The 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University

Transcription

1 The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, / 112

2 Outline I 1 Introduction 2 Ideal MHD equations 3 Fully dissipative MHD equations 4 Dissipation only Small global solution and decay rates Small solutions for a system with damping 5 Magnetic diffusion only MHD equations with ( ) β b with β > 1 6 Vertical dissipation and horizontal magnetic diffusion 2 / 112

3 Outline II 7 Horizontal dissipation and vertical magnetic diffusion 8 Horizontal dissipation and horizontal magnetic diffusion 9 2D MHD with fractional dissipation A summary of current results 10 The 2D Compressible MHD with velocity dissipation 3 / 112

4 Introduction Introduction The standard 2D incompressible MHD equations can be written as u t + u u = p + ν u + b b, b t + u b = η b + b u, (1) u = 0, b = 0, where u denotes the velocity field, b the magnetic field, p the pressure, ν 0 the viscosity and η 0 the magnetic diffusivity (resistivity). 4 / 112

5 Introduction The MHD equations model electrically conducting fluid in the presence of a magnetic field. The first equation is the Navier-Stokes equation with the Lorentz force generated by the magnetic field and the second equation is the induction equation for the magnetic field. The MHD equations model many phenomena in physics, especially, in geophysics and astrophysics. The MHD equations have been studied analytically, numerically and experimentally. Many books and papers have been written on them. 5 / 112

6 Introduction Mathematically the 2D MHD equations may serve as a lower-dimensional model of the 3D hydrodynamics equations. They are naturally the next level of equations to study after the 2D Boussinesq equations. Due to the strong nonlinear coupling, it can be extremely challenging to deal with some of mathematical issues even in the 2D case. 6 / 112

7 Introduction We will focus on the 2D MHD equations with partial or fractional dissipation. For this purpose, we write the MHD equations in more general forms. The first one is the anisotropic MHD equations, u t + u u = p + ν 1 u xx + ν 2 u yy + b b, b t + u b = η 1 b xx + η 2 b yy + b u, u = 0, b = 0, (2) where ν 1 0, ν 2 0, η 1 0 and η 2 0. (2) will be called anisotropic MHD equations. When ν 1 = ν 2 and η 1 = η 2, (2) becomes the standard MHD equations. 7 / 112

8 Introduction Another generalized form is the MHD equations with fractional dissipation, replacing the Laplacian by fractional Laplacians, namely u t + u u = p ν( ) α u + b b, b t + u b = η( ) β b + b u, u = 0, b = 0, (3) where 0 < α, β 1, and the fractional Laplacian can be defined by the Fourier transform (or through Riesz potential), ( ) α f (ξ) = ξ 2α f (ξ). This system will be called fractional MHD equations. 8 / 112

9 Introduction We consider the initial-value problems of the MHD equations with the initial data u(x, 0) = u 0 (x), b(x, 0) = b 0 (x). What we care about is the global regularity issue: Do these IVPs have a global solution for sufficiently smooth data (u 0, b 0 )? The global regularity problem on the 2D MHD equations has attracted considerable attention recently from the PDE community and progress has been made. 9 / 112

10 Introduction Consider the following seven cases: ν 1 = ν 2 = η 1 = η 2 = 0, ideal MHD ν 1 > 0, ν 2 > 0, η 1 > 0 and η 2 > 0, MHD with dissipation and magnetic diffusion ν 1 = ν 2 > 0, η 1 = η 2 = 0. dissipation but no magnetic diffusion η 1 > 0, η 2 > 0, ν 1 = ν 2 = 0. magnetic diffusion but no dissipation 10 / 112

11 Introduction ν 1 > 0 and η 2 > 0 horizontal dissipation and vertical magnetic diffusion ν 2 > 0 and η 1 > 0 vertical dissipation and horizontal magnetic diffusion ν 1 > 0 and η 1 > 0 horizontal dissipation and horizontal magnetic diffusion 11 / 112

12 Introduction One general idea for proving the global (in time) existence and uniqueness. This is divided into two steps: 1) Local existence and uniqueness. This is in general done by the contraction mapping principle for f (t) = G(f (t)) f (0) + t 0 N(f (τ)) dτ. This usually requires that the time interval is small. 2) Global bounds and the extension theorem. The nonlinear term is treated as bad part and the dissipation as good part. 12 / 112

13 Ideal MHD equations Ideal MHD equations Ideal MHD equations: u t + u u = p + b b, b t + u b = b u, u = 0, b = 0, u(x, y, 0) = u 0 (x, y), b(x, y, 0) = b 0 (x, y). (4) The global regularity problem remains open, although we do have local well-posedness and regularity criteria. 13 / 112

14 Ideal MHD equations Theorem Given (u 0, b 0 ) H s (R 2 ) with s > 2. Then there exists a unique local classical solution (u, b) C([0, T 0 ); H s ) for some T 0 > 0. In addition, if T 0 ( ω + j ) dt < for T > T 0, then the solution remains in H s for any t T. The L -norm can replaced by BMO or B, / 112

15 Ideal MHD equations Why is the global regularity problem hard? The global L 2 -bound for (u, b) follows directly from the MHD equations u(t) 2 L 2 + b(t) 2 L 2 = u 0 2 L 2 + b 0 2 L 2. But global bounds for any Sobolev-norm appear to be impossible, for example, the H 1 -norm. Consider the equations of ω = u and j = b, { ωt + u ω = b j, j t + u j = b ω + 2 x b 1 ( y u 1 + x u 2 ) 2 x u 1 ( y b 1 + x b 2 ) 15 / 112

16 Ideal MHD equations Clearly, 1 d ( ω 2 2 dt L 2 + j 2 ) L = 2 2 j x b 1 y u 1 + Since there is no dissipation, we need T 0 ω dt < or T 0 j dt < in order for this differential inequality to be closed. To bound Sobolev norms with more than one derivative, we need both. 16 / 112

