Global well-posedness of the partially damped 2D MHD equations via a direct normal mode method for the anisotropic linear operator Min Jun Jo Junha Kim Jihoon Lee

2025-05-06 0 0 538.19KB 37 页 10玖币

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Global well-posedness of the partially damped 2D MHD equations

via a direct normal mode method for the anisotropic linear operator

Min Jun Jo, Junha Kim, Jihoon Lee

October 20, 2022

Abstract

We prove the global well-posedness of the 2D incompressible non-resistive MHD equations

with a velocity damping term near the non-zero constant background magnetic ﬁeld. To this

end, we newly design a normal mode method of eﬀectively leveraging the anisotropy of the linear

propagator that encodes both the partially dissipative nature of the non-resistive MHD system

and the stabilizing mechanism of the underlying magnetic ﬁeld. Isolating new key quantities

and estimating them with themselves in an entangling way via the eigenvalue analysis based on

Duhamel’s formulation, we establish the global well-posedness for any initial data (v0, B0) that

is suﬃciently small in a space rougher than H4∩L1. This improves the recent work in SIAM J.

Math. Anal. 47, 2630–2656 (2015) where the similar result was obtained provided that (v0, B0)

was small enough in a space strictly embedded in H20 ∩W6,1.

1 Introduction

Plasma, the fourth state of matter after solid, liquid, and gas, accounts for the state of most

visible matter in space - investigating the dynamics of plasma is crucial for our understanding of

physics. The most famous example of the plasma entities is the Sun in our solar system. Coronal

mass ejection from the Sun, which is known to be huge release of plasma, can trigger magnetic

storms that would damage the communication satellites. There was an incident that a coronal mass

ejection actually landed on the earth and stretched out the auroral zone, see [26]. In astrophysics,

plasma dynamics is an important subject of research.

The motion of plasmas can be eﬀectively modeled [2, 5, 20] by the incompressible MHD (mag-

netohydrodynamics) equations











∂tv+κ(−∆)αv+ (v· ∇)v−(B· ∇)B− ∇p= 0,

∂tB+µ(−∆)αB+ (v· ∇)B−(B· ∇)v= 0,

∇ · v=∇ · B= 0,

(1.1)

in R2with the divergence-free initial data (v0,B0) for α≥0.Here v,B, and pdenote the velocity

vector ﬁeld, the magnetic vector ﬁeld, and the scalar pressure, respectively. The numbers κ≥0

and µ≥0 are the diﬀusion coeﬃcients, called the ﬂuid viscosity and the ohmic resistivity.

arXiv:2210.10283v1 [math.AP] 19 Oct 2022

1.1 Partial dissipation and inviscid damping for the MHD system

It has been conjectured that energy of the system (1.1) for α= 1, equipped with a nontrivial

background magnetic ﬁeld, would be dissipated at a rate independent of the resistivity µ≥0; in

[7], the conjecture was numerically backed up for the 2D case. This suggests that the stabilizing

mechanism, which stems from the presence of the background magnetic ﬁeld, be strong enough to

allow us to ignore the eﬀect of resistivity in view of energy dissipation. To see the heart of matter,

we investigate the extreme case µ= 0 of (1.1) which is the following non-resisitve MHD system











∂tv+κ(−∆)αv+ (v· ∇)v−(B· ∇)B− ∇p= 0,

∂tB+ (v· ∇)B−(B· ∇)v= 0,

∇ · v=∇ · B= 0.

(1.2)

The above system is partially dissipative, meaning that only certain types of motion are damped

within the system. Here only the ﬂuid velocity is damped by the diﬀusion term (−∆)αv. In

contrast, when κ > 0 and µ > 0, both the ﬂuid velocity and the magnetic ﬁeld are damped and

so the original MHD system (1.1) is fully dissipative. While it is well-known that the system (1.1)

is globally well-posed [28] as long as we ensure its fully dissipative nature by setting up κ > 0

and µ > 0, it remains open whether the classical solutions to the non-resistive (and so partially

dissipative) system (1.2) develop ﬁnite time singularities or not.

