[Paper Review] Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

Xianpeng