17 Ideal MHD equations In fact, for Y = u(t) 2 H s + b 2 H s, d dt Y (t) C ( u L + b L ) Y (t). Then one uses the logarithmic inequality, u L C (1+ ω L 2+ ω L log + u W s+1,p), p (1, ), s > d/p Then d dt Y (t) C ( ω L + j L ) Y (t) log+ Y (t). To get a stronger regularity criterion, one uses BMO or B 0,, f L C(1 + f BMO log + f W s,p), p (1, ), s > d/p. 17 / 112

18 Fully dissipative MHD equations Dissipative MHD equations Fully dissipative MHD equations: u t + u u = p + ν u + b b, b t + u b = η b + b u, u = 0, b = 0. The global regularity can be easily established. Theorem Let (u 0, b 0 ) H 1 (R 2 ). Then there exists a unique global strong solution (u, b) satisfying, for any T > 0, u, b L ([0, T ; H 1 (R 2 )) L 2 ([0, T ]; H 2 (R 2 )) 18 / 112

19 Fully dissipative MHD equations The global regularity problem for the ideal MHD equations is extremely difficult while the problem for the fully dissipative MHD equations is very easy. Naturally we would like to consider the cases with intermediate dissipation. 19 / 112

20 Dissipation only Dissipation only The 2D MHD equations with no magnetic diffusion: u t + u u = p + ν u + b b, b t + u b = b u, u = 0, b = 0, where ν > 0. The global regularity problem remains open. The dissipation is NOT enough for global bounds in Sobolev spaces. Again we have the global L 2 -bound u(t) 2 L 2 + b(t) 2 L 2 + t 0 u(τ) 2 L 2 dτ = u 0 2 L 2 + b 0 2 L / 112

21 Dissipation only To consider the H 1 -norm, we use the equations for ω = u and j = b, { ωt + u ω = ν ω + b j, j t + u j = b ω + 2 x b 1 ( y u 1 + x u 2 ) 2 x u 1 ( y b 1 + x b 2 ). 1 d 2 dt ( ω 2 L 2 + j 2 ) L + ν ω 2 2 L 2 = 2 To close the inequality, we need j x b 1 y u 1 + T 0 u L dt < or T 0 ω dt <. 21 / 112

22 Dissipation only Even the global existence of weak solutions is unknown. The standard idea does not appear to work: 1) Mollify the equations and the data to obtain a global smooth solution (u N, b N ); 2) Obtain uniform bounds, for any fixed T > 0, s > 2, u N L (0, T ; L 2 ) L 2 (0, T ; H 1 ), t u N L (0, T ; H s ), b N L (0, T ; L 2 ), t b N L (0, T ; H s ); 22 / 112

23 Dissipation only 3) Apply the local version of the Aubin-Lions Lemma u N u in L 2 (0, T ; L 2 loc ), b N b in L 2 (0, T ; H δ loc ) (δ > 2) The trouble is that this does not allow us to pass to the limit in b N b N b b in the distributional sense. 23 / 112

24 Dissipation only Global solutions near equilibrium Some very recent efforts are devoted to global solutions near an equilibrium. Progress has been made: F. Lin, L. Xu, and P. Zhang, Global small solutions to 2-D incompressible MHD system, arxiv: v2 [math.ap] 4 Jun X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Functional Anal. 267 (2014), / 112

25 Dissipation only J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, arxiv: v1 [math.ap] 24 Nov T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, arxiv: v1 [math.ap] 23 Apr X. Hu and F. Lin, Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity, arxiv: v1 [math.ap] 1 May / 112

26 Dissipation only Why near an equilibrium? Mathematically, the lack of magnetic diffusion makes it extremely difficult to obtain global solutions, even small global solutions. Rewriting the equations near equilibrium generates favorable terms. Since b = 0, write b = φ = ( y, x )φ and u t + u u = p + ν u + φ φ, φ t + u φ = 0, u = 0. Clearly, (u, φ) = (0, y) is a steady solution. 26 / 112

27 Dissipation only Setting φ = y + ψ yields t ψ + u ψ + u 2 = 0, t u 1 + u u 1 ν u ψ = 1 p ( 1 ψ ψ), t u 2 + u u 2 ν u ψ = 2p ( 2 ψ ψ). (5) The aim is to look for global small solutions of this system. 27 / 112

28 Dissipation only The work of Lin, Xu and Zhang reformulated the system in Lagrangian coordinates. More precisely, they define Y (x, y, t) = X (x, y, t) (x, y), where X = X (x, y, t) be the particle trajectory determined by u. Y satisfies Y tt Y t 2 x Y = f (Y, q(y)), q = p + ψ 2. They then estimate the Lagrangian velocity Y t in L 1 t Lip x, using anisotropic Littlewood-Paley theory and anisotropic Besov space techniques. 28 / 112

29 Dissipation only Due to their use of the Lagrangian coordinates, they need to impose a compatibility condition on the initial data ψ 0, more precisely, y ψ 0 and (1 + y ψ 0, x ψ 0 ) are admissible on 0 R and supp y ψ 0 (, y) [ K, K] for some K. y ψ 0 and (1 + y ψ 0, x ψ 0 ) are admissible on 0 R if y ψ 0 (X (a, t))dt = 0 for all a 0 R. R where X is the particle trajectory defined by (1 + y ψ 0, x ψ 0 ). 29 / 112

30 Dissipation only Theorem Given u 0 and ψ 0 satisfying (u 0, ψ 0 ) H s Ḣ s 2 with s 1 > 1, s 2 ( 1, 1 2 ) and s > s 1 + 2, and ψ 0 H s , ( ψ 0, u 0 ) Ḣs 1 +1 Ḣs 2 + y ψ 0 H s 1 +2 ɛ 0 for some ɛ 0 small. Assume that y ψ 0 and (1 + y ψ 0, x ψ 0 ) are admissible on 0 R and y ψ 0 (, y) [ K, K] for some K. Then the 2D MHD equations with no magnetic diffusion has a unique global solution (ψ, u, p). 30 / 112

