Existence theorems for two-fluid magnetohydrodynamics

Existence theorems for two-fluid magnetohydrodynamics Manuel Núñez Citation: Journal of Mathematical Physics 46, 083101 (2005); doi: 10.1063/1.1985009 View online: https://doi.org/10.1063/1.1985009 View Table of Contents: http://aip.scitation.org/toc/jmp/46/8 Published by the American Institute of Physics Articles you may be interested in General theory of the plasmoid instability Physics of Plasmas 23, 100702 (2016); 10.1063/1.4964481 Existence theorems for two-fluid magnetohydrodynamics Manuel Núñeza兲 Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain 共Received 5 April 2005; accepted 1 June 2005; published online 27 July 2005兲 The description of a plasma as composed by two types of fluids, formed by ions and electrons, is more complete than the classical one-fluid magnetohydrodynamics 共MHD兲model and it has proved necessary to explain the phenomena of fast magnetic reconnection. We prove a finite-time theorem of existence and uniqueness of solutions for this system for either Dirichlet or periodic boundary conditions in dimension three. It turns out that the regularity estimates for the magnetic field are finer than the MHD ones. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1985009兴 I. INTRODUCTION AND MATHEMATICAL SETTING The evolution of a collection of charged particles can often be modeled as the motion of several fluid species linked by electromagnetic forces. The most important instance is probably the two-fluid case, where positive ions and electrons form a quasineutral plasma. The equations of motion of both species, under reasonable approximations1–3 may be combined to yield ␳ 冉 ⳵ v ⳵ t+v· ⵜv 冊 = ␯ ⌬v+JÃB−ⵜp+f 共1兲 E+v⫻B= ␩ J+1 enJÃB−1 en ⵜpe+m e2n ⳵ J ⳵ t, where the variables are explained as follows: 共1兲vis the mass velocity v=Mvi+mve M+m, where viis the ions velocity, Mis its mass, veis the electrons velocity, and mis its mass. Since MⰇm,v⬇vi. 共2兲eis the electron charge. 共3兲nis the electrons number density, approximately equal to the ions one, ␳ =n共M+m兲the mass density. We will assume the fluid incompressible and will scale ␳ to 1. 共4兲 ␯ is the kinematic viscosity and ␩ is the resistivity. Both are taken as constant 共positive兲 scalars, meaning that these plasma properties are homogeneous and isotropic. 共5兲p=pi+peis the total pressure, sum of the ion and electron ones. 共6兲Eis the electric field, Bis the magnetic one, and J=⵱ÃBis the current density. This is also equal to 共vi−ve兲en. 