It turned out the diﬃculty due to the absence of the damping for Bin (1.2) can be overcome

by exploiting the stabilizing eﬀect of the underlying constant magnetic ﬁeld. Speciﬁcally, Lin, Xu,

and Zhang (2015) in [22] proved the small global well-posedness of such system (1.2) with α= 1

near the stationary state (v,B) = (0, e1). For the 3D case, see [23]. After these breakthroughs,

there have been many results regarding the eﬀect of the underlying ﬁeld (0, e1). One may refer to

[1, 6, 8, 9, 14, 18, 27, 31, 32] for the various related results.

The stabilizing mechanism of the constant background magnetic ﬁeld on Bcan be compared

with inviscid damping for the Euler equations near the Couette (or shear) ﬂow in the sense that

Bequations themselves in (1.3) are diﬀusion-less but the perturbation around (0, e1) mixes the

phase as a whole, yielding the exponential time decay on the linearized level. See in (1.4) how

the extra linear structure of ∂1Band ∂1vis obtained in a system-entangling way by adopting the

perturbative regime. One may also see [6] for the ideal MHD case.

In this paper, we focus on the particular model with a damping velocity











∂tv+v+ (v· ∇)v−(B· ∇)B− ∇p= 0

∂tB+ (v· ∇)B−(B· ∇)v= 0

∇ · v=∇ · B= 0,

(1.3)

which is an end-case (α, µ) = (0,0) of the original MHD system (1.2). One notices that the L2

energy estimate for (1.3) does not give the control over ∇vanymore unlike the α= 1 case of (1.1).

Despite such obstruction, near the stationary solution (v,B) = (0, e1), not only the global small

existence of the solutions to (1.3) but also the temporal decay of such solutions were obtained in

[29] for the relatively higher-order Sobolev spaces. The goal of this paper is to improve the result

of [29] by establishing both the small global wellposedness and the time decay of the solutions in

the lower-order Sobolev spaces.

1.2 Magnetic relaxation conjecture

To put such energy dissipation of the partially dissipative MHD system into perspective, we intro-

duce another phenomenon called magnetic relaxation, which was originally conjectured for (1.2) by

Arnol’d (1974) in [3] and then carefully discussed by Moﬀatt (1985) in [24]. The magnetic relax-

ation conjecture basically tells that the magnetic ﬁeld Bwould asymptotically converge to some

stationary Euler ﬂow while the ﬂuid particles will eventually stop due to the kinetic dissipation,

and that during such convergence the core topology of Bwould be preserved as the one of the

initial ﬁeld B0.

As justiﬁcation of our target model (1.3) in view of the relaxation conjecture, we emphasize

that even though originally the case α= 1 was the main target of [24], in the same paper, Moﬀatt

still expected that various type of dissipation, including the velocity damping case α= 0, would

do equally well in terms of the relaxation. As brifely noted in the previous section, the case α= 0

does not allow us to have control over the velocity gradient anymore unlike the full Laplacian case

α= 1; so the extreme case α= 0 might be more suitable to study magnetic relaxation, preventing

any heavy reliance on the presence of the diﬀusion term. Such a view is well-aligned with our

consideration of the speciﬁc model (1.3).

The heuristic argument suggested in [24] for the relaxation problem simply follows from the

fact that suﬃciently regular solutions to (1.1) with µ= 0 should satisfy the energy equality

dt(kvk2

L2+kBk2

L2) + κk(−∆)α

2vk2

L2= 0

which means that the ﬂuid viscosity dissipates the total energy of the solutions. But physically the

zero resistivity, i.e. the perfect conductivity, allows the magnetic ﬁeld to preserve the nontrivial

topological structure of the initial data B0over time, and thus, according to Moﬀatt in [24], the

magnetic ﬁeld energy should enjoy certain lower bound due to such inherent conﬁguration of the

magnetic ﬁeld while the velocity vgoes to 0.The magnetic ﬁeld Bwill be ﬁnally frozen once the

ﬂuid particles stop, and that moment will be the time that Battains its minimum energy because if

Bis still unfrozen then Lorentz force make the ﬂuid particles move again and so the corresponding

kinetic energy stemming from the movement will be dissipated by the viscosity anyway until the

ﬂuid particles are ultimately immobilized. We can formalize such behavior as the following.