31 Dissipation only Small global solution and decay rates Work of X. Ren, J. Wu, Z. Xiang and Z. Zhang X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Functional Anal. 267 (2014), The aim here is twofold: 1) to do direct energy estimates without Lagrangian coordinates and remove the compatibility assumption; 2) to confirm the numerical observation that the energy of the MHD equations is dissipated at a rate as that for the linearized equations. 31 / 112

32 Dissipation only Small global solution and decay rates Definition Let σ, s R. The anisotropic Sobolev space H σ,s (R 2 ) is defined by H σ,s (R 2 ) = { } f S (R 2 ) : f Hσ,s < +, where f H = σ,s { 2 js 2 σk j h k f L }j,k 2. l 2 or [ f H = σ,s R 2 ] 1 ξ 2s ξ1 2σ f (ξ) 2 2 dξ. 32 / 112

33 Dissipation only Small global solution and decay rates Theorem Assume ( ψ 0, u 0 ) H 8 (R 2 ). Let s (0, 1 2 ). There exists a small positive constant ε such that, if, ( ψ 0, u 0 ) H s, s H s,8 (R 2 ), and ( ψ 0, u 0 ) H 8 + ( ψ 0, u 0 ) H s, s + ( ψ 0, u 0 ) H s,8 ε, then (5) has a unique global solution (ψ, u) satisfying ( ψ, u) C([0, + ); H 8 (R 2 )). 33 / 112

34 Dissipation only Small global solution and decay rates Theorem Moreover, the solution decays at the same rate as that for the linearized solutions, k x ψ L 2 + k x u L 2 Cε(1 + t) s+k 2, for any t [0, + ) and k = 0, 1, / 112

35 Dissipation only Small global solution and decay rates Ideas in the proof: First, we consider the linearized equation Proposition Consider the linearized equation t u 1 u 1 x1 x 2 ψ = 0, t u 2 u 2 + x1 x 1 ψ = 0, t ψ + u 2 = 0, u(x, 0) = u 0 (x), ψ(x, 0) = ψ 0 (x). Assume (u 0, ψ 0 ) H 4 and D 1 s u 0 H 1+s and D 1 s ψ 0 H 1+s for s > 0, then, for k = 0, 1, 2, k x 1 u L 2 + k x 1 ψ L 2 C(1 + t) k+s / 112

36 Dissipation only Small global solution and decay rates Proof. For ε 1 > 0, define D 0 (t) = u 2 L 2 + u 2 L 2 + ψ 2 L ψ 2 L 2 + 2ε 1 u 2, ψ, H 0 (t) = u 2 L u 2 L 2 + ε 1 1 ψ 2 L 2 ε 1 u 2 2 L 2 ε 1 u 2, ψ. E s (t) = D 1 s u 2 L 2 + D 1 s ψ 2 L 2 + D 1+s D 1 s u 2 L 2 + D 1+s D 1 s ψ 2 L 2. We can show d dt D 0(t) + C H 0 (t) 0, d dt E s(t) / 112

37 Dissipation only Small global solution and decay rates By interpolation inequalities, D 0 (t) E s (t) 1 1+s H0 (t) s 1+s, H0 (t) E s (0) 1 s D0 (t) 1+ 1 s. Thus, d dt D 0(t) + C E s (0) 1 s D0 (t) 1+ 1 s 0. E(t) (E(0) 1 s + C(s)t) s = E 0 (E 1 s 0 C(s)t + 1 ) s. 37 / 112

38 Dissipation only Small global solution and decay rates We return to the full nonlinear system t ψ + u ψ + u 2 = 0, t u 1 + u u 1 ν u ψ = 1 p ( 1 ψ ψ), t u 2 + u u 2 ν u ψ = 2p ( 2 ψ ψ). (6) The frame work to prove the global existence of small solutions is the Bootstrap Principle. 38 / 112

39 Dissipation only Small global solution and decay rates T. Tao, Local and global analysis of dispersive and wave equations, p.21. Lemma (Abstract Bootstrap Principle) Let I be an interval. Let C(t)and H(t) be two statements related to t I. If C(t) and H(t)satisfy (a) If H(t) is true, then C(t) is true for the same t, (b) If C(t 1 ) is true, then H(t) is true for t in a neighborhood of t 1, (c) If C(t k ) is true for a sequence t k t, then C(t)is true, (d) C(t) is true for at least one t 0 I, then, C(t) is true for all t I. 39 / 112

40 Dissipation only Small global solution and decay rates What we do here is to: 1) obtain decay rates under the assumption that the solution is small; 2) show that the solution is even smaller if the initial data is small. Then the Bootstrap principle would imply that the solution remain small for all time. 40 / 112

41 Dissipation only Small global solution and decay rates We use anisotropic Sobolev and Besov spaces due to the anisotropicity, u tt u t x 2 u = 0 The characteristic equation satisfies λ 2 + ξ 2 λ + ξ1 2 = 0, which has two roots λ ± = ξ 2 ± ξ 4 4ξ As ξ, { λ (ξ) ξ2 1 1, ξ ξ 2 ξ1, 0, ξ ξ 1 The dissipation is weak in the case of ξ ξ / 112

42 Dissipation only Small global solution and decay rates Proposition If (u, ψ)satisfies (u(t), ψ(t)) H 4 δ for some δ > 0 and for t [0, T ], then we can show d dt D 0 + CH 0 0 for t [0, T ]. 42 / 112

43 Dissipation only Small global solution and decay rates Define For l = 1, 2, we define D l (t) = j,k 2 2lk ( j h k u 2 L 2 + j h k u 2 L 2 + j h k ψ 2 L 2 + j h k 2 ψ 2 L 2 + 2ε 1 j h k u 2, j h k ψ ), H l (t) = j,k 2 2lk ( j h k u 2 L 2 + j h k 2 u 2 L 2 + ε 1 j h k 1ψ 2 L 2 j h k u 2 2 L 2 ε 1 j h k u 2, j h k ψ ). 43 / 112