共7兲fis an arbitrary forcing on the momentum equation, given, e.g., by gravitational forces. a兲Electronic mail: [email protected] JOURNAL OF MATHEMATICAL PHYSICS 46, 083101 共2005兲 46, 083101-10022-2488/2005/46共8兲/083101/12/$22.50 © 2005 American Institute of Physics It is apparent that the coefficient m/共e2n兲is very small. Neglecting it we are left with the Hall magnetohydrodynamics 共MHD兲system. By neglecting also the current displacement in Maxwell’s equations, an admissible procedure in low-frequency phenomena, and using Faraday’s law ⳵ B ⳵ t=− ⵱ÃE, we are left with the Hall induction equation ⳵ B ⳵ t=− ⵜ⫻共 ␩ J兲+ⵜÃ共vÃB兲−1 en ⵜÃ共JÃB兲.共2兲 The Hall current term 共1/en兲ⵜÃ共JÃB兲is small in dense plasmas, and in general at large scales its effect is less important than the one of the velocity. Omitting it and using certain vector analysis identities, we obtain the classical incompressible MHD equations ⳵ v ⳵ t= ␯ ⌬v−v· ⵱v+B· ⵱B−ⵜ 冉 p+1 2B2 冊 +f, ⳵ B ⳵ t= ␩ ⌬B+ⵜÃ共vÃB兲,共3兲 ⵜ·v=ⵜ·B=0. Its study is well known and similar in many aspects to the one of the Navier–Stokes equation.4 However, it has become apparent that in certain important physical phenomena, in particular those involving magnetic reconnection, the one-fluid description given by 共3兲is not appropriate. The extremely rapid conversion rate from magnetic to kinetic energy present in such spectacular phenomena as solar flares could not be adequately explained by any MHD model, and there was no lack of effort in this sense. There exists an enormous literature on this problem: see, e.g., Refs. 5–7, and references therein. It is generally accepted now that near the current sheets where reconnection occurs, electrons and ions cease to move together: it is the high electron velocity which prevents the throttling of MHD reconnection schemes and allows fast energy conversion. It is clear that a two-fluid description is necessary and therefore it seems natural to ask for some existence theorem for the relevant equation. Although admittedly the electron inertia term m/共e2n兲 is small, its presence is necessary to obtain a mathematically consistent model. By taking the curl of 共1兲and using Faraday’s law, we obtain the two-fluid MHD system ⳵ v ⳵ t= ␯ ⌬v−v· ⵜv+B· ⵜB−ⵜ 冉 p+1 2B2 冊 +f,共4兲 m e2n ⳵ ⵜÃJ ⳵ t+ ⳵ B ⳵ t= ␩ ⌬B+ⵜÃ共vÃB兲−1 en ⵜÃ共JÃB兲,共5兲 which will be our object of study. We will abbreviate the electron inertia term m/共e2n兲to ␣ , the Hall coefficient 1/共en兲to h. In addition to these equations, both vand Bmust be solenoidal. This follows automatically for Bfor all time if it happens for the initial condition at time zero, because the divergence of the induction equation yields 083101-2 Manuel Núñez J. Math. Phys. 46, 083101 共2005兲 ⳵ 共⵱·B兲 ⳵ t=0. As for the momentum one, the task of keeping vsolenoidal is given to the pressure: psatisfies an elliptic equation to this end. Indeed, physically the pressure compensates for the tensions created by the fluid incompressibility. Finally, boundary conditions must be added. Although other possibilities exist, we will concentrate for simplicity on two cases: 共1兲The domain ⍀is smooth, and both vand Bsatisfy homogeneous Dirichlet conditions at its boundary ⳵ ⍀. 