Conjecture 1.1. Any suﬃciently regular solution (v,B)to (1.2) exhibits the magnetic relaxation,

i.e., Bconverges to some stationary Euler ﬂow ˚

Bas t→ ∞ while |v|L2decays to 0as t→ ∞.

The diﬃculty of proving the conjecture arises in the fact that we do not even know whether the

global existence result can be shown for the system (1.2) in both two and three dimensions. For

the recent local well-posedness results with α= 1, see [12], [16], and [17].

However, once we restrict ourselves to the vicinity of the simplest stationary solution (v,B) =

(0, e1), we can establish not only the global existence for our main model (1.3) but also the temporal

decay of the corresponding solutions. See Theorem A and (1.6). Both results heavily depend on

the following perturbation method. Looking at the unknowns as small perturbation around (0, e1),

we write

v= 0 + v, B=e1+B,

which transforms the original system (1.3) into our main target system











∂tv+v+ (v· ∇)v−(B· ∇)B− ∇p=∂1B,

∂tB+ (v· ∇)B−(B· ∇)v=∂1v,

∇ · v=∇ · B= 0.

(1.4)

Note that the above equations have the additional linear terms ∂1Band ∂1v. Such extra linear

structure of the equations allows us to prove the small global existence in our main theorem, and

we can get even the temporal decay of the solutions (1.5). In other words, we witness that the

solutions as small deviation from the equilibrium (0, e1) converge to the equilibrium asymptotically.

This appears to be the magnetic relaxation phenomena of [3, 24] and simultaneously it is also the

resistivity-independent energy dissipation conjectured in [7]. Then we reach the following question:

Can one actually view such resistivity-independent energy dissipation of the solutions to (1.2)

(including (1.3)) as a part of the magnetic relaxation conjecture? Equivalently, for any background

magnetic ﬁeld that is a stationary solution to the system (1.1) with zero velocity ﬁeld, will the

corresponding solutions to (1.2) be always dissipated and converge to the speciﬁed background

ﬁeld? The answer is not known, so far only the nonzero constant magnetic ﬁeld was considered.

Very recently, Beekie, Friedlander, and Vicol in [4] proved that the steady state (0, e1) is asymp-

totically stable in the so-called magnetic relaxation equations (MRE) with respect to the norm of

Hmfor m≥14. The MRE was introduced by Moﬀatt in [24, 25] to guarantee the energy dissipation

in view of the relaxation phenomena of the non-resistive MHD equations. Note that our velocity

damping case (1.3) corresponds to the MRE with γ= 0, which was the main target for the stability

analysis in [4]. See Chapter 5 in [4]. This corroborates our perspective on Conjecture 1.1 as the

generalized statement for the stabilizing mechanism induced by the presence of certain underlying

nontrivial magnetic ﬁeld. As a ﬁnal remark, such a perturbative regime is consistent with the

geometric view of [11] by Choﬀrut and ˇ

Sver´ak where the richness of the nearby steady states for

the 2D Euler equations is shown. See also [15].

1.3 Previous work

To the best of the authors’ knowledge, currently [29] is the only available result regarding the

velocity damping case (1.3). In their paper [29], Wu, Wu, and Xu (2015) attained the small global

unique existence for (1.4) even with the temporal decay properties. To discuss their result more

precisely, writing B=∇⊥ψ, one may deﬁne the norm for the class Y0for initial data by

k(v0, ψ0)kY0:= kh∇iN(v0, ψ0)kL2+kh∇i6+(v0, ψ0)kL1+kh∇i6+(v1, ψ1)kL1

where N≥20 is a large number, v1and ψ1are deﬁned in an involved way: they both contain several

nonlinear terms that concerned some wave-type linear operators. Now we state the existence part

of [29]. For simplicity, we omit the statement for the properties of the low-order derivatives. The

below is a slightly rougher version of their original theorem.

