The 2D Magnetohydrodynamic Equations with Partial …helper.ipam.ucla.edu/publications/mtws1/mtws1_12215.pdfuniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv:1404.5681v1

2D MHD Equations

The 2D Magnetohydrodynamic Equations withPartial Dissipation

Jiahong Wu

Oklahoma State University

IPAM Workshop “Mathematical Analysis of Turbulence”IPAM, UCLA, September 29-October 3, 2014

1 / 112

2D MHD Equations

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 (−∆)βb with β > 1

6 Vertical dissipation and horizontal magnetic diffusion2 / 112

2D MHD Equations

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

2D MHD Equations

Introduction

Introduction

The standard 2D incompressible MHD equations can be written asut + u · ∇u = −∇p + ν∆u + b · ∇b,bt + u · ∇b = η∆b + b · ∇u,∇ · u = 0, ∇ · b = 0,

(1)

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

2D MHD Equations

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

2D MHD Equations

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

2D MHD Equations

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,ut + u · ∇u = −∇p + ν1uxx + ν2uyy + b · ∇b,bt + u · ∇b = η1bxx + η2byy + b · ∇u,∇ · u = 0, ∇ · b = 0,

(2)

where ν1 ≥ 0, ν2 ≥ 0, η1 ≥ 0 and η2 ≥ 0. (2) will be called

anisotropic MHD equations. When ν1 = ν2 and η1 = η2, (2)

becomes the standard MHD equations.

7 / 112

2D MHD Equations

Introduction

Another generalized form is the MHD equations with fractional

dissipation, replacing the Laplacian by fractional Laplacians,

namely ut + u · ∇u = −∇p − ν(−∆)αu + b · ∇b,bt + u · ∇b = −η(−∆)βb + 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),

(−∆)αf (ξ) = |ξ|2αf (ξ).

This system will be called fractional MHD equations.

8 / 112

2D MHD Equations

Introduction

We consider the initial-value problems of the MHD equations with

the initial data

u(x , 0) = u0(x), b(x , 0) = b0(x).

What we care about is the global regularity issue: Do these IVPs

have a global solution for sufficiently smooth data (u0, b0)?

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

2D MHD Equations

Introduction

Consider the following seven cases:

ν1 = ν2 = η1 = η2 = 0, ideal MHD

ν1 > 0, ν2 > 0, η1 > 0 and η2 > 0,

MHD with dissipation and magnetic diffusion

ν1 = ν2 > 0, η1 = η2 = 0.

dissipation but no magnetic diffusion

η1 > 0, η2 > 0, ν1 = ν2 = 0.

magnetic diffusion but no dissipation

10 / 112

2D MHD Equations

Introduction

ν1 > 0 and η2 > 0

horizontal dissipation and vertical magnetic diffusion

ν2 > 0 and η1 > 0

vertical dissipation and horizontal magnetic diffusion

ν1 > 0 and η1 > 0

horizontal dissipation and horizontal magnetic diffusion

11 / 112

2D MHD Equations

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

2D MHD Equations

Ideal MHD equations

• Ideal MHD equations

Ideal MHD equations:ut + u · ∇u = −∇p + b · ∇b,bt + u · ∇b = b · ∇u,∇ · u = 0, ∇ · b = 0,u(x , y , 0) = u0(x , y), b(x , y , 0) = b0(x , y).

(4)

The global regularity problem remains open, although we do have

local well-posedness and regularity criteria.

13 / 112

2D MHD Equations

Ideal MHD equations

Theorem

Given (u0, b0) ∈ Hs(R2) with s > 2. Then there exists a unique

local classical solution (u, b) ∈ C ([0,T0); Hs) for some T0 > 0. In

addition, if ∫ T

0(‖ω‖∞ + ‖j‖∞) dt <∞

for T > T0, then the solution remains in Hs for any t ≤ T .

The L∞-norm can replaced by BMO or B0∞,∞.

14 / 112

2D MHD Equations

Ideal MHD equations

Why is the global regularity problem hard? The global L2-bound

for (u, b) follows directly from the MHD equations

‖u(t)‖2L2 + ‖b(t)‖2

L2 = ‖u0‖2L2 + ‖b0‖2

L2 .

But global bounds for any Sobolev-norm appear to be impossible,

for example, the H1-norm. Consider the equations of ω = ∇× u

and j = ∇× b,{ωt + u · ∇ω = b · ∇j ,jt + u · ∇j = b · ∇ω + 2∂xb1(∂yu1 + ∂xu2)− 2∂xu1(∂yb1 + ∂xb2)

15 / 112

2D MHD Equations

Ideal MHD equations

Clearly,

1

2

d

dt

(‖ω‖2

L2 + ‖j‖2L2

)= 2

∫j ∂xb1∂yu1 + · · ·

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

2D MHD Equations

Ideal MHD equations

In fact, for Y = ‖u(t)‖2Hs + ‖b‖2

Hs ,

d

dtY (t) ≤ C (‖∇u‖L∞ + ‖∇b‖L∞) Y (t).

Then one uses the logarithmic inequality,

‖∇u‖L∞ ≤ C (1+‖ω‖L2+‖ω‖L∞ log+ ‖u‖W s+1,p), p ∈ (1,∞), s > d/p

Then

d

dtY (t) ≤ C (‖ω‖L∞ + ‖j‖L∞) Y (t) log+ Y (t).

To get a stronger regularity criterion, one uses BMO or B0∞,∞,

‖f ‖L∞ ≤ C (1 + ‖f ‖BMO log+ ‖f ‖W s,p), p ∈ (1,∞), s > d/p.

17 / 112

2D MHD Equations

Fully dissipative MHD equations

• Dissipative MHD equations

Fully dissipative MHD equations:ut + u · ∇u = −∇p + ν∆u + b · ∇b,bt + u · ∇b = η∆b + b · ∇u,∇ · u = 0, ∇ · b = 0.

The global regularity can be easily established.

Theorem

Let (u0, b0) ∈ H1(R2). Then there exists a unique global strong

solution (u, b) satisfying, for any T > 0,

u, b ∈ L∞([0,T ; H1(R2)) ∩ L2([0,T ]; H2(R2))

18 / 112

2D MHD Equations

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

2D MHD Equations

Dissipation only

• Dissipation only

The 2D MHD equations with no magnetic diffusion:ut + u · ∇u = −∇p + ν∆u + b · ∇b,bt + 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 L2-bound

‖u(t)‖2L2 + ‖b(t)‖2

L2 +

∫ t

0‖∇u(τ)‖2

L2 dτ = ‖u0‖2L2 + ‖b0‖2

L2 .