44 Dissipation only Small global solution and decay rates Proposition Let e(t) = (u, ψ) H 8 H s, s H s,8. If sup e(t) δ t [0,T ] for some sufficiently small δ, then d dt D l(t) + CH l (t) 0, t [0, T ]. 44 / 112

45 Dissipation only Small global solution and decay rates We further define E s,s1 = (u, ψ) 2 Ḣ + (u, s,s 1 ψ) 2, Ḣ s,s 1 +1 ε s,k (t) = E s,0 (t) + E s,s+k (t). Proposition Assume, for k = 0, 1, 2, sup e(t) δ, t [0,T ] sup ε s,k (t) Cɛ 2, t [0,T ] then l x 1 (u, ψ) L 2 + l x 1 ( u, 2 ψ) L 2 C(1 + t) l+s / 112

46 Dissipation only Small global solution and decay rates Proposition If e(0) = (u 0, ψ 0 ) H 8 Ḣ s,8 Ḣ s, s r 0, then (u, ψ) satisfies e(t) = (u, ψ) H 8 Ḣ s,8 Ḣ s, s 2r 0 We can choose r 0 to be sufficiently small so that 2r 0 < δ. 46 / 112

47 Dissipation only Small solutions for a system with damping Global small solutions for a damped system J. Wu, Yifei Wu and Xiaojing Xu, Global small solution to the 2D MHD system with a velocity damping term, arxiv: [math.ap] 24 Nov Consider the following 2D MHD equation t u + u u + u + P = div( φ φ), (t, x, y) R + R 2, t φ + u φ = 0, u = 0, u t=1 = u 0 (x, y), φ t=1 = φ 0 (x, y), (7) where u = (u, v). 47 / 112

48 Dissipation only Small solutions for a system with damping Letting φ = y + ψ in (6) yields t u + u x u + v y u + u + x P = ψ x ψ, t v + u x v + v y v + v + y P = ψ ψ y ψ, t ψ + u x ψ + v y ψ + v = 0, x u + y v = 0, (8) where P = P φ 2. By u = 0, P = ( u u) ( ψ ψ) y ψ. 48 / 112

49 Dissipation only Small solutions for a system with damping Therefore, (32) can be written as t u + u xy ψ = N 1, (9) t v + v+ xx ψ = N 2, (10) t ψ + v = u x ψ v y ψ, (11) where N 1 = u u + x 1 ( u u) ψ x ψ + x 1 ( ψ ψ), N 2 = u v + y 1 ( u u) ψ y ψ + y 1 ( ψ ψ). 49 / 112

50 Dissipation only Small solutions for a system with damping Taking the time derivative leads to tt u + t u xx u = F 1, tt v + t v xx v = F 2, tt ψ + t ψ xx ψ = F 0, u t=1 = u 0 (x, y), u t t=1 = u 1 (x, y) ψ t=1 = ψ 0 (x, y), ψ t t=1 = ψ 1 (x, y), (12) where u 1 = (u 1 (x, y), v 1 (x, y)), ψ 0 = φ 0 y, and u 1 = ( u + xy ψ + N 1 ) t=1, v 1 = ( v xx ψ + N 2 ) t=1, ψ 1 = ( u x ψ v y ψ v) t=1, 50 / 112

51 Dissipation only Small solutions for a system with damping and F 0 = u ψ t ( u ψ) N 2, F 1 = t N 1 xy ( u ψ), F 2 = t N 2 + xx ( u ψ). 51 / 112

52 Dissipation only Small solutions for a system with damping We consider the linear equation tt Φ + t Φ xx Φ = 0, (13) with the initial data Φ(0, x, y) = Φ 0 (x, y), Φ t (0, x, y) = Φ 1 (x, y). Taking the Fourier transform on the equation (13), we have tt ˆΦ + t ˆΦ + ξ 2 ˆΦ = 0, (14) where the Fourier transform ˆΦ is defined as ˆΦ(t, ξ, η) = e ixξ+iyη Φ(t, x, y) dxdy. R 2 52 / 112

53 Dissipation only Small solutions for a system with damping Solving (14) by a simple ODE theory, we have ˆΦ(t, ξ, η) = 1 2 ( ) ( ( e ξ2 t + e (e ( ξ ξ2 )t ) Φ0 (ξ, η) 4 ξ2 )t e ( 1 2 ) 1 4 ξ2 )t ( 1 2 Φ 0 (ξ, η) + Φ 1 (ξ, η)). 53 / 112

54 Dissipation only Small solutions for a system with damping Definition Let the operators K 0 (t, x ), K 1 (t, x ) be defined as K 0 (t, x )f (t, ξ, η) = 1 ( ) ( ( e ξ2 t + e ξ2 )t )ˆf (t, ξ, η); and K 1 (t, x )f (t, ξ, η) = ξ2 ˆf (t, ξ, η). ( ) ( ( e ξ2 t e ) ) ξ2 t where 1 = i. 54 / 112

55 Dissipation only Small solutions for a system with damping Therefore, the solution Φ of the equation (13) is written as Φ(t, x, y) = K 0 (t, x )Φ 0 + K 1 (t, x ) ( 1 2 Φ 0 + Φ 1 ). Moreover, consider the inhomogeneous equation, tt Φ + t Φ xx Φ = F, (15) with initial data Φ(1, x) = Φ 0, t Φ(1, x) = Φ / 112

56 Dissipation only Small solutions for a system with damping Then we have the following standard Duhamel formula, Φ(t, x, y) =K 0 (t, x )Φ 0 + K 1 (t, x ) ( 1 2 Φ ) 0 + Φ 1 + t 1 (16) K 1 (t s, x )F (s, x, y) ds. (17) The rest of the proof is to apply this formula to rewrite (12) and then verify the continuity principle. We will need the following estimates on K 0 and K / 112