Theorem 1.2 (J. Wu, Y. Wu, X. Xu).Let N≥20 and ε < 0.01. There exists a suﬃciently small

δ > 0such that the following holds. Suppose k(v0, ψ0)kY0≤δ. Then there exists a unique global

solution pair (v, ψ)to (1.4) with B=∇⊥ψsuch that

sup

t≥1

t−εkh∇iN(v, ∇ψ)kL2<∞.

This was the ﬁrst breakthrough for the case α= 0 for (1.1) in 2D. We point out that the requirement

N≥20 appears to stem from the method they used, which will be presented shortly. The authors of

[29] utilized the wave-type linearized equations that were derived by taking the time derivative. This

method originated in [23] where Lagrangian coordinates were adopted to show anisotropic regularity

propagation for the free transport equation that emerged in the stream function formulation of (1.2)

with α= 1. Rewriting (1.2) in the corresponding Lagrangian coordinates, certain damped wave

equations of the form

∂ttΦ−∆∂tΦ−∂2

1Φ=0

were obtained as the linearized system of (1.2). In [29], although Lagrangian formulation was not

directly used unlike in [22], the linear kernel estimates were done based on a similar wave-type

system

∂ttΦ + ∂tΦ−∂2

1Φ=0

as the linearized system of (1.2).

1.4 Summary of main result

We summarize the contributions of our main result as follows.

•Leveraging anisotropy. The key diﬃculty lies in the inherent anisotropy of the linear propa-

gator. This hampers the direct analysis of the linearized equations of (1.4) via a normal mode

method; the corresponding eigenvectors are not orthogonal and so one cannot recover the orig-

inal representation of a vector from the inner products of the vector with the eigenvectors. By

introducing the inverse matrix of the eigenvector matrix, we produce the anisotropic decompo-

sitions that are opted for bringing the temporal decay out of the speciﬁc linear propagator, see

(2.5) and (2.6).

•Identiﬁcation of the key quantities. We perform a speciﬁc type of Hmenergy estimate

that pinpoints the key quantities we need to control, which are k∂1vkL∞and kB2kL∞. Those

key quantities encode the anisotropic nature of the problem; a careful use of the incompressibil-

ity condition, combined with certain calculus inequalities, allows for clarifying the directional

information more suitably in view of partial dissipation.

•Reduction of the Sobolev exponents. In [29], the required Sobolev exponent N≥20 for

the initial data was distant from the optimal Sobolev exponents that were naturally expected

for the local well-posendess, cf. [16, 17]. See also Proposition 2.7 in the next section. More

precisely, the initial data was assumed to belong to a space strictly embedded in H20 ∩W7,1;

another notable condition is the L1assumption on the higher-order derivatives, which generally

helps one leverage the linear kernel. In this work, we prove that it is suﬃcient to require the

initial data to be in a space rougher than H4∩L1.

•Non-turbulent solution. The highest H20+ norm of the solution (v, B) established in [29]

possibly grows in time with the order tε.Such growth in time implies that there could be

turbulence, which means energy transfer from low to high frequencies. Our main theorem

guarantees that the highest Sobolev norm Hmof (v, B) does not grow in time, preventing

energy transfer from low frequencies to high frequencies in the perturbative regime near the

background magnetic ﬁeld.

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Globalwell-posednessofthepartiallydamped2DMHDequationsviaadirectnormalmodemethodfortheanisotropiclinearoperatorMinJunJo,JunhaKim,JihoonLeeOctober20,2022AbstractWeprovetheglobalwell-posednessofthe2Dincompressiblenon-resistiveMHDequationswithavelocitydampingtermnearthenon-zeroconstantbackgroundmagneti...

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Global well-posedness of the partially damped 2D MHD equations via a direct normal mode method for the anisotropic linear operator Min Jun Jo Junha Kim Jihoon Lee

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