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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
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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
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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
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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).
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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.
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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∞,∞.
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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)
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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.
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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.
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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))
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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.
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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 .
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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 <∞.
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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);
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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
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2D MHD Equations
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), 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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
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 (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
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2D MHD Equations
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
Dissipation only
Small global solution and decay rates
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
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2D MHD Equations
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: 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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
and
F0 = −~u · ∇ψ − ∂t(~u · ∇ψ)−N2,
F1 = ∂tN1 − ∂xy (~u · ∇ψ),
F2 = ∂tN2 + ∂xx(~u · ∇ψ).
51 / 112
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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 .
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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
}.
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2D MHD Equations
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 ‖(~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 .
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2D MHD Equations
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 (~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.
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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)
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
Lemma
For any s ≥ 1,
∥∥〈∇〉5|∇| 12−εF1(s, ·)∥∥L1xy. s−
32−ε ‖(~u, ψ)‖2
Y . (20)
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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 .
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2D MHD Equations
Dissipation only
Small solutions for a system with damping
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.
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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.
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2D MHD Equations
Magnetic diffusion only
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).
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2D MHD Equations
Magnetic diffusion only
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).
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2D MHD Equations
Magnetic diffusion only
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,
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2D MHD Equations
Magnetic diffusion only
‖∇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
).
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2D MHD Equations
Magnetic diffusion only
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.
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2D MHD Equations
Magnetic diffusion only
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.
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2D MHD Equations
Magnetic diffusion only
MHD equations with (−∆)βb with β > 1
• 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)
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2D MHD Equations
Magnetic diffusion only
MHD equations with (−∆)βb 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, 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.
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2D MHD Equations
Magnetic diffusion only
MHD equations with (−∆)βb with β > 1
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.
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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.
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2D MHD Equations
Vertical dissipation and horizontal magnetic diffusion
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).
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2D MHD Equations
Vertical dissipation and horizontal magnetic diffusion
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 .
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2D MHD Equations
Vertical dissipation and horizontal magnetic diffusion
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.
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2D MHD Equations
Vertical dissipation and horizontal magnetic diffusion
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.
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2D MHD Equations
Horizontal dissipation and vertical magnetic diffusion
• 2D MHD with ν1 > 0 and η2 > 0
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.
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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.
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
• 2D MHD with ν1 > 0 and η1 > 0
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.
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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)).
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2D MHD Equations
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, 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.
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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),
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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.
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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.
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2D MHD Equations
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), 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)
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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 .
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2D MHD Equations
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
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
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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
),
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2D MHD Equations
Horizontal dissipation and horizontal magnetic diffusion
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.
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2D MHD Equations
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 ∈ 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
Horizontal dissipation and horizontal magnetic diffusion
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
Horizontal dissipation and horizontal magnetic diffusion
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
Horizontal dissipation and horizontal magnetic diffusion
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
Horizontal dissipation and horizontal magnetic diffusion
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
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 | ≤ 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
Horizontal dissipation and horizontal magnetic diffusion
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
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
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 < 1.
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2D MHD Equations
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 > 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.
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2D MHD Equations
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:1306.2762v1 [math.AP] 13 Jun 2013.
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2D MHD Equations
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 < 1
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2D MHD Equations
2D MHD with fractional dissipation
A summary of current results
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.
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2D MHD Equations
The 2D Compressible MHD with velocity dissipation
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
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 (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.
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2D MHD Equations
The 2D Compressible MHD with velocity dissipation
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
The 2D Compressible MHD with velocity dissipation
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
The 2D Compressible MHD with velocity dissipation
Thank You Very Much!
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