20 / 112

2D MHD Equations

Dissipation only

To consider the H1-norm, we use the equations for ω = ∇× u and

j = ∇× b,{ωt + u · ∇ω = ν∆ω + b · ∇j ,jt + u · ∇j = b · ∇ω + 2∂xb1(∂yu1 + ∂xu2)− 2∂xu1(∂yb1 + ∂xb2).

1

2

d

dt

(‖ω‖2

L2 + ‖j‖2L2

)+ ν‖∇ω‖2

L2 = 2

∫j ∂xb1∂yu1 + · · ·

To close the inequality, we need∫ T

0‖∇u‖L∞ dt <∞ or

∫ T

0‖ω‖∞ dt <∞.

21 / 112

2D MHD Equations

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 (uN , bN);

2) Obtain uniform bounds, for any fixed T > 0, s > 2,

uN ∈ L∞(0,T ; L2) ∩ L2(0,T ; H1), ∂tuN ∈ L∞(0,T ; H−s),

bN ∈ L∞(0,T ; L2), ∂tbN ∈ L∞(0,T ; H−s);

22 / 112

2D MHD Equations

Dissipation only

3) Apply the local version of the Aubin-Lions Lemma

uN → u in L2(0,T ; L2loc),

bN → b in L2(0,T ; H−δloc ) (δ > 2)

The trouble is that this does not allow us to pass to the limit in

bN · ∇bN → b · ∇b

in the distributional sense.

23 / 112

2D MHD Equations

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:1302.5877v2 [math.AP] 4

Jun 2013.

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), 503-541.

24 / 112

2D MHD Equations

Dissipation only

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D

MHD system with a velocity damping term,

arXiv:1311.6185v1 [math.AP] 24 Nov 2013.

T. Zhang, An elementary proof of the global existence and

uniqueness theorem to 2-D incompressible non-resistive MHD

system, arXiv:1404.5681v1 [math.AP] 23 Apr 2014.

X. Hu and F. Lin, Global Existence for Two Dimensional

Incompressible Magnetohydrodynamic Flows with Zero

Magnetic Diffusivity, arXiv: 1405.0082v1 [math.AP] 1 May

2014.

25 / 112

2D MHD Equations

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 andut + u · ∇u = −∇p + ν∆u +∇⊥φ · ∇∇⊥φ,φt + u · ∇φ = 0,∇ · u = 0.

Clearly, (u, φ) = (0, y) is a steady solution.

26 / 112

2D MHD Equations

Dissipation only

Setting φ = y + ψ yields∂tψ + u · ∇ψ + u2 = 0,∂tu1 + u · ∇u1 − ν∆u1 + ∂1∂2ψ = −∂1p −∇ · (∂1ψ∇ψ),∂tu2 + u · ∇u2 − ν∆u2 + ∂2

1ψ = −∂2p −∇ · (∂2ψ∇ψ).(5)

The aim is to look for global small solutions of this system.

27 / 112

2D MHD Equations

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

Ytt −∆Yt − ∂2xY = f (Y , q(y)), q = p + |∇ψ|2.

They then estimate the Lagrangian velocity Yt in L1t Lipx , using

anisotropic Littlewood-Paley theory and anisotropic Besov space

techniques.

28 / 112

2D MHD Equations

Dissipation only

Due to their use of the Lagrangian coordinates, they need to

impose a compatibility condition on the initial data ψ0, more

precisely, ∂yψ0 and (1 + ∂yψ0,−∂xψ0) are admissible on 0×R and

supp∂yψ0(·, y) ⊂ [−K ,K ] for some K .

∂yψ0 and (1 + ∂yψ0,−∂xψ0) are admissible on 0× R if∫R∂yψ0(X (a, t))dt = 0 for all a ∈ 0× R.

where X is the particle trajectory defined by (1 + ∂yψ0,−∂xψ0).

29 / 112

2D MHD Equations

Dissipation only

Theorem

Given u0 and ψ0 satisfying (u0,∇ψ0) ∈ Hs ∩ Hs2 with s1 > 1,

s2 ∈ (−1,−12 ) and s > s1 + 2, and

‖∇ψ0‖Hs1+2 ≤ 1, ‖(∇ψ0, u0)‖Hs1+1∩Hs2 + ‖∂yψ0‖Hs1+2 ≤ ε0

for some ε0 small. Assume that ∂yψ0 and (1 + ∂yψ0,−∂xψ0) are

admissible on 0× R and ∂yψ0(·, 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

2D MHD Equations

Dissipation only


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), 503-541.

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

2D MHD Equations

Dissipation only


Definition

Let σ, s ∈ R. The anisotropic Sobolev space Hσ,s(R2) is defined by

Hσ,s(R2) ={

f ∈ S ′(R2) : ‖f ‖Hσ,s < +∞},

where

‖f ‖Hσ,s =∥∥∥{2js2σk‖∆j∆

hk f ‖L2

}j ,k

∥∥∥`2.

or

‖f ‖Hσ,s =

[∫R2

|ξ|2s ξ2σ1 |f (ξ)|2 dξ

] 12

.

32 / 112

2D MHD Equations

Dissipation only


Theorem

Assume (∇ψ0, u0) ∈ H8(R2). Let s ∈ (0, 12 ). There exists a small

positive constant ε such that, if, (∇ψ0, u0) ∈ H−s,−s ∩ H−s,8(R2),

and

‖(∇ψ0, u0)‖H8 + ‖(∇ψ0, u0)‖H−s,−s + ‖|(∇ψ0, u0)‖H−s,8 ≤ ε,

then (5) has a unique global solution (ψ, u) satisfying

(∇ψ, u) ∈ C ([0,+∞); H8(R2)).

33 / 112

2D MHD Equations

Dissipation only


Theorem

Moreover, the solution decays at the same rate as that for the

linearized solutions,

‖∂kx∇ψ‖L2 + ‖∂kx u‖L2 ≤ Cε(1 + t)−s+k

2 ,

for any t ∈ [0,+∞) and k = 0, 1, 2.

