γx(γ)−∂3 γx(γ+(r−1)η))dr allows us to write (∂γx−−∂2 γxη +1 2∂3 γxη2)·∂γx =η21 0 r∂3 γx(γ+(r−1)η)) ·(∂γx(γ+(r−1)η)−∂γx(γ))dr +η21 0 r(∂3 γx(γ)·∂γx(γ)−∂3 γx(γ+(r−1)η)·∂γx(γ+(r−1)η))dr. The use of equality (4) and integration by parts in ryield (∂γx−−∂2 γxη +1 2∂3 γxη2)·∂γx =η21 0 r∂3 γx(γ+(r−1)η)) ·(∂γx(γ+(r−1)η)−∂γx(γ))dr −η31 0 r2∂2 γx(γ+(r−1)η)·∂3 γx(γ+(r−1)η)dr, and, therefore, |(∂γx−−∂2 γxη +1 2∂3 γxη2)·∂γx|≤2|η|3xC21 0|∂3 γx(γ+(r−1)η)|dr.
UNIQUENESS FOR SQG PATCH SOLUTIONS 23 Hence |∂γ∂Ω2,1|≤4F(x)3 L∞xC21 0|∂3 γx(γ+(r−1)η)|dr +|l.o.t.|. Finally an integration in γgives the desired property: ∂γ∂Ω2,1L2≤C(t).Analogously we have ∂γ∂Ω2,jL2≤C(t)forj=2,3,4 and therefore the same bound holds for ∂γ∂ηΩ2: ∂γ∂ηΩ2L2≤C(t). We achieve the desired estimate (35). Regarding D, we first integrate by parts and then split D=∂η(∂2 γφ−)∂ηΩ2dη =−∂2 γφ−∂2 ηΩ2dη +∂2 γφ−∂ηΩ2 η=π η=−π=D1+D2. Then formulas (36) show that the functions ∂ηΩ2(γ,±π) are regular enough to get an appropriate bound for D2: D2L2≤C(t)∂2 γφL2≤C(t)∂log∂2 γφL2. Following the decomposition for ∂ηΩ2in (36), let us introduce ∂η∂Ω2,1=∂2Ω1 2,1+ ∂2Ω2 2,1+∂2Ω3 2,1+∂2Ω4 2,1,where ∂2Ω1 2,1=4 η3 (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η||x−|,∂ 2Ω2 2,1=−2 η2 (−∂γx−+∂2 γxη)·∂γx |∂γx||∂γx|+|x− η||x−|, ∂2Ω3 2,1=2 η2 (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η||x−|2 x−·∂γx |x−|, and ∂2Ω4 2,1=2 η2 (x−−∂γxη +1 2∂2 γxη2)·∂γx |∂γx||∂γx|+|x− η|2|x−| x− η·∂γxη−x− η2 x− η. As was shown before, we have x−−∂γxη +1 2∂2 γxη2=1 2η31 0 r2∂3 γx(γ+(r−1)η)dr and, therefore, |∂2Ω1 2,1|≤ 2 |η|F(x)2 L∞1 0|∂3 γx(γ+(r−1)η)|dr. Analogously, we obtain 4 j=2 |∂2Ωj 2,1| ≤2 |η|(F(x)2 L∞+F(x)3 L∞xC1+F(x)2 L∞xC1) 1 0|∂3 γx(γ+(r−1)η)|dr, implying |∂η∂Ω2,1|≤ 1 |η|C(t)1 0|∂3 γx(γ+(r−1)η)|dr.