57 Dissipation only Small solutions for a system with damping Lemma Let K 0, K 1 be defined in Definition (4.10), then 1) ξ α Ki (t, ) L q ξ ( ξ 1 2 ) t 1 2 ( 1 q +α), for any α 0, 1 q, i = 0, 1. 2) t Ki (t, ) 1 L q ξ ( ξ 1 t 1 2q, i = 0, 1. ) 2 3) Ki (t, ξ) e 1 2 t, for any ξ 1 2, i = 0, 1. 4) ξ 1 t K0 (t, ξ), t K1 (t, ξ) e 1 2 t, for any ξ / 112

58 Dissipation only Small solutions for a system with damping Let X 0 be the Banach space defined by the following norm ( u 0, ψ 0 ) X0 = N ( u 0, ψ 0 ) L 2 xy ( u 0, ψ 0 ) L 1 xy + 6+ ( u 1, ψ 1 ) L 1 xy, where = (I ) 1 2, N 1and a+ denotes a + ɛ for small ɛ > 0. The solution spaces X is defined by { ( u, ψ) X = sup t ε N ( u(t), ψ(t)) 2 + t ψ 2 t 1 + t ψ 2 + t 3 2 xx ψ + t xx ψ 2 + t 3 2 xxx ψ 2 +t 3 2 t u + t 5 4 t u 2 + t x u + t 3 2 x t v 2 }. 58 / 112

59 Dissipation only Small solutions for a system with damping Our main result can be stated as follows: Theorem There exists a small constant ε > 0 such that, if the initial data satisfies ( u 0, ψ 0 ) X0 ε, then (6) possesses a unique global solution (u, v, ψ) X. Moreover, the following decay estimates hold u(t) L x εt 1 ; v(t) L x εt 3 2 ; ψ(t) L x εt / 112

60 Dissipation only Small solutions for a system with damping The proof of this theorem relies on the continuity argument. Lemma (Continuity Argument) Suppose that ( u, ψ) with the initial data ( u 0, ψ 0 ), satisfies ( u, ψ) X ( u 0, ψ 0 ) X0 + C u, ψ) β X (18) with β > 1. Then, there exists r 0 such that, if ( u 0, ψ 0 ) X0 r 0, then ( u, ψ) X 2r / 112

61 Dissipation only Small solutions for a system with damping Proposition Let K(t, x ) be a Fourier multiplier operator satisfying α x K(t, ξ) L 1 ξ ( ξ 1 2 ) <, K(t, ξ) L <, α 0. ξ ( ξ 1 ) 2 Then, for any space-time Schwartz function f, α x K(t, x )f L xy ( α x K(t, ξ) L 1 ξ ( ξ 1 2 ) + K(t, ξ) L ξ ( ξ 1 2 ) ) α+1+ɛ y f L 1 xy. (19) 61 / 112

62 Dissipation only Small solutions for a system with damping Lemma For any s 1, ɛ F 1 (s, ) L 1 xy s 3 2 ε ( u, ψ) 2 Y. (20) 62 / 112

63 Dissipation only Small solutions for a system with damping Using the Duhamel formula, namely (16), ψ(t, x, y) = K 0 (t, x )ψ 0 + K 1 (t, x )( 1 2 ψ 0 + ψ 1 ) + Therefore, t 1 K 1 (t s, x )F 0 (s xx ψ xx K 0 (t)ψ 0 + xx K 1 (t)( 1 2 ψ 0 + ψ 1 ) + t By Corollary 4.13 and Lemma 2, xx K 0 (t)ψ 0 1 xx K 1 (t s)f 0 (s)ds. ( xx K 0 (t, ξ) L 1 ξ ( ξ 1 2 ) + K 0 (t, ξ) L ξ ( ξ 1 2 ) ) 2+ε xx y ψ 0 L 1 xy ( t e t ) 5+ε ψ 0 L 1 xy t ε ψ 0 X0. 63 / 112

64 Dissipation only Small solutions for a system with damping Since the estimates for K 0 and K 1 are the same, we also have xx K 1 (t)( 1 2 ψ 0 + ψ 1 ) t ε ( 1 2 ψ 0 + ψ 1 ) X0. Moreover, t xx K 1 (t s) F 0 (s) ds 1 t 1 t 2 1 xx K 1 (t s) F 0 (s) ds t xx K 1 (t s) F 0 (s) ds + x K 1 (t s) x F 0 (s) ds t 2 64 / 112

65 Magnetic diffusion only Magnetic Diffusion only, ν 1 = ν 2 = 0, η 1 = η 2 > 0 The 2D MHD equations with no dissipation: u t + u u = p + b b, b t + u b = η b + b u, u = 0, b = 0, (21) The global regularity problem remains open. 65 / 112

66 Magnetic diffusion only Global weak solutions have been established Theorem Let κ > 0. Let (u 0, b 0 ) H 1. Then (21) has a global weak solution (u, b) with (u, b) L ([0, ); H 1 ). 66 / 112

67 Magnetic diffusion only In this case, we can show that (u, b) admits global H 1 -bound. { ωt + u ω = ν ω + b j, j t + u j = b ω + 2 x b 1 ( y u 1 + x u 2 ) 2 x u 1 ( y b 1 + x b 2 ). 67 / 112

68 Magnetic diffusion only 1 d ω 2 2 = b j ω dxdy, 2 dt 1 d j η j 2 2 = b ω j dxdy dt j x b 1 x u 2 dxdy + Since b j ω dxdy + b ω j dxdy = 0, we have, for X (t) = ω(t) j(t) 2 2, d X (t) dt + 2η j 2 2 C u 2 b 4 j 4, 68 / 112

69 Magnetic diffusion only u 2 = ω 2, b 4 j 4, j 2 4 j 2 j 2 and Young s inequality, we find In particular, d X (t) dt d X (t) dt By Gronwall s inequality, X (t) + η t 0 + 2η j 2 2 C η ω 2 2 j η j η j 2 2 C η j 2 2 X (t). j(τ) 2 2 dτ X (0) exp ( C η t 0 ) j 2 2 dτ. 69 / 112