34 / 112

2D MHD Equations

Dissipation only


Ideas in the proof: First, we consider the linearized equation

Proposition

Consider the linearized equation∂tu1 −∆u1 − ∂x1x2ψ = 0,

∂tu2 −∆u2 + ∂x1x1ψ = 0,

∂tψ + u2 = 0,

u(x , 0) = u0(x), ψ(x , 0) = ψ0(x).

Assume (u0,∇ψ0) ∈ H4 and |D1|−su0 ∈ H1+s and |D1|−s∇ψ0 ∈

H1+s for s > 0, then, for k = 0, 1, 2,

‖∂kx1u‖L2 + ‖∂kx1

∇ψ‖L2 ≤ C (1 + t)−k+s

2 .

35 / 112

2D MHD Equations

Dissipation only


Proof. For ε1 > 0, define

D0(t) =‖u‖2L2 + ‖∇u‖2

L2 + ‖∇ψ‖2L2 + ‖∇2ψ‖2

L2 + 2ε1〈u2,∇ψ〉,

H0(t) =‖∇u‖2L2 + ‖∇2u‖2

L2 + ε1‖∇∂1ψ‖2L2 − ε1‖∇u2‖2

L2 − ε1〈∆u2,∆ψ〉.

Es(t) =‖|D1|−su‖2L2 + ‖|D1|−s∇ψ‖2

L2

+ ‖|D|1+s |D1|−su‖2L2 + ‖|D|1+s |D1|−s∇ψ‖2

L2 .

We can show

d

dtD0(t) + C H0(t) ≤ 0,

d

dtEs(t) ≤ 0.

36 / 112

2D MHD Equations

Dissipation only


By interpolation inequalities,

D0(t) ≤ Es(t)1

1+s H0(t)s

1+s , H0(t) ≥ Es(0)−1s D0(t)1+ 1

s .

Thus,

d

dtD0(t) + C Es(0)−

1s D0(t)1+ 1

s ≤ 0.

E (t) ≤ (E (0)−1s + C (s)t)−s = E0

(E

1s

0 C (s)t + 1

)−s.

37 / 112

2D MHD Equations

Dissipation only


We return to the full nonlinear system∂tψ + u · ∇ψ + u2 = 0,∂tu1 + u · ∇u1 − ν∆u1 + ∂1∂2ψ = −∂1p −∇ · (∂1ψ∇ψ),∂tu2 + u · ∇u2 − ν∆u2 + ∂2

1ψ = −∂2p −∇ · (∂2ψ∇ψ).(6)

The frame work to prove the global existence of small solutions is

the Bootstrap Principle.

38 / 112

2D MHD Equations

Dissipation only


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 (t1) is true, then H(t) is true for t in a neighborhood of t1,

(c) If C (tk) is true for a sequence tk → t, then C(t)is true,

(d) C(t) is true for at least one t0 ∈ I ,

then, C(t) is true for all t ∈ I .

39 / 112

2D MHD Equations

Dissipation only


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

2D MHD Equations

Dissipation only


We use anisotropic Sobolev and Besov spaces due to the

anisotropicity,

utt −∆ut − ∂2xu = 0

The characteristic equation satisfies

λ2 + |ξ|2λ+ ξ21 = 0,

which has two roots

λ± = −|ξ|2 ±

√|ξ|4 − 4ξ2

1

2.

As |ξ| → ∞,

λ−(ξ)→ − ξ21

|ξ|2∼{−1, |ξ| ∼ |ξ1|,0, |ξ| � |ξ1|

The dissipation is weak in the case of |ξ| � |ξ1|.41 / 112

2D MHD Equations

Dissipation only


Proposition

If (u,∇ψ)satisfies

‖(u(t),∇ψ(t))‖H4 ≤ δ

for some δ > 0 and for t ∈ [0,T ], then we can show

d

dtD0 + CH0 ≤ 0 for t ∈ [0,T ].

42 / 112

2D MHD Equations

Dissipation only


Define For l = 1, 2, we define

Dl(t) =∑j ,k

22lk(‖∆j∆hku‖2

L2 + ‖∆j∆hk∇u‖2

L2 + ‖∆j∆hk∇ψ‖2

L2

+‖∆j∆hk∇2ψ‖2

L2 + 2ε1〈∆j∆hku2,∆j∆

hk∆ψ〉),

Hl(t) =∑j ,k

22lk(‖∆j∆hk∇u‖2

L2 + ‖∆j∆hk∇2u‖2

L2 + ε1‖∆j∆hk∇∂1ψ‖2

L2

−‖∆j∆hk∇u2‖2

L2 − ε1〈∆j∆hk∆u2,∆j∆

hk∆ψ〉).

43 / 112

2D MHD Equations

Dissipation only


Proposition

Let e(t) = ‖(u,∇ψ)‖H8∩H−s,−s∩H−s,8 . If

supt∈[0,T ]

e(t) ≤ δ

for some sufficiently small δ, then

d

dtDl(t) + CHl(t) ≤ 0, t ∈ [0,T ].

44 / 112

2D MHD Equations

Dissipation only


We further define

Es,s1 = ‖(u,∇ψ)‖2H−s,s1

+ ‖(u,∇ψ)‖2H−s,s1+1 ,

εs,k(t) = Es,0(t) + Es,s+k(t).

Proposition

Assume, for k = 0, 1, 2,

supt∈[0,T ]

e(t) ≤ δ, supt∈[0,T ]

εs,k(t) ≤ Cε2,

then

‖∂lx1(u,∇ψ)‖L2 + ‖∂ lx1

(∇u,∇2ψ)‖L2 ≤ C (1 + t)−l+s

2 .

45 / 112

2D MHD Equations

Dissipation only


Proposition

If

e(0) = ‖(u0,∇ψ0)‖H8∩H−s,8∩H−s,−s ≤ r0,

then (u,∇ψ) satisfies

e(t) = ‖(u,∇ψ)‖H8∩H−s,8∩H−s,−s ≤ 2r0

We can choose r0 to be sufficiently small so that 2r0 < δ.

46 / 112

2D MHD Equations

Dissipation only


• 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: 1311.6185 [math.AP] 24 Nov 2013.

