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Low Mach Number Limit of Viscous Compressible Magnetohydrodynamic Flows
Xianpeng Hu, Dehua Wang
TL;DR
The paper addresses whether compressible isentropic viscous MHD weak solutions converge to incompressible viscous MHD weak solutions in the low Mach number regime. It proves this convergence as density approaches a constant and ε → 0 across periodic, whole-space, and bounded domains, with domain-specific analytical tools. The bounded-domain results have additional assumptions associated with boundary effects.
Problem
The paper investigates the relationship between compressible and incompressible MHD when density approaches a constant and the Mach number tends to zero.
Method
The authors analyze global weak solutions using group methods, Strichartz estimates, weak convergence, and dissipative-wave spectral analysis with Duhamel’s principle.
Results
Weak solutions of the compressible equations converge to weak solutions of the incompressible MHD equations in periodic, whole-space, and bounded domains.
Takeaways & Limitations
The incompressible MHD equations are rigorously justified as the low Mach number limit of compressible isentropic viscous MHD in the stated spatial settings.
Takeaways & Limitations
In bounded domains, the stronger velocity convergence requires assumption (A), which holds for bounded simply connected planar domains with Lipschitz boundary.
Abstract
from arXiv · showhide
The relationship between the compressible magnetohydrodynamic flows with low Mach number and the incompressible magnetohydrodynamic flows is investigated. More precisely, the convergence of weak solutions of the compressible isentropic viscous magnetohydrodynamic equations to the weak solutions of the incompressible viscous magnetohydrodynamic equations is proved as the density becomes constant and the Mach number goes to zero, that is, the corresponding incompressible limits are justified when the spatial domain is a periodic domain, the whole space, or a bounded domain.
1. Introduction
The paper rigorously justifies the low Mach number limit from compressible isentropic viscous MHD to incompressible viscous MHD as density approaches a constant. It treats periodic, whole-space, and bounded spatial domains using domain-specific convergence methods.
- Scaling: The low Mach number scaling makes density variations, velocity, and magnetic field enter on different scales, with ε also related to the reciprocal Alfvén number.The viscosity and magnetic-viscosity coefficients are scaled so the limit system remains viscous rather than becoming an Euler system.
- Contribution: The paper proves convergence of weak solutions from compressible isentropic MHD to incompressible MHD as density becomes constant and ε → 0.The hydrostatic pressure in the incompressible system is identified as the limit of the scaled compressible pressure term.
- Methods: Bounded domains require spectral analysis of a dissipative wave semigroup and Duhamel’s principle because boundary layers couple dissipation, wave propagation, and magnetic effects.These boundary interactions prevent direct application of the whole-space method.
- Methods: Periodic and whole-space proofs use group methods, Strichartz estimates, and weak convergence to control oscillations and velocity components.In the whole space, the gradient velocity component converges strongly to zero, while the incompressible component has only local strong convergence.
2. Main Results
The paper states incompressible-limit results for weak solutions of compressible isentropic MHD in periodic, whole-space, and bounded domains. Convergence properties differ by domain and, in bounded domains, depend on a geometric condition controlling boundary layers.
- 2.1. The periodic case.: In periodic domains, up to a subsequence, compressible weak solutions converge to incompressible weak solutions for finite times.The projected velocity converges strongly in L2([0, T]; Lp(T)) for 1 ≤ p < 2N, while the magnetic field converges strongly in L2([0, T]; L2(T)) and weakly in L2([0, T]; H1(T)).
- 2.2. The whole space case.: In the whole space, up to a subsequence, compressible weak solutions converge to incompressible weak solutions for every finite time.The acoustic component Q uε converges strongly to zero in L2([0, T]; Lq(RN)) for 2 < q < 2N, and Hε converges strongly in L2([0, T]; L2(RN)) and weakly in L2([0, T]; H1(RN)).
- 2.3. The bounded domain case.: Assumption (A) requires every solution of the associated over-determined problem to be trivial; every bounded simply connected planar Lipschitz domain satisfies it.This condition determines whether boundary layers generate persistent oscillatory modes that prevent strong velocity convergence.
- 2.3. The bounded domain case.: In bounded domains, uε converges weakly to u and strongly when the domain satisfies assumption (A), while Hε has strong L2 and weak H1 convergence.The limiting fields satisfy no-slip boundary conditions for velocity and homogeneous boundary conditions for the magnetic field.