共2兲⍀is a box 共0,L1兲⫻共0,L2兲⫻共0,L3兲, and both vand Bare periodic at opposite faces. Moreover, in this case the mean of both in ⍀is zero. Incidently, notice that, in contrast to the MHD and Navier–Stokes cases, there is no possibility here of studying two-dimensional problems, because even when both velocity and magnetic field are horizontal, the current density and the Hall term are not. We first recall the definitions of the classical Lebesgue and Sobolev spaces: Lp共⍀兲is the space of measurable functions fdefined in ⍀whose Lp-norm 储f储p= 冉 冕 ⍀ 兩f兩pdV 冊 1/p is bounded. L⬁共⍀兲is the space of measurable functions defined in ⍀such that they are bounded outside a set of zero measure E, and 储f储⬁is the maximum of 兩f共x兲兩 when x苸E.Hp共⍀兲is the space of functions whose differentials up to the order pare square integrable 关i.e., they belong to L2共⍀兲兴. The norm 储f储Hp共⍀兲is the sum of the L2-norms of all the partial derivatives of fup to the order p. We will only use H1共⍀兲and H2共⍀兲.H0 1共⍀兲is the subspace of H1共⍀兲formed 共roughly兲by the functions vanishing in the boundary ⳵ ⍀.共The correct definition is somewhat more involved, but essentially amounts to this兲. We will use a number of standard notations:4,8 In the Dirichlet case, H=兵B苸L2共⍀兲3:⵱·B=0,B·n兩 ⳵ ⍀=0其.共6兲 Although this divergence and trace term must be understood in the sense of distributions, it is known that His a closed subspace of L2共⍀兲3. V=H0 1共⍀兲3艚H, D共A兲=H2共⍀兲3艚V,共7兲 A:D共A兲→H,A=PH⌬, where PHis the orthogonal projection on H. In the periodic case, H= 再 u苸L2共⍀兲3: 冕 ⍀ udV=0,⵱·u=0, u·nantiperiodic at opposite sides of ⍀ 冎 . 共8兲 Again His a closed subspace of L2共⍀兲3 V=H1共⍀兲3艚H, D共A兲=H2共⍀兲3艚V,共9兲 083101-3 Two-fluid MHD J. Math. Phys. 46, 083101 共2005兲 A:D共A兲→H,A=⌬. In connection with these spaces, we will use the following inequality 共see Ref. 8兲: For all f 苸V,g苸D共A兲,h苸H, 冏 冕 ⍀ f· ⵱g·hdV冏艋c储f储H1储g储H1 1/2储Ag储2 1/2储h储2.共10兲 cis a constant depending only on ⍀. To avoid an unmanageable proliferation of constants, we will always denote such parameters by c: thus, e.g., 3c+c1/2=c. Constants not depending on ⍀, such as the diffusivity terms or bounds depending on specific functions will be given different names. We intend to prove local existence and uniqueness of the solution to the problem 共4兲and 共5兲 with initial condition v共0兲苸V,B共0兲苸D共A兲. We will consider first the induction equation, which in