Consider the following 2D MHD equation∂t~u + ~u · ∇~u + ~u +∇P = −div(∇φ⊗∇φ), (t, x , y) ∈ R+ × R2,

∂tφ+ ~u · ∇φ = 0,

∇ · ~u = 0,

~u|t=1 = ~u0(x , y), φ|t=1 = φ0(x , y),

(7)

where ~u = (u, v).47 / 112

2D MHD Equations

Dissipation only


Letting φ = y + ψ in (6) yields∂tu + u ∂xu + v∂yu + u + ∂x P = −∆ψ∂xψ,

∂tv + u ∂xv + v∂yv + v + ∂y P = −∆ψ −∆ψ∂yψ,

∂tψ + u∂xψ + v∂yψ + v = 0,

∂xu + ∂yv = 0,

(8)

where P = P + 12 |∇φ|

2. By ∇ · ~u = 0,

∆P = −∇ · (~u · ∇~u)−∇ · (∆ψ∇ψ)−∆∂yψ.

48 / 112

2D MHD Equations

Dissipation only


Therefore, (32) can be written as

∂tu + u − ∂xyψ = N1, (9)

∂tv + v+∂xxψ = N2, (10)

∂tψ + v = −u∂xψ − v∂yψ, (11)

where

N1 = −~u · ∇u + ∂x∆−1∇ · (~u · ∇~u)−∆ψ ∂xψ + ∂x∆−1∇ · (∆ψ∇ψ),

N2 = −~u · ∇v + ∂y∆−1∇ · (~u · ∇~u)−∆ψ ∂yψ + ∂y∆−1∇ · (∆ψ∇ψ).

49 / 112

2D MHD Equations

Dissipation only


Taking the time derivative leads to

∂ttu + ∂tu − ∂xxu = F1,

∂ttv + ∂tv − ∂xxv = F2,

∂ttψ + ∂tψ − ∂xxψ = F0,

~u|t=1 = ~u0(x , y), ~ut |t=1 = ~u1(x , y)

ψ|t=1 = ψ0(x , y), ψt |t=1 = ψ1(x , y),

(12)

where ~u1 = (u1(x , y), v1(x , y)), ψ0 = φ0 − y , and

u1 = (−u + ∂xyψ + N1)|t=1,

v1 = (−v − ∂xxψ + N2)|t=1,

ψ1 = (−u∂xψ − v∂yψ − v)|t=1,

50 / 112

2D MHD Equations

Dissipation only


and

F0 = −~u · ∇ψ − ∂t(~u · ∇ψ)−N2,

F1 = ∂tN1 − ∂xy (~u · ∇ψ),

F2 = ∂tN2 + ∂xx(~u · ∇ψ).

51 / 112

2D MHD Equations

Dissipation only


We consider the linear equation

∂ttΦ + ∂tΦ− ∂xxΦ = 0, (13)

with the initial data

Φ(0, x , y) = Φ0(x , y), Φt(0, x , y) = Φ1(x , y).

Taking the Fourier transform on the equation (13), we have

∂ttΦ + ∂tΦ + ξ2Φ = 0, (14)

where the Fourier transform Φ is defined as

Φ(t, ξ, η) =

∫R2

e ixξ+iyηΦ(t, x , y) dxdy .

52 / 112

2D MHD Equations

Dissipation only


Solving (14) by a simple ODE theory, we have

Φ(t, ξ, η) =1

2

(e

(− 1

2+√

14−ξ2)t

+ e

(− 1

2−√

14−ξ2)t)

Φ0(ξ, η)

+1

2√

14 − ξ2

(e

(− 12

+√

14−ξ2)t − e

(− 12−√

14−ξ2)t

)(1

2Φ0(ξ, η) + Φ1(ξ, η)

).

53 / 112

2D MHD Equations

Dissipation only


Definition

Let the operators K0(t, ∂x),K1(t, ∂x) be defined as

K0(t, ∂x)f (t, ξ, η) =1

2

(e

(− 1

2+√

14−ξ2)t

+ e

(− 1

2−√

14−ξ2)t)

f (t, ξ, η);

and

K1(t, ∂x)f (t, ξ, η) =1

2√

14 − ξ2

(e

(− 1

2+√

14−ξ2)t − e

(− 1

2−√

14−ξ2)t)

f (t, ξ, η).

where√−1 = i .

54 / 112

2D MHD Equations

Dissipation only


Therefore, the solution Φ of the equation (13) is written as

Φ(t, x , y) = K0(t, ∂x)Φ0 + K1(t, ∂x)(1

2Φ0 + Φ1

).

Moreover, consider the inhomogeneous equation,

∂ttΦ + ∂tΦ− ∂xxΦ = F , (15)

with initial data Φ(1, x) = Φ0, ∂tΦ(1, x) = Φ1.

55 / 112

2D MHD Equations

Dissipation only


Then we have the following standard Duhamel formula,

Φ(t, x , y) =K0(t, ∂x)Φ0 + K1(t, ∂x)(1

2Φ0 + Φ1

)(16)

+

∫ t

1K1(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 K0 and K1.

56 / 112

2D MHD Equations

Dissipation only


Lemma

Let K0,K1 be defined in Definition (4.10), then

1)∥∥|ξ|αKi (t, ·)

∥∥Lqξ(|ξ|≤ 1

2). t−

12

( 1q

+α), for any α ≥ 0,

1 ≤ q ≤ ∞, i = 0, 1.

2)∥∥∂tKi (t, ·)

∥∥Lqξ(|ξ|≤ 1

2). t−1− 1

2q , i = 0, 1.

3)∣∣Ki (t, ξ)

∣∣ . e−12t , for any |ξ| ≥ 1

2 , i = 0, 1.

4)∣∣〈ξ〉−1∂tK0(t, ξ)

∣∣, ∣∣∂tK1(t, ξ)∣∣ . e−

12t , for any |ξ| ≥ 1

2 .

57 / 112

2D MHD Equations

Dissipation only


Let X0 be the Banach space defined by the following norm

‖(~u0, ψ0)‖X0 = ‖〈∇〉N(~u0, ∇ψ0)‖L2xy

+ +‖〈∇〉6+(~u0, ψ0)‖L1xy

+ ‖〈∇〉6+(~u1, ψ1)‖L1xy,

where 〈∇〉 = (I −∆)12 , N � 1and a+ denotes a + ε for small

ε > 0.

