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Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions
Song Jiang, Qiangchang Ju, Fucai Li
TL;DR
The paper studies when weak solutions of compressible MHD approach strong solutions of incompressible MHD in a periodic domain. It proves this convergence for well-prepared and general initial data, obtaining convergence rates in the well-prepared case.
Problem
The paper addresses the incompressible limit of compressible MHD with periodic boundary conditions for both well-prepared and general initial data.
Source: §1 · p. 2Method
The proof uses energy arguments to handle loss of spatial compactness and imposes stronger regularity and a diffusion-related bound on oscillations for general data.
Source: §1 · pp. 2–3Results
Weak compressible MHD solutions converge to strong ideal or viscous incompressible solutions for well-prepared data, and to viscous or partially viscous incompressible solutions for general data.
Source: §1 · pp. 2, 4Takeaways & Limitations
The convergence results apply over intervals where the corresponding strong incompressible solution exists, and rates are obtained for well-prepared data.
Source: §1 · pp. 2–3The stated setup assumes positive shear viscosity and magnetic diffusivity, with 2µ+dλ>0.
Source: §1 · p. 1
Abstract
from arXiv · showhide
This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.
1. Introduction
The paper extends incompressible-limit results for compressible MHD to periodic domains, treating both well-prepared and general initial data. It proves convergence to strong incompressible solutions, with convergence rates for well-prepared data.
1. Introduction
- For well-prepared data, compressible weak solutions converge to strong ideal and viscous incompressible MHD solutions, with rates obtained for the ideal limit.The convergence holds on intervals where the corresponding strong solution exists.Source: §1 · pp. 2–3
- For general data, convergence is proved to incompressible MHD solutions, including the partially viscous case, under conditions controlling the initial oscillations.The oscillating parts require higher regularity than L2, and their Sobolev norm must be comparable to magnetic diffusion.Source: §1 · pp. 2, 4
- The paper extends prior Cauchy-problem results to the periodic setting.Source: §1 · p. 2
- Vanishing viscosity and magnetic diffusion remove spatial compactness, so the proof uses energy arguments to establish convergence.Source: §1 · p. 3
2. main results
The section presents incompressible-limit results for well-prepared and general initial data, with ideal, viscous, and partially viscous targets. It also describes the proof strategies and extensions to compressible Navier–Stokes equations.
2. main results
- Theorem 2.3 treats the ideal incompressible target for 0 < α + β < 2, with α, β > 0, over intervals before the strong solution’s maximal existence time.The proof combines modulated energy, weak convergence, and refined energy analysis, addressing strong coupling between fluid motion and the magnetic field.Source: §2 · p. 6
- For well-prepared data, the viscous-limit result gives strong L2-in-time, L2-in-space convergence of both velocity and magnetic-field gradients.The result holds on intervals before the maximal existence time of the incompressible solution.Source: §2 · p. 6
- For the viscous-limit proof, the authors modulate total energy and partial dissipative energy simultaneously and exploit viscous dissipation.They report better convergence rates than in Theorem 2.3 when shear viscosity and magnetic diffusion tend to positive constants.Source: §2 · p. 7
- For general initial data, the extension requires higher regularity of the oscillating parts and their Sobolev norm to be comparable to magnetic diffusion.The authors state that magnetic diffusion can balance the oscillations’ influence on the magnetic field.Source: §2 · p. 7
- For general initial data, the stated results include strong convergence of the velocity and magnetic-field gradients and strong L∞-in-time, L2-in-space convergence of the magnetic field.The theorem applies up to any time before the maximal existence time of the incompressible solution.Source: §2 · p. 7
- For the partially viscous target, the shear viscosity tends to zero while magnetic diffusion tends to a positive constant; the stated result includes strong convergence of magnetic-field gradients and the field.The result is stated for intervals before the target solution’s maximal existence time.Source: §2 · pp. 7–8
- The authors state that their arguments also yield compressible Navier–Stokes convergence to incompressible Euler or Navier–Stokes equations for general initial data.This extension is obtained by taking the magnetic field to be identically zero.Source: §2 · p. 8
3. Proof of Theorems 2.3 and 2.4
The proofs use modulated-energy estimates to establish convergence to strong incompressible MHD solutions. The viscous case additionally yields strong convergence of the velocity and magnetic-field gradients.
3. Proof of Theorems 2.3 and 2.4
- The well-prepared-data proofs combine modulated energy, weak convergence, and refined energy analysis; the viscous proof also modulates the dissipation energy.The viscous case assumes constant coefficients in the stated proof, while the case (1.7) is said to be treatable similarly.Source: §3 · pp. 8, 14
3. Proof of Theorem 2.3
- For the ideal case, the limits K and J equal H and u in L∞(0,T;L²), respectively.The proof then derives inequality (2.13) from (2.10) and (3.21).Source: §3 · pp. 9, 14
4. Proof of Theorem 2.4
- The viscous-case proof establishes strong convergence of ∇uᵋ to ∇u and ∇Hᵋ to ∇H in L²(0,T;L²(Tᵈ)).The proof combines the resulting estimate with (2.15) to obtain (2.17).Source: §3 · p. 16
4. Proof of Theorem 2.5
For general initial data, the proof targets convergence to the strong viscous incompressible MHD solution by controlling oscillations caused by the initial data. It uses filtering and establishes strong convergence of the magnetic field and its gradient.
4. Proof of Theorem 2.5
- The proof seeks to identify the limiting velocity and magnetic field with the strong viscous incompressible MHD solution by controlling initial-data oscillations.The authors use the filtering method to describe these oscillations.Source: §4 · p. 17
- Hᵋ converges strongly to H in L∞(0,T;L²(Tᵈ)), and ∇Hᵋ converges strongly to ∇H in L²(0,T;L²(Tᵈ)).These conclusions finish the proof of Theorem 2.5.Source: §4 · p. 26