the MHD case models the so-called kinematic dynamo, where the velocity is taken as a datum. The momentum equation in a classical Navier–Stokes one with a forcing given by the Lorentz force, for which well-known estimates are available. Combining the bounds for both equations we will obtain a contractive operator whose fixed point is the unique solution. Finally we will obtain some energy inequalities emphasizing the difference with the single-fluid MHD model. All these results will hold even in infinitely conducting plasmas 共 ␩ =0兲, in contrast with classical MHD. II. INDUCTION EQUATION Definition 2.1: For B苸D共A兲we will denote by wthe function w=− ␣ ⌬B+B.共11兲 Notice that the properties of elliptic systems guarantee that w苸Hand 储B储H2艋c储w储2. Lemma 2.2:Consider the term ␩ ⌬B+⵱Ã共vÃB兲−hⵜÃ共JÃB兲as a (nonlocal) function F共w,v兲.Then, for every v苸V,F is a continuous function from H into H.Moreover, for any v 苸V,w苸H, 储F共w,v兲储2艋c共 ␩ +储v储H1+储w储2兲储w储2,共12兲 and for any v1,v2苸V,w1,w2苸H, 储F共w1,v1兲−F共w2,v2兲储2 艋c共 ␩ +储v1储H1+储v2储H1+储w1储2+储w2储2兲储w1−w2储2+c共储w1储2+储w2储兲储v1−v2储H1.共13兲 Proof: Recall that ⵜÃ共vÃB兲=−v· ⵱B+B· ⵱v, ⵜÃ共JÃB兲=−J· ⵱B+B· ⵱J. By standard Sobolev inequalities: ␩ 储⌬B1−⌬B2储2艋c ␩ 储B1−B2储H2艋c ␩ 储w1−w2储2, and 储ⵜÃ共v1ÃB1兲−ⵜÃ共v2ÃB2兲储2 =储−v1·⵱共B1−B2兲−共v1−v2兲·⵱B2+B1·⵱共v1−v2兲+共B1−B2兲·⵱v2储2 艋储v1储4储⵱共B1−B2兲储4+储v1−v2储4储⵱B2储4+储B1储⬁储⵱共v1−v2兲储2+储B1−B2储⬁储⵱v2储2. Since for dimension 3, V傺L4共⍀兲,D共A兲傺L⬁共⍀兲, this amount may be bounded by 083101-4 Manuel Núñez J. Math. Phys. 46, 083101 共2005兲 c共储v1储H1储B1−B2储H2+储B2储H2储v1−v2储H1+储B1储H2储v1−v2储H1+储v2储H1储B1−B2储H2 艋c共储v1储H1+储v2储H1兲储w1−w2储2+c共储w1储2+储w储2兲储v1−v2储H1. Finally, by changing in the previous inequalities vito Jiwe find 储ⵜÃ共J1ÃB1兲−ⵜÃ共J2ÃB2兲储2 艋储J1储4储ⵜ共B1−B2兲储4+储J1−J2储4储⵱B2储4+储B1储⬁储ⵜ共J1−J2兲储2+储B1−B2储⬁储⵱J2储2 艋c共储J1储H1+储B2储H2+储B1储H2+储J2储H1兲储B1−B2储H2艋c共储w1储2+储w2储2兲储w1−w2储2. The inequality 共12兲is a simplified version of the previous calculation, with only a single vand w. This concludes the proof. 䊏 Theorem 2.3:For fixed v苸L2共共0,S兲,V兲,Eq. (5), which may be written in the form dw dt=F共w,v兲, 共14兲 w共0兲苸H, admits a unique a.e. differentiable solution w:共0,T兲→H,for some T艋S,T⬎0. We may take T so that this solution satisfies 储w共t兲储2艋储w共0兲储2+1 for all t苸共0,T兲.Moreover, the interval 共0,T兲 depends only on 储w共0兲储2and 储v储L2共共0,T兲,V兲. Proof: Let us prove that the integral operator, defined in the closed ball of C共共0,T兲,H兲of center 0and radius 储w共0兲储2+1 given by w→w共0兲+ 冕 0 t F共w共s兲,v共s兲兲ds, takes the ball to itself and is contractive for Tsmall enough. Using 共12兲, for any wbounded by 储w共0兲储2+1, 冕 0 t 储F共w,v兲储2ds艋c 冕 0 t 共 ␩ +储v储H1+储w储2兲储w储2ds 艋c 冉 冑T 冉 冕 0 T 储v储H1 2ds 冊 1/2 +T共 ␩ +储w共0兲储2+1兲 冊 共储w共0兲储2+1兲. Clearly we can take Tso that