The solution spaces X is defined by

‖(~u, ψ)‖X = supt≥1

{t−ε‖〈∇〉N(~u(t), ∇ψ(t))‖2 + t

14 ‖〈∇〉3ψ‖2

+ t14 ‖〈∇〉3ψ‖2 + t

32 ‖∂xxψ‖∞ + t

54 ‖〈∇〉2∂xxψ‖2 + t

32 ‖∂xxxψ‖2

+t32 ‖∂t~u‖∞ + t

54 ‖〈∇〉∂t~u‖2 + t‖〈∇〉∂x~u‖∞ + t

32 ‖∂x∂tv‖2

}.

58 / 112

2D MHD Equations

Dissipation only


Our main result can be stated as follows:

Theorem

There exists a small constant ε > 0 such that, if the initial data

satisfies ‖(~u0, ψ0)‖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−32 ; ‖ψ(t)‖L∞x . εt−

12 .

59 / 112

2D MHD Equations

Dissipation only


The proof of this theorem relies on the continuity argument.

Lemma (Continuity Argument)

Suppose that (~u, ψ) with the initial data (~u0, ψ0), satisfies

‖(~u, ψ)‖X . ‖(~u0, ψ0)‖X0 + C ‖~u, ψ)‖βX (18)

with β > 1. Then, there exists r0 such that, if

‖(~u0, ψ0)‖X0 . r0,

then ‖(~u, ψ)‖X . 2r0.

60 / 112

2D MHD Equations

Dissipation only


Proposition

Let K (t, ∂x) be a Fourier multiplier operator satisfying

∥∥∂αx K (t, ξ)∥∥L1ξ(|ξ|≤ 1

2)<∞,

∥∥K (t, ξ)∥∥L∞ξ (|ξ|≥ 1

2)<∞, α ≥ 0.

Then, for any space-time Schwartz function f ,

∥∥∂αx K (t, ∂x)f∥∥L∞xy

.(∥∥∂αx K (t, ξ)

∥∥L1ξ(|ξ|≤ 1

2)

+∥∥K (t, ξ)

∥∥L∞ξ (|ξ|≥ 1

2)

)×∥∥〈∇〉α+1+ε∂y f

∥∥L1xy. (19)

61 / 112

2D MHD Equations

Dissipation only


Lemma

For any s ≥ 1,

∥∥〈∇〉5|∇| 12−εF1(s, ·)∥∥L1xy. s−

32−ε ‖(~u, ψ)‖2

Y . (20)

62 / 112

2D MHD Equations

Dissipation only


Using the Duhamel formula, namely (16),

ψ(t, x , y) = K0(t, ∂x)ψ0 + K1(t, ∂x)(1

2ψ0 + ψ1) +

∫ t

1K1(t − s, ∂x)F0(s) ds.

Therefore,

‖〈∇〉∂xxψ‖∞ . ‖〈∇〉∂xxK0(t)ψ0‖∞ + ‖〈∇〉∂xxK1(t)(1

2ψ0 + ψ1)‖∞

+ ‖∫ t

1〈∇〉∂xxK1(t − s)F0(s)ds‖∞.

By Corollary 4.13 and Lemma 2,

‖〈∇〉∂xxK0(t)ψ0‖∞

.(‖∂xxK0(t, ξ)‖L1

ξ(|ξ|≤ 12

) + ‖K0(t, ξ)‖L∞ξ (|ξ|≥ 12

)

)‖〈∇〉2+ε∂xx∂yψ0‖L1

xy

.(t−

32 + e−t

)‖〈∇〉5+εψ0‖L1

xy. t−

32 ‖〈∇〉5+εψ0‖X0 .

63 / 112

2D MHD Equations

Dissipation only


Since the estimates for K0 and K1 are the same, we also have

‖〈∇〉∂xxK1(t)(1

2ψ0 + ψ1)‖∞ . t−

32

∥∥〈∇〉5+ε(1

2ψ0 + ψ1)

∥∥X0.

Moreover,∥∥∥∥∫ t

1〈∇〉∂xxK1(t − s) F0(s) ds

∥∥∥∥∞

.∫ t

1‖∂xxK1(t − s) 〈∇〉F0(s)‖∞ ds

.∫ t

2

1‖∂xxK1(t − s) 〈∇〉F0(s)‖∞ ds +

∫ t

t2

‖∂xK1(t − s) 〈∇〉∂xF0(s)‖∞ ds.

64 / 112

2D MHD Equations

Magnetic diffusion only

• Magnetic Diffusion only, ν1 = ν2 = 0, η1 = η2 > 0

The 2D MHD equations with no dissipation:ut + u · ∇u = −∇p + b · ∇b,bt + u · ∇b = η∆b + b · ∇u,∇ · u = 0, ∇ · b = 0,

(21)

The global regularity problem remains open.

65 / 112

2D MHD Equations


Global weak solutions have been established

Theorem

Let κ > 0. Let (u0, b0) ∈ H1. Then (21) has a global weak

solution (u, b) with

(u, b) ∈ L∞([0,∞); H1).

66 / 112

2D MHD Equations


In this case, we can show that (u, b) admits global H1-bound.{ωt + u · ∇ω = ν∆ω + b · ∇j ,jt + u · ∇j = b · ∇ω + 2∂xb1(∂yu1 + ∂xu2)− 2∂xu1(∂yb1 + ∂xb2).

67 / 112

2D MHD Equations


1

2

d ‖ω‖22

dt=

∫b · ∇j ω dxdy ,

1

2

d ‖j‖22

dt+ η‖∇j‖2

2 =

∫b · ∇ω j dxdy + 2

∫j ∂xb1 ∂xu2 dxdy + · · ·

Since ∫b · ∇j ω dxdy +

∫b · ∇ω j dxdy = 0,

we have, for X (t) = ‖ω(t)‖22 + ‖j(t)‖2

2,

d X (t)

dt+ 2η ‖∇j‖2

2 ≤ C ‖∇u‖2 ‖∇b‖4 ‖j‖4,

68 / 112

2D MHD Equations


‖∇u‖2 = ‖ω‖2, ‖∇b‖4 ≤ ‖j‖4, ‖j‖24 ≤ ‖j‖2 ‖∇j‖2

and Young’s inequality, we find

d X (t)

dt+ 2η ‖∇j‖2

2 ≤C

η‖ω‖2

2 ‖j‖22 + η ‖∇j‖2

2.