70 Magnetic diffusion only Remark. It remains open whether or not two H 1 -weak solutions must coincide. Remark. It remains open whether or not the H 1 -weak solution becomes regular when (u 0, b 0 ) is more regular, say (u 0, b 0 ) H 2. The global regularity problem for the 2D MHD equations with only Laplacian magnetic diffusion remains open. 70 / 112

71 Magnetic diffusion only The main difficulty is the lack of the global bound for ω L, although we do have global L p -bound. Proposition For any p (2, ) and q (2, ), the solution (u, b) obeys, for any T > 0, ω L (0,T ;L p ) C, b L q (0,T ;W 2,p ) C, where C is a constant depending on p, q, T and the initial data only. It is not clear how ω L p depends on p. 71 / 112

72 Magnetic diffusion only MHD equations with ( ) β b with β > 1 Global regularity for MHD equation with ( ) β b Consider t u + u u = p + b b, x R 2, t > 0, t b + u b + ( ) β b = b u, x R 2, t > 0, u = 0, b = 0, x R 2, t > 0, u(x, 0) = u 0 (x), b(x, 0) = b 0 (x), x R 2, (22) 72 / 112

73 Magnetic diffusion only MHD equations with ( ) β b with β > 1 C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J Math Anal., 46 (2014), No. 1, Q. Jiu and J. Zhao, A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations, arxiv: [math.ap] 12 Jun / 112

74 Magnetic diffusion only MHD equations with ( ) β b with β > 1 Theorem (C. Cao, J. Wu and B. Yuan) Consider (22) with β > 1. Assume that (u 0, b 0 ) H s (R 2 ) with s > 2, u 0 = 0, b 0 = 0 and j 0 = b 0 satisfying j 0 L <. Then (22) has a unique global solution (u, b) satisfying, for any T > 0, (u, b) L ([0, T ]; H s (R 2 )), j L 1 ([0, T ]; L (R 2 )) where j = b. 74 / 112

75 Vertical dissipation and horizontal magnetic diffusion 2D MHD with ν 2 > 0 and η 1 > 0 The 2D MHD equations with vertical dissipation and horizontal magnetic diffusion u t + u u = p + ν u yy + b b, b t + u b = η b xx + b u, u = 0, b = 0. C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Advances in Mathematics 226 (2011), / 112

76 Vertical dissipation and horizontal magnetic diffusion Theorem Assume u 0 H 2 (R 2 ) and b 0 H 2 (R 2 ) with u 0 = 0 and b 0 = 0. Then the aforementioned MHD equations have a unique global classical solution (u, b). In addition, (u, b) satisfies (u, b) L ([0, ); H 2 ), ω y L 2 ([0, ); H 1 ), j x L 2 ([0, ); H 1 ). 76 / 112

77 Vertical dissipation and horizontal magnetic diffusion The main efforts are devoted to global a priori bounds in H 1 and H 2. We need a lemma. Lemma Assume that f, g, g y, h and h x are all in L 2 (R 2 ). Then, f g h dxdy C f L 2 g 1/2 L 2 g y 1/2 L 2 h 1/2 L 2 h x 1/2 L / 112

78 Vertical dissipation and horizontal magnetic diffusion H 1 -bounds Proposition If (u, b) is a solution of the aforementioned MHD equations, then ω(t) j(t) ν t 0 ω y (τ) 2 2 dτ + η C(ν, η) ( ω j 0 2 ) 2 t 0 j x (τ) 2 2 dτ where C(ν, η) denotes a constant depending on ν and η only, ω 0 = u 0 and j 0 = b / 112

79 Vertical dissipation and horizontal magnetic diffusion H 2 -bounds Proposition If (u, b) is a solution of the aforementioned MHD equations, then ω(t) j(t) ν t 0 ω y (τ) 2 2 dτ + η C(ν, η, t) ( ω j 0 2 ) 2 where C(ν, η, t) depends on ν, η and t only. t 0 j x (τ) 2 2 dτ 79 / 112

80 Horizontal dissipation and vertical magnetic diffusion 2D MHD with ν 1 > 0 and η 2 > 0 The 2D MHD equations with horizontal dissipation and vertical magnetic diffusion u t + u u = p + ν u xx + b b, b t + u b = η b yy + b u, u = 0, b = 0. For any initial data (u 0, b 0 ) H 2, this system of equations also possess a unique global solution. 80 / 112

81 Horizontal dissipation and vertical magnetic diffusion In fact, the case ν u xx and ηb yy can be converted into the case νu yy and ηb xx. Set U 1 (x, y, t) = u 2 (y, x, t), B 2 (x, y, t) = b 1 (y, x, t), U 2 (x, y, t) = u 1 (y, x, t), B 1 (x, y, t) = b 2 (y, x, t), P(x, y, t) = p(y, x, t). Then U = (U 1, U 2 ), P and B = (B 1, B 2 ) satisfy U t + U U = P + ν U yy + B B, B t + U B = η B xx + B U, U = 0, B = / 112

82 Horizontal dissipation and horizontal magnetic diffusion 2D MHD with ν 1 > 0 and η 1 > 0 The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion t u + u u = p + xx u + b b, t b + u b = xx b + b u, u = 0, b = 0, (23) where we have set ν 1 = η 1 = 1. The global regularity for this case is almost obtained, but this case appears to be very difficult. 82 / 112

83 Horizontal dissipation and horizontal magnetic diffusion Theorem (Expected Theorem) Assume that (u 0, b 0 ) H 2 (R 2 ), u 0 = 0 and b 0 = 0. Then, (23) has a unique global solution (u, b) satisfying, for any T > 0 and t T, u, b L ([0, T ]; H 2 (R 2 )), x u, x b L 2 ([0, T ]; H 2 (R 2 )). 83 / 112