24 ANTONIO C ´ ORDOBA, DIEGO C ´ ORDOBA, AND FRANCISCO GANCEDO The same approach for ∂η∂Ω2,j with j=2,3,4 yields |∂2 ηΩ2|≤ 1 |η|C(t)1 0|∂3 γx(γ+(r−1)η)|dr. Therefore, we get the estimate |D1|≤C(t)∂2 γφC1 31 |η|2/31 0|∂3 γx(γ+(r−1)η)|drdη, and consequently D1L2≤C(t)∂2 γφC1 3∂3 γxL2≤C(t)∂2 γφH11 12 ≤C(t)∂log∂2 γφL2 by Sobolev embedding. Putting all those estimates together we obtain DL2≤C(t)∂log∂2 γφL2, which together with (35) allows us to get finally the needed estimate for J2,1 1,2in (32) using (33). We are then done with J2 1,2. For the less singular kernel Ω1in (30) a similar analysis yields J1 1,2≤C(t)∂log∂2 γφ2 L2. Hence the same estimate is achieved for J1,2and accordingly for J1: J1≤C(t)∂log∂2 γφ2 L2. Next we estimate J2=J2,1+J2,2given by J2,1=−2∂log∂2 γφ∂log∂2 γφx−·∂γx− |x−|3dηdγ =2∂log∂2 γφ∂log∂2 γφ∂γAdγ, and J2,2=2∂log∂2 γφ∂log∂2 γφx−·∂γx− |x−|3dηdγ =2∂log∂2 γφ∂log∂2 γφΩ3dηdγ, where ∂γAwas introduced in (29) and the kernel Ω3can be rewritten as Ω3=x−·∂γx−−∂γx·∂2 γxη2 |x−|3. Observe that J2,1=2J3 1,1and, therefore, we already know the estimate of that term. The other J2,2is similar to J2,1 1,2because the kernel Ω3is of degree 0 as ∂ηΩ2, and has the same loss of regularity in the tangential direction. Then, as before we obtain |∂γΩ3|≤C(t)1 0|∂3 γx(γ+(r−1)η)|dr, |∂ηΩ3|≤ 1 |η|C(t)1 0|∂3 γx(γ+(r−1)η)|dr, helping to estimate J2,2,and J2≤C(t)∂log∂2 γφ2 L2.
UNIQUENESS FOR SQG PATCH SOLUTIONS 25 Finally, to deal with J3, we proceed as follows: J3≤∂log∂2 γφL2∂log∂γφ−∂γx−·∂γx− |x−|3dηL2 ≤∂log∂2 γφL2∂γ∂γφ−∂γx−·∂γx− |x−|3dηL2, that is, J3≤∂log∂2 γφL2∂2 γφ−∂γx−·∂γx− |x−|3dηL2+∂γφ−∂2 γx−·∂γx− |x−|3dηL2. Next let us observe that the two inequalities ∂2 γφ−∂γx−·∂γx− |x−|3 ≤∂2 γφC1 3 |η|2 3F(x)2 L∞1 0|∂3 γx(γ+(r−1)η)|dr+4F(x)3 L∞x2 C2, ∂γφ−∂2 γx−·∂γx− |x−|3 ≤C(t)∂2 γφL∞|∂3 γx|+|∂3 γx|+1 0|∂3 γx(γ+(r−1)η)|dr +1 , together with Sobolev embedding yield J3≤C(t)∂log∂2 γφL2(∂2 γφC1 3+∂2 γφL∞)≤C(t)∂log∂2 γφ2 L2, giving us the control: J≤C(t)∂log∂2 γφ2 L2. To finish, it remains to deal with K. First we will show the regularity of (37) λ(x)∈C([0,T]; H3 log )forx∈C([0,T]; H3). To do that we begin observing that λ(x)∈C([0,T]; L2). Next we continue showing that Ilog(∂3 γλ(x)) ∈C([0,T]; L2) with Ilog given in (22). We use the following decomposition ∂3 γλ(x)=E1+E2+E3,where E1=−∂3 γx |∂γx|2·∂γ∂γx− |x−|dη,E 2=−2∂2 γx |∂γx|2·∂2 γ∂γx− |x−|dη, and E3=−∂γx |∂γx|2·∂3 γ∂γx− |x−|dη. The inequality E1≤(F(x)3 L∞∂2 γxC1 2+F(x)4 L∞∂2 γx2 L∞)|∂3 γx| gives E1∈C([0,T]; L2). For E2we consider E2=E2,1+E2,2+E2,3,where E2,1=−2∂2 γx |∂γx|2·∂3 γx− |x−|dη,E 2,2=−2∂2 γx |∂γx|2·∂2 γx− |x−|3x−·∂γx−dη, and E2,3=−2∂2 γx |∂γx|2·∂2 γx−∂γx−·∂γx− |x−|3dη.