In particular,

d X (t)

dt+ η ‖∇j‖2

2 ≤C

η‖j‖2

2 X (t).

By Gronwall’s inequality,

X (t) + η

∫ t

0‖∇j(τ)‖2

2 dτ ≤ X (0) exp

(C

η

∫ t

0‖j‖2

2 dτ

).

69 / 112

2D MHD Equations


Remark. It remains open whether or not two H1-weak solutions

must coincide.

Remark. It remains open whether or not the H1-weak solution

becomes regular when (u0, b0) is more regular, say (u0, b0) ∈ H2.

The global regularity problem for the 2D MHD equations with only

Laplacian magnetic diffusion remains open.

70 / 112

2D MHD Equations


The main difficulty is the lack of the global bound for ‖ω‖L∞ ,

although we do have global Lp-bound.

Proposition

For any p ∈ (2,∞) and q ∈ (2,∞), the solution (u, b) obeys, for

any T > 0,

‖ω‖L∞(0,T ;Lp) ≤ C , ‖b‖Lq(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‖Lp depends on p.

71 / 112

2D MHD Equations



• Global regularity for MHD equation with (−∆)βb

Consider∂tu + u · ∇u = −∇p + b · ∇b, x ∈ R2, t > 0,

∂tb + u · ∇b + (−∆)βb = b · ∇u, x ∈ R2, t > 0,

∇ · u = 0, ∇ · b = 0, x ∈ R2, t > 0,

u(x , 0) = u0(x), b(x , 0) = b0(x), x ∈ R2,

(22)

72 / 112

2D MHD Equations



C. Cao, J. Wu, B. Yuan, The 2D incompressible

magnetohydrodynamics equations with only magnetic diffusion,

SIAM J Math Anal., 46 (2014), No. 1, 588-602.

Q. Jiu and J. Zhao, A Remark On Global Regularity of 2D

Generalized Magnetohydrodynamic Equations, arXiv:1306.2823

[math.AP] 12 Jun 2013.

73 / 112

2D MHD Equations



Theorem (C. Cao, J. Wu and B. Yuan)

Consider (22) with β > 1. Assume that (u0, b0) ∈ Hs(R2) with

s > 2, ∇ · u0 = 0, ∇ · b0 = 0 and j0 = ∇× b0 satisfying

‖∇j0‖L∞ <∞.

Then (22) has a unique global solution (u, b) satisfying, for any

T > 0,

(u, b) ∈ L∞([0,T ]; Hs(R2)), ∇j ∈ L1([0,T ]; L∞(R2))

where j = ∇× b.

74 / 112

2D MHD Equations

Vertical dissipation and horizontal magnetic diffusion

• 2D MHD with ν2 > 0 and η1 > 0

The 2D MHD equations with vertical dissipation and horizontal

magnetic diffusion

ut + u · ∇u = −∇p + ν uyy + b · ∇b,

bt + u · ∇b = η bxx + 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), 1803-1822.

75 / 112

2D MHD Equations


Theorem

Assume u0 ∈ H2(R2) and b0 ∈ H2(R2) with ∇ · u0 = 0 and

∇ · b0 = 0. Then the aforementioned MHD equations have a

unique global classical solution (u, b). In addition, (u, b) satisfies

(u, b) ∈ L∞([0,∞); H2),

ωy ∈ L2([0,∞); H1), jx ∈ L2([0,∞); H1).

76 / 112

2D MHD Equations


The main efforts are devoted to global a priori bounds in H1 and

H2. We need a lemma.

Lemma

Assume that f , g , gy , h and hx are all in L2(R2). Then,∫∫|f g h| dxdy ≤ C ‖f ‖L2 ‖g‖1/2

L2 ‖gy‖1/2L2 ‖h‖

1/2L2 ‖hx‖1/2

L2 .

77 / 112

2D MHD Equations


H1-bounds

Proposition

If (u, b) is a solution of the aforementioned MHD equations, then

‖ω(t)‖22 + ‖j(t)‖2

2 + ν

∫ t

0‖ωy (τ)‖2

2 dτ + η

∫ t

0‖jx(τ)‖2

2 dτ

≤ C (ν, η)(‖ω0‖2

2 + ‖j0‖22

)where C (ν, η) denotes a constant depending on ν and η only,

ω0 = ∇× u0 and j0 = ∇× b0.

78 / 112

2D MHD Equations


H2-bounds

Proposition

If (u, b) is a solution of the aforementioned MHD equations, then

‖∇ω(t)‖22 +‖∇j(t)‖2

2 +ν

∫ t

0‖∇ωy (τ)‖2

2 dτ +η

∫ t

0‖∇jx(τ)‖2

2 dτ

≤ C (ν, η, t)(‖∇ω0‖2

2 + ‖∇j0‖22

)where C (ν, η, t) depends on ν, η and t only.

79 / 112

2D MHD Equations

Horizontal dissipation and vertical magnetic diffusion


The 2D MHD equations with horizontal dissipation and vertical

magnetic diffusion

ut + u · ∇u = −∇p + ν uxx + b · ∇b,

bt + u · ∇b = η byy + b · ∇u,

∇ · u = 0, ∇ · b = 0.

For any initial data (u0, b0) ∈ H2, this system of equations also

possess a unique global solution.

80 / 112

2D MHD Equations

Horizontal dissipation and vertical magnetic diffusion

In fact, the case ν uxx and ηbyy can be converted into the case

νuyy and ηbxx . Set

U1(x , y , t) = u2(y , x , t), U2(x , y , t) = u1(y , x , t),

B2(x , y , t) = b1(y , x , t), B1(x , y , t) = b2(y , x , t),

P(x , y , t) = p(y , x , t).

Then U = (U1,U2), P and B = (B1,B2) satisfy

Ut + U · ∇U = −∇P + ν Uyy + B · ∇B,

Bt + U · ∇B = η Bxx + B · ∇U,

∇ · U = 0, ∇ · B = 0.

81 / 112

2D MHD Equations

Horizontal dissipation and horizontal magnetic diffusion


The 2D MHD equations with horizontal dissipation and horizontal

magnetic diffusion∂tu + u · ∇u = −∇p + ∂xxu + b · ∇b,∂tb + u · ∇b = ∂xxb + b · ∇u,∇ · u = 0, ∇ · b = 0,

(23)

where we have set ν1 = η1 = 1.

The global regularity for this case is almost obtained, but this case

appears to be very difficult.