this amount is less than 1 for all t苸共0,T兲. Taking now 共13兲, 冕 0 t 储F共w1,v兲−F共w2,v兲储2ds艋c 冕 0 t 共 ␩ +2储v储H1+储w1储2+储w2储2兲储w1−w2储2ds, so that 储w1−w2储C共0,T兲,H艋c 冉 冕 0 T 2储v储H1+储w1储2+储w2储2ds 冊 储w1−w2储C共0,T兲,H 艋c 冉 2冑T 冉 冕 0 T 储v储H1 2ds 冊 1/2 +2T共储w共0兲储2+1兲 冊 储w1−w2储C共0,T兲,H. Obviously by taking Tsmall enough we can make this functional to take the ball of radius 储w共0兲储2+1 in itself in a contractive manner. The classical proof follows. 䊏 Let us study now the dependence of the solutions upon the velocity: Theorem 2.4:Let w1,w2be the respective solutions of 083101-5 Two-fluid MHD J. Math. Phys. 46, 083101 共2005兲 dwi dt=F共wi,vi兲 共15兲 wi共0兲=w共0兲, for vi苸L2共共0,S兲,V兲,and let 共0,T兲be a common interval of existence of w1,w2.Then there exists T1艋T,depending only on 储w共0兲储2,储vi储L2共共0,T兲,V兲,such that 储w1−w2储C共共0,T1兲,H兲艋1 2储v1−v2储L2共共0,T1兲,V兲.共16兲 Proof: Using inequality 共13兲, and taking an interval short enough for Therom 2.3 to hold for both wi, 储w1共t兲−w2共t兲储2= 冐 冕 0 t F共w1,v1兲−F共w2,v2兲ds 冐 2 艋c 冕 0 t 共 ␩ +储v1储H1+储v2储H1+储w1储2+储w2储2兲储w1 −w2储2+共储w1储2+储w2储2兲储v1−v2储H1ds艋c关冑T共储v1储L2共共0,T兲,V兲+储v2储L2共共0,T兲,V兲兲+T ␩ +2T共储w共0兲储2+1兲兴储w1−w2储C共共0,T兲,H兲+2 冑T共储w共0兲储2+1兲储v1−v2储L2共共0,T兲,V兲. Hence it is enough to take T1small enough for c关冑T1共储v1储L2共共0,T1兲,V兲+储v2储L2共共0,T1兲,V兲兲+T1 ␩ +2T1共储w共0兲储2+1兲兴 ⬍1 2,2冑T1共储w共0兲储2+1兲⬍1 4. III. MOMENTUM EQUATION This is the Navier–Stokes one with two forcings: the independent one fand the Lorentz force JÃB. We will denote the last one by G共w兲. Notice that 储JÃB储2艋储J储4储B储4艋c储w储2 2.共17兲 We may apply the classical results 共see, e.g., Ref. 8兲and conclude the following. 共a兲For every initial condition v共0兲苸V, there exists a unique solution for an interval 共0,T兲 satisfying 储v共t兲储H1艋2共储v共0兲储H1+1兲,共18兲 where Tdepends only on 储v共0兲储H1,储G共w兲储C共共0,T兲,H兲, sup共0,T兲储f储2. 共b兲v苸L2共共0,T兲,D共A兲兲, and the norm of vin this space is bounded by a constant depending only on the same quantities given in 共a兲. We will first estimate the norm of vin the space L2共共0,T兲,V兲. To emphasize its value, it is convenient to single out 共only in this instance兲the Poincaré constant in V,储v储H1 2艋kp储ⵜv储2 2. Proposition 3.1.: If vis the solution of 共4兲in the interval 共0,T兲, 储v储L2共共0,T兲,V兲 2艋kp ␯ 储v共0兲储2 2+cT ␯ 2储w储C共共0,T兲,H兲 4+cT ␯ 2sup 共0,T兲储f储2 2.共19兲 Proof: By making the scalar product of 共4兲with v, and using standard inequalities 共Cauchy– Schwarz, Poincaré, and Young’s兲, 083101-6 Manuel Núñez J. Math. Phys. 46, 083101 共2005兲 1 2 d储v储2 2 dt 艋− ␯ 储ⵜv储2 2+储G共w兲储2储v储2+储f储2储v储2艋− ␯ 储ⵜv储2 2+c储G共w兲储2储ⵜv储2+c储f储2储ⵜv储2 艋− ␯ 储ⵜv储2 2+c2 ␯ 储G共w兲储2 2+ ␯ 4储ⵜv储2 2+c2 ␯ 储f储2 2+ ␯ 4储ⵜv储2 2. Integrating in 共0,T兲and omitting the term in 储v共t兲储2, ␯ 2 冕 0 T 储ⵜv储2 2ds艋1 2储v共0兲储2 2+c2 ␯ 冕 0 T 储G共w兲储2 2ds+c2 ␯ 冕 0 T 储f储2 2ds, from which, in view of 共17兲, the result follows. 