- 2.3. The bounded domain case.: If assumption (A) fails, boundary-generated eigenmodes can remain oscillatory, so velocity convergence is generally only weak; in bounded simply connected planar domains, uε strongly converges to zero.The bounded-domain analysis addresses boundary-layer effects absent from the periodic and whole-space settings.
3. The Periodic Case
The periodic-case proof establishes convergence of density, velocity components, and magnetic fields to incompressible MHD limits using compactness and weak-convergence arguments.
- A priori bounds and consequences: The magnetic field converges strongly in L2([0, T ]; L2(T)) and weakly in L2([0, T ]; H1(T)).Aubin-Lions compactness applies after obtaining bounds on the field and its time derivative.
- The weak convergence of Qu: The limiting velocity and magnetic field satisfy the incompressible induction and momentum equations in the sense of distributions.The magnetic nonlinear term converges distributionally, and the periodic-case proof is completed.
- A priori bounds and consequences: The density converges strongly to 1, while the divergence-free velocity component converges strongly to the incompressible velocity.The strong convergence of P uε is obtained through projected equations, bounds, and a compactness lemma.
- The weak convergence of Qu: The principal difficulty is controlling the nonlinear compressible convective term, especially its gradient-velocity component, during passage to the limit.The argument separates the divergence-free and gradient parts and uses the continuity equation to identify the limiting convection term.
- The weak convergence of Qu: A wave-group and compactness argument shows the gradient component's nonlinear contribution converges to a gradient, allowing the incompressible momentum equation to emerge.The group generated by the wave operator is an isometry on the relevant Sobolev spaces; the remaining oscillatory contribution is handled through distributional convergence.
4. The Whole Space Case
For the whole space, the proof combines uniform estimates, wave dispersive estimates, and compactness to establish convergence toward incompressible MHD.
- Strong convergence of Quε: The gradient velocity component converges strongly to 0 in L2([0, T ]; Lp(RN)) for all 2 < p < 2N/(N−2).This is obtained using the wave formulation, Duhamel’s formula, Strichartz estimates, and limits in auxiliary parameters.
- Convergence of the incompressible variables: The density converges strongly to 1 locally, while the incompressible velocity component converges weakly globally and strongly locally.Specifically, P uε converges weakly in L2([0, T ]; H1(RN)) and strongly in L2([0, T ]; L2(BR)) for every finite-radius ball BR.
- Convergence of the magnetic field: The magnetic field converges strongly locally in L2([0, T ]; L2(RN)) and weakly in L2([0, T ]; H1(RN)).Uniform energy and gradient bounds yield compactness, while Sobolev interpolation provides additional integrability.
- Convergence of the magnetic field: The limiting velocity and magnetic field satisfy the incompressible induction equation, and the magnetic Lorentz term converges distributionally.The strong magnetic-field convergence supports passage to the nonlinear term (∇×Hε) × Hε.
5. The Bounded Domain Case
In bounded smooth domains, the proof handles boundary-induced difficulties through spectral analysis of a dissipative wave operator and separates damped from nondamped modes.
- The convergence of Quε: The bounded-domain analysis uses the semigroup generated by the dissipative wave operator and spectral information for the Neumann Laplacian.An orthonormal eigenbasis is constructed to decompose the gradient velocity component into spectral modes.
- The bounded domain case: The bounded-domain limits include strong convergence of the divergence-free velocity under condition (A), and strong magnetic-field convergence in L2([0, T ]; L2(Ω)).The velocity also converges weakly in L2(Ω×(0,T)), while the magnetic field converges weakly in L2([0,T];H1(Ω)).
- The case k ∈J: The analysis distinguishes modes with instantaneous acoustic damping from modes that create no significant boundary layer or enhanced dissipation.The sets I and J are defined according to the real parts of the approximating eigenvalues.
- The convergence of Quε: The gradient velocity is split into damped and nondamped spectral terms, with the damped component converging strongly to 0.For nondamped modes, oscillatory interactions are analyzed separately and their nonlinear contribution is shown to converge to a gradient.
- The case k ∈J: For nondamped modes, the oscillatory finite sums converge to a gradient in the sense of distributions, neutralizing their contribution to the incompressible projection.The proof treats equal and distinct eigenfrequency cases using spectral structure and oscillation arguments.