82 / 112

2D MHD Equations


Theorem (Expected Theorem)

Assume that (u0, b0) ∈ H2(R2), ∇ · u0 = 0 and ∇ · b0 = 0. Then,

(23) has a unique global solution (u, b) satisfying, for any T > 0

and t ≤ T ,

u, b ∈ L∞([0,T ]; H2(R2)), ∂xu, ∂xb ∈ L2([0,T ]; H2(R2)).

83 / 112

2D MHD Equations


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, 2661-2681.

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

2D MHD Equations


The major effort is devoted to obtaining global bounds. We have

global bound for the L2-norm:

‖u(t)‖22 + ‖b(t)‖2

2 + 2

∫ t

0‖∂xu(τ)‖2

2dτ + 2

∫ t

0‖∂xb(τ)‖2

2dτ

= ‖u0‖22 + ‖b0‖2

2,

The trouble arises when we try to obtain the global H1-bound. If

we resort to the equations of ω and j = ∇× b,∂tω + u · ∇ω = ∂2

xω + b · ∇j ,∂t j + u · ∇j = ∂2

x j + b · ∇ω+2∂xb1(∂xu2 + ∂yu1)− 2∂xu1(∂xb2 + ∂yb1),

85 / 112

2D MHD Equations


1

2

d

dt

(‖ω‖2

2 + ‖j‖22

)+ ‖∂xω‖2

2 + ‖∂x j‖22

= 2

∫j (∂xb1(∂xu2 + ∂yu1)− 2∂xu1(∂xb2 + ∂yb1)) dxdy .

If we use the anisotropic Sobolev inequalities stated in the previous

lemma, ∫∫|f g h| dxdy ≤ C ‖f ‖2 ‖g‖

122 ‖gy‖

122 ‖h‖

122 ‖hx‖

122 ,

two terms can be bounded suitably. But∫

j∂xb1∂yu1 and∫j∂xu1 ∂yb1 can not be controlled.

86 / 112

2D MHD Equations


If we do know that ∫ T

0‖(u1, b1)‖2

∞ dt <∞, (24)

then ∣∣∣∣∫ j∂xb1∂yu1 +

∫j∂xu1 ∂yb1

∣∣∣∣ ≤ 1

2

(‖∂xω‖2

2 + ‖∂x j‖22

)+C ‖(u1, b1)‖2

∞(‖ω‖2

2 + ‖j‖22

).

Then we can close the differential inequality and get a global

bound for ‖ω‖22 + ‖j‖2

2. (24) also allows us to get a global bound

for ‖∇ω‖22 + ‖∇j‖2

2.

87 / 112

2D MHD Equations


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), 985-1004,

we bound the Lr -norm of (u1, b1) suitably.

Theorem

Let (u, b) be a solution of (23). Let 2 < r <∞. Then,

‖(u1, b1)(t)‖Lr ≤ B0

√r log r + B1(t), (25)

88 / 112

2D MHD Equations


The proof of this bound uses the symmetric structure of (23),

namely

w± = u ± b

satisfies ∂tw

+ + (w− · ∇)w + = −∇p + ∂2xw +,

∂tw− + (w + · ∇)w− = −∇p + ∂2

xw−,

∇ · w + = 0, ∇ · w− = 0.

(26)

We then bound ‖w±1 ‖Lr .

89 / 112

2D MHD Equations


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

2r

d

dt‖w +

1 ‖2r2r + (2r − 1)

∫|∂xw +

1 |2|w +

1 |2r−2

= (2r − 1)

∫p ∂xw +

1 |w+1 |

2r−2. (27)

The main effort is devoted to bounding the pressure. If we knew

that ∫ T

0‖p‖L∞ dt <∞,

then we can easily show that

‖w +1 ‖2r ≤ C

√r

90 / 112

2D MHD Equations


In order to get the bounds for the pressure, we showed that

‖(u1, b1)‖2r ≤ C1eC2 r3for any r

‖(u2, b2)(t)‖L2r ≤ 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

2D MHD Equations


We can also show that, for any s ∈ (0, 1),∫ T

0‖p(τ)‖2

Hs dτ < C .

Since

‖Λsp‖2 ≤ ‖Λs(−∆)−1∂x(w−1 ∂xw +1 + w +

1 ∂xw−1 )‖2

+‖Λs(−∆)−1∂y (w +1 ∂xw−2 + w−1 ∂xw +

2 )‖2

≤ C(‖∂xw +‖2 + ‖∂xw−‖2

) (‖w +

1 ‖ 21−s

+ ‖w−1 ‖ 21−s

),

92 / 112

2D MHD Equations


To start, we fix R > 0 (to be specified later) and write

(2r − 1)

∫p ∂xw +

1 |w+1 |

2r−2 = J1 + J2,

where

J1 = (2r−1)

∫p ∂xw +

1 |w+1 |

2r−2, J2 = (2r−1)

∫p ∂xw +

1 |w+1 |

2r−2

with p and p as defined as follows.

93 / 112

2D MHD Equations


As for the 2D Boussinesq equations, we decompose the pressure

into low and high frequency parts and bound each part accordingly.

Lemma

Let f ∈ Hs(R2) with s ∈ (0, 1). Let R ∈ (0,∞). Denote by

B(0,R) the box centered at zero with each side R and by χB(0,R)

the characteristic function on B(0,R). Write

f = f +f with f = F−1(χB(0,R)F f ) and f = F−1((1−χB(0,R))F f ),

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 .

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2D MHD Equations


Lemma

(1) For a pure constant C0 (independent of s),

‖f ‖∞ ≤C0√1− s

R1−s ‖f ‖Hs(R2),

(2) For any 2 ≤ q <∞ satisfying 1− s − 2q < 0, there is a

constant C1 independent of s, q, R and f such that

‖f ‖q ≤ C1 q R1−s− 2q ‖f ‖Hs(R2).

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2D MHD Equations


By Holder’s and Young’s inequalities, we find

|J1| ≤ (2r − 1)‖p‖∞‖|w +1 |

r−1‖2‖∂xw +1 (w +

1 )r−1‖2

≤ (2r − 1)‖p‖2∞‖|w +

1 |r−1‖2

2 +2r − 1

4‖∂xw +

1 (w +1 )r−1‖2

2.