䊏 Corollary 3.2.: There exists a time T1艋T, depending only on 储w储C共共0,T1兲,H兲, such that 储v储L2共共0,T1兲,V兲艋kp 冑 ␯ 储v共0兲储2+1艋kp 冑 ␯ 储v共0兲储H1+1. 共20兲 We will consider now the difference between two solutions corresponding to different wiand the same initial condition: Theorem 3.2:Let v1,v2be the respective solutions of Eq. (4) in 共0,T兲corresponding to Lorentz forces w1,w2苸C共共0,T兲,H兲and the same initial condition. Then there exists T2艋T,depending only on 储wi储C共共0,T兲,H兲,such that 储v1−v2储L2共共0,T2兲,V兲艋1 2储w1−w2储C共共0,T2兲,H兲.共21兲 Proof: The scalar product of 共 ⳵ 共v1−v2兲兲/共 ⳵ t兲with v1−v2yields 1 2 ⳵ ⳵ t储v1−v2储2 2+ ␯ 储ⵜ共v1−v2兲储2 2=共−v1·⵱v1+v2·⵱v2,v1−v2兲+共G共w1兲−G共w2兲,v1−v2兲 −共ⵜ共p1−p2兲,v1−v2兲. The last term is zero. We have 共−v1·ⵜv1+v2·⵱v2,v1−v2兲=共−v1·ⵜ共v1−v2兲+共−v1+v2兲·⵱v2,v1−v2兲, and 共−v1·ⵜ共v1−v2兲,v1−v2兲=0. By using inequality 共10兲,wefind 兩共−v1·ⵜv1+v2·ⵜv2,v1−v2兲兩 艋储v1−v2储H1储v1−v2储2储v2储H1 1/2储Av2储2 1/2. Finally 兩共G共w1兲−G共w2兲,v1−v2兲兩 艋储G共w1兲−G共w2兲储2储v1−v2储2. In order to abbreviate the notation, let us denote by 储g储2,⬁the supremum of 储g储2when t苸共0,T兲. Property 共a兲of the solutions of the Navier–Stokes equations guarantees that 储v2储H1is bounded in 共0,T兲by an amount depending only on 储v2共0兲储H1,储G共w2兲储2,⬁, and 储f储2,⬁. Let Mbe an upper bound of 储v2储H1 1/2 in 共0,T兲. Integrating the resulting inequality in time, 083101-7 Two-fluid MHD J. Math. Phys. 46, 083101 共2005兲 1 2储共v1−v2兲共t兲储2 2+ ␯ 冕 0 t 储ⵜ共v1−v2兲储2 2ds 艋cM 冕 0 t 储v1−v2储2储v1−v2储H1储v2储H2 1/2ds+c 冕 0 t 储G共w1兲−G共w2兲储2储v1−v2储2ds 艋cM储v1−v2储2,⬁ 冉 冕 0 T 储v1−v2储H1 2ds 冊 1/2 冉 冕 0 T 储v2储H2ds 冊 1/2 +cT储G共w1兲−G共w2兲储2,⬁储v1−v2储2,⬁.共22兲 Since this is valid for any t苸共0,T兲, by considering only the first term in the left-hand side of 共22兲 and using Young’s inequality, 1 2储v1−v2储2,⬁ 2艋c2M2 2 冉 冕 0 T 储v2储H2ds 冊 储v1−v2储2,⬁ 2+1 2 冕 0 T 储v1−v2储H1 2ds+T 2储G共w1兲−G共w2兲储2,⬁ 2 +c2 2T储v1−v2储2,⬁ 2. Since by property 共b兲 冕 0 T 储v2储H2ds艋冑T 冉 冕 0 T 储v2储H2 2ds 冊 1/2 艋N冑T, where Nis a constant depending on the same functions stated before for 储v2储H1,wefind 共1−c2M2N冑T−c2T兲储v1−v2储2,⬁ 2艋 冕 0 T 储v1−v2储H1 2ds+T储G共w1兲−G共w2兲储2,⬁ 2. Taking Tsmall enough for the constant before 储v1−v2储2,⬁ 2to be at least 1/2, we find 储v1−v2储2,⬁ 2艋2 冕 0 T 储v1−v2储H1 2ds+2T储G共w1兲−G共w2兲储2,⬁ 2, which implies 储v1−v2储2,⬁艋冑2 冉 冕 0 T 储v1−v2储H1 2ds 冊 1/2 +冑2T储G共w1兲−G共w2兲储2,⬁.共23兲 Let us now return to 共22兲omitting now the term 储共v1−v2兲共t兲储2,⬁ 2, and bounding the last term in a slightly different form: 冕 0 T 储G共w1兲−G共w2兲储2储v1−v2储ds艋储G共w1兲−G共w2兲储2,⬁ 冕 0 T 储v1−v2储2ds 艋储G共w1兲−G共w2兲储2,⬁ 冕 0 T 储v1−v2储H1ds. By using again Poincaré’s inequality on v1−v2,weget 083101-8 Manuel Núñez J. Math. Phys. 46, 083101 共2005兲