84 Horizontal dissipation and horizontal magnetic diffusion References C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations 254 (2013), No.7, C. Cao, D. Regmi, J. Wu and X. Zheng, Global regularity for the 2D magnetohydrodynamics equations with horizontal dissipation and horizontal magnetic diffusion, preprint. 84 / 112

85 Horizontal dissipation and horizontal magnetic diffusion The major effort is devoted to obtaining global bounds. We have global bound for the L 2 -norm: t t u(t) b(t) x u(τ) 2 2dτ + 2 x b(τ) 2 2dτ 0 0 = u b 0 2 2, The trouble arises when we try to obtain the global H 1 -bound. If we resort to the equations of ω and j = b, t ω + u ω = x 2 ω + b j, t j + u j = x 2 j + b ω +2 x b 1 ( x u 2 + y u 1 ) 2 x u 1 ( x b 2 + y b 1 ), 85 / 112

86 Horizontal dissipation and horizontal magnetic diffusion 1 d 2 dt ( ω j 2 ) 2 + x ω x j 2 2 = 2 j ( x b 1 ( x u 2 + y u 1 ) 2 x u 1 ( x b 2 + y b 1 )) dxdy. If we use the anisotropic Sobolev inequalities stated in the previous lemma, f g h dxdy C f 2 g g y h h x 1 2 2, two terms can be bounded suitably. But j x b 1 y u 1 and j x u 1 y b 1 can not be controlled. 86 / 112

87 Horizontal dissipation and horizontal magnetic diffusion If we do know that T 0 (u 1, b 1 ) 2 dt <, (24) then j x b 1 y u 1 + j x u 1 y b ( x ω x j 2 2) +C (u 1, b 1 ) 2 ( ω j 2 2). Then we can close the differential inequality and get a global bound for ω j 2 2. (24) also allows us to get a global bound for ω j / 112

88 Horizontal dissipation and horizontal magnetic diffusion It appears to be extremely hard to prove (24) directly. Motivated by our recent work on the 2D Boussinesq equations with partial dissipation, C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal. 208 (2013), , we bound the L r -norm of (u 1, b 1 ) suitably. Theorem Let (u, b) be a solution of (23). Let 2 < r <. Then, (u 1, b 1 )(t) L r B 0 r log r + B1 (t), (25) 88 / 112

89 Horizontal dissipation and horizontal magnetic diffusion The proof of this bound uses the symmetric structure of (23), namely w ± = u ± b satisfies t w + + (w )w + = p + x 2 w +, t w + (w + )w = p + x 2 w, w + = 0, w = 0. (26) We then bound w ± 1 L r. 89 / 112

90 Horizontal dissipation and horizontal magnetic diffusion Multiplying the first component of the first equation of (26) by w + 1 w + 1 2r 2 and integrating with respect to space variable, we obtain, after integration by parts, 1 d 2r dt w 1 + 2r + (2r 1) x w w 1 + 2r 2 = (2r 1) p x w 1 + w 1 + 2r 2. (27) The main effort is devoted to bounding the pressure. If we knew that T then we can easily show that 0 p L dt <, w + 1 2r C r 90 / 112

91 Horizontal dissipation and horizontal magnetic diffusion In order to get the bounds for the pressure, we showed that (u 1, b 1 ) 2r C 1 e C 2 r 3 for any r (u 2, b 2 )(t) L 2r C, r = 2, 3, (28) Since p = (w w + ) and p q C w 2q w + 2q, we can show that, for any 1 < q 3, p(t) q C, (29) where C is a constant depending on T and the initial data. 91 / 112

92 Horizontal dissipation and horizontal magnetic diffusion We can also show that, for any s (0, 1), Since T 0 p(τ) 2 Hs dτ < C. Λ s p 2 Λ s ( ) 1 x (w 1 xw w + 1 xw 1 ) 2 + Λ s ( ) 1 y (w + 1 xw 2 + w 1 xw + 2 ) 2 C ( x w x w 2 ) ( w s + w s ), 92 / 112

93 Horizontal dissipation and horizontal magnetic diffusion To start, we fix R > 0 (to be specified later) and write (2r 1) p x w + 1 w + 1 2r 2 = J 1 + J 2, where J 1 = (2r 1) p x w + 1 w + 1 2r 2, J 2 = (2r 1) p x w + 1 w + 1 2r 2 with p and p as defined as follows. 93 / 112

94 Horizontal dissipation and horizontal magnetic diffusion As for the 2D Boussinesq equations, we decompose the pressure into low and high frequency parts and bound each part accordingly. Lemma Let f H s (R 2 ) with s (0, 1). Let R (0, ). Denote by B(0, R) the box centered at zero with each side R and by χ B(0,R) the characteristic function on B(0, R). Write f = f + f with f = F 1 (χ B(0,R) Ff ) and f = F 1 ((1 χ B(0,R) )Ff ) where F and F 1 denote the Fourier transform and the inverse Fourier transform, respectively. Then we have the following estimates for f and f. 94 / 112

95 Horizontal dissipation and horizontal magnetic diffusion Lemma (1) For a pure constant C 0 (independent of s), f C 0 1 s R 1 s f H s (R 2 ), (2) For any 2 q < satisfying 1 s 2 q < 0, there is a constant C 1 independent of s, q, R and f such that f q C 1 q R 1 s 2 q f H s (R 2 ). 95 / 112

96 Horizontal dissipation and horizontal magnetic diffusion By Hölder s and Young s inequalities, we find J 1 (2r 1) p w + 1 r 1 2 x w + 1 (w + 1 )r 1 2 (2r 1) p 2 w 1 + r r 1 x w (w 1 + )r Applying Lemma 5, we have p C 0 1 s R 1 s p H s, (30) We will skip more details. 96 / 112