Applying Lemma 5, we have

‖p‖∞ ≤C0√1− s

R1−s ‖p‖Hs , (30)

We will skip more details.

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2D MHD Equations


We need the bound for the L∞-norm and we have a suitable bound

for Lr -norm. The bridge is the following interpolation inequality.

Proposition

Let s > 1 and f ∈ Hs(R2). Then there exists a constant C

depending on s only such that

‖f ‖L∞(R2) ≤ C supr≥2

‖f ‖r√r log r[

log(e + ‖f ‖Hs(R2)) log log(e + ‖f ‖Hs(R2))] 1

2 .

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2D MHD Equations


Proof of the proposition on interpolation inequality: By the

Littlewood-Paley decomposition, we can write

f = SN+1f +∞∑

j=N+1

∆j f ,

where ∆j denotes the Fourier localization operator and

SN+1 =N∑

j=−1

∆j .

The definitions of ∆j and SN are now standard. Therefore,

‖f ‖∞ ≤ ‖SN+1f ‖∞ +∞∑

j=N+1

‖∆j f ‖∞.

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2D MHD Equations


We denote the terms on the right by I and II . By Bernstein’s

inequality, for any q ≥ 2,

|I | ≤ 22Nq ‖SN+1f ‖q ≤ 2

2Nq ‖f ‖q.

Taking q = N, we have

|I | ≤ 4‖f ‖N ≤ 4√

N log N supr≥2

‖f ‖r√r log r

.

By Bernstein’s inequality again, for any s > 1,

|II | ≤∞∑

j=N+1

2j‖∆j f ‖2 =∞∑

j=N+1

2−j(s−1) 2sj‖∆j f ‖2

= C 2−(N+1)(s−1) ‖f ‖Bs2,2.

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2D MHD Equations


where C is a constant depending on s only. By identifying Bs2,2

with Hs , we obtain

‖f ‖∞ ≤ 4√

N log N supr≥2

‖f ‖r√r log r

+ C 2−(N+1)(s−1) ‖f ‖Hs .

We obtain the desired inequality (31) by taking

N =

[1

s − 1log2(e + ‖f ‖Hs )

],

where [a] denotes the largest integer less than or equal to a.

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2D MHD Equations

2D MHD with fractional dissipation


Consider the 2D fractional MHD equationsut + u · ∇u + ν(−∆)αu = −∇p + b · ∇b,bt + u · ∇b + η(−∆)βb = b · ∇u,∇ · u = 0, ∇ · b = 0,u(x , 0) = u0(x), b(x , 0) = b0(x).

(31)

where

(−∆)αf (ξ) = |ξ|2αf (ξ).

The aim is at the smallest α and β for which (31) has a global

regular solution.

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2D MHD Equations


The results we have indicate three cases:

The subcritical case: α + β > 1;

The critical case: α + β = 1;

The supercritical case: α + β < 1.

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2D MHD Equations



The global regularity results we currently have are for subcritical

cases.

1 ν = 0 and β > 1C. Cao, J. Wu, B. Yuan, The 2D incompressible

magnetohydrodynamics equations with only magnetic

diffusion, SIAM J Math Anal., 46 (2014), No. 1, 588-602.

Q. Jiu and J. Zhao, Global Regularity of 2D Generalized MHD

Equations with Magnetic Diffusion, arXiv:1309.5819

[math.AP] 23 Sep 2013.

103 / 112

2D MHD Equations



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:1306.2762v1 [math.AP] 13 Jun 2013.

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2D MHD Equations



Open problems:

1) ν = 0 and β = 1

2)η = 0 and 1 ≤ α < 2

3)1 < α + β < 2, 0 < β < 1

105 / 112

2D MHD Equations



J. Wu, Generalized MHD equations, J. Differential Equations 195

(2003), 284-312.

C. Trann, X. Yu and Z. Zhai, On global regularity of 2D

generalized magnetohydrodynamic equations, J. Differential

Equations 254 (2013), 4194-4216.

B. Yuan and L. Bai, Remarks on global regularity of 2D generalized

MHD equations, arXiv:1306.2190v1 [math.AP] 11 Jun 2013.

Q. Jiu and J. Zhao, A Remark On Global Regularity of 2D

Generalized Magnetohydrodynamic Equations, arXiv:1306.2823

[math.AP] 12 Jun 2013.

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2D MHD Equations

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, 2014.

Xianpeng Hu, Global Existence for Two Dimensional Compressible

Magnetohydrodynamic Flows with Zero Magnetic Diffusivity,

arXiv:1405.0274v1 [math.AP] 1 May 2014.

107 / 112

2D MHD Equations


The 2D compressible MHD system can be written as∂tρ+∇ · (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 = ρ0(x , y), ~u|t=0 = ~u0(x , y), ~b|t=0 = ~b0(x , y).

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2D MHD Equations


This paper of Wu and Wu achieves three goals:

It establishes the global well-posedness of smooth solutions of

when the initial data (ρ0, ~u0,~b0) is smooth and close to the

equilibrium state (1,~0,~e1), where we denote ~0 = (0, 0) and

~e1 = (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

2D MHD Equations


In the 2D case, ∇ · ~b = 0 implies that for a scalar function φ,

~b = ∇⊥φ ≡(∂yφ,−∂xφ

).

With this substitution, (6) becomes∂tρ+∇ · (ρ~u) = 0, (t, x , y) ∈ R+ × R× R,∂t(ρ~u) +∇ · (ρ~u ⊗ ~u)−∆~u − λ∇(∇ · ~u) + ρ2∇ρ = −∇φ∆φ,

∂tφ+ ~u · ∇φ = 0.

(32)

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2D MHD Equations


Theorem

Assume |λ| ≤ c0 for some absolute constant c0 > 0, and let

n = ρ− 1, ψ = φ− y, n0 = ρ0 − 1, ψ0 = φ0 − y. Then there exists

a small constant δ > 0 such that, if the initial data (n0, ~u0, φ0)

satisfies ‖(n0, ~u0, ψ0)‖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−12 ; ‖~u(t)‖L∞xy . δ t−1; ‖∇ψ(t)‖L∞xy . δ t−

12 .

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2D MHD Equations


Thank You Very Much!

112 / 112

Documents

The 2D Magnetohydrodynamic Equations with Partial …helper.ipam.ucla.edu/publications/mtws1/mtws1_12215.pdfuniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv:1404.5681v1