97 Horizontal dissipation and horizontal magnetic diffusion We need the bound for the L -norm and we have a suitable bound for L r -norm. The bridge is the following interpolation inequality. Proposition Let s > 1 and f H s (R 2 ). Then there exists a constant C depending on s only such that f f L (R 2 ) C sup r r 2 r log r [ log(e + f H s (R 2 )) log log(e + f H s (R 2 )) ] / 112

98 Horizontal dissipation and horizontal magnetic diffusion Proof of the proposition on interpolation inequality: By the Littlewood-Paley decomposition, we can write f = S N+1 f + j f, j=n+1 where j denotes the Fourier localization operator and N S N+1 = j. j= 1 The definitions of j and S N are now standard. Therefore, f S N+1 f + j f. j=n+1 98 / 112

99 Horizontal dissipation and horizontal magnetic diffusion We denote the terms on the right by I and II. By Bernstein s inequality, for any q 2, I 2 2N q S N+1 f q 2 2N q f q. Taking q = N, we have I 4 f N 4 f N log N sup r. r 2 r log r By Bernstein s inequality again, for any s > 1, II 2 j j f 2 = 2 j(s 1) 2 sj j f 2 j=n+1 j=n+1 = C 2 (N+1)(s 1) f B s 2,2. 99 / 112

100 Horizontal dissipation and horizontal magnetic diffusion where C is a constant depending on s only. By identifying B s 2,2 with H s, we obtain f 4 f N log N sup r + C 2 (N+1)(s 1) f H s. r 2 r log r We obtain the desired inequality (31) by taking [ ] 1 N = s 1 log 2(e + f H s ), where [a] denotes the largest integer less than or equal to a. 100 / 112

101 2D MHD with fractional dissipation 2D MHD with fractional dissipation Consider the 2D fractional MHD equations u t + u u + ν( ) α u = p + b b, b t + u b + η( ) β b = b u, u = 0, b = 0, u(x, 0) = u 0 (x), b(x, 0) = b 0 (x). (31) where ( ) α f (ξ) = ξ 2α f (ξ). The aim is at the smallest α and β for which (31) has a global regular solution. 101 / 112

102 2D MHD with fractional dissipation The results we have indicate three cases: The subcritical case: α + β > 1; The critical case: α + β = 1; The supercritical case: α + β < / 112

103 2D MHD with fractional dissipation A summary of current results The global regularity results we currently have are for subcritical cases. 1 ν = 0 and β > 1 C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J Math Anal., 46 (2014), No. 1, Q. Jiu and J. Zhao, Global Regularity of 2D Generalized MHD Equations with Magnetic Diffusion, arxiv: [math.ap] 23 Sep / 112

104 2D MHD with fractional dissipation A summary of current results 1 α > 0 and β = 1 J. Fan, G. Nakamura, Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, preprint. 2 α 2 (or with logarithmic improvement): K. Yamazaki, Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation, arxiv: v1 [math.ap] 13 Jun / 112

105 2D MHD with fractional dissipation A summary of current results Open problems: 1) ν = 0 and β = 1 2)η = 0 and 1 α < 2 3)1 < α + β < 2, 0 < β < / 112

106 2D MHD with fractional dissipation A summary of current results J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), C. Trann, X. Yu and Z. Zhai, On global regularity of 2D generalized magnetohydrodynamic equations, J. Differential Equations 254 (2013), B. Yuan and L. Bai, Remarks on global regularity of 2D generalized MHD equations, arxiv: v1 [math.ap] 11 Jun Q. Jiu and J. Zhao, A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations, arxiv: [math.ap] 12 Jun / 112

107 The 2D Compressible MHD with velocity dissipation 2D Compressible MHD Two recent papers are devoted to the compressible MHD with only velocity dissipation: Jiahong Wu and Yifei Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, preprint, April, Xianpeng Hu, Global Existence for Two Dimensional Compressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity, arxiv: v1 [math.ap] 1 May / 112

108 The 2D Compressible MHD with velocity dissipation The 2D compressible MHD system can be written as t ρ + (ρ u) = 0, (t, x, y) R + R R, t (ρ u) + (ρ u u) u λ ( u) + P = 1 2 ( b 2) + b b, t b + u b = b u, b = 0. with the initial data ρ t=0 = ρ 0 (x, y), u t=0 = u 0 (x, y), b t=0 = b 0 (x, y). 108 / 112

109 The 2D Compressible MHD with velocity dissipation This paper of Wu and Wu achieves three goals: It establishes the global well-posedness of smooth solutions of when the initial data (ρ 0, u 0, b 0 ) is smooth and close to the equilibrium state (1, 0, e 1 ), where we denote 0 = (0, 0) and e 1 = (1, 0); It offers a new way of diagonalizing a complex system of linearized equations; It obtains explicit and sharp large-time decay rates for the solutions in various Sobolev spaces. 109 / 112

110 The 2D Compressible MHD with velocity dissipation In the 2D case, b = 0 implies that for a scalar function φ, b = φ ( y φ, x φ ). With this substitution, (6) becomes t ρ + (ρ u) = 0, (t, x, y) R + R R, t (ρ u) + (ρ u u) u λ ( u) + ρ 2 ρ = φ φ, t φ + u φ = 0. (32) 110 / 112

111 The 2D Compressible MHD with velocity dissipation Theorem Assume λ c 0 for some absolute constant c 0 > 0, and let n = ρ 1, ψ = φ y, n 0 = ρ 0 1, ψ 0 = φ 0 y. Then there exists a small constant δ > 0 such that, if the initial data (n 0, u 0, φ 0 ) satisfies (n 0, u 0, ψ 0 ) X0 δ, then there exists a unique global solution (ρ, u, v, φ) X to the MHD system. Moreover, (n, u, v, φ) X δ. Especially, the following decay estimates hold n(t) L xy δ t 1 2 ; u(t) L xy δ t 1 ; ψ(t) L xy δ t / 112

112 The 2D Compressible MHD with velocity dissipation Thank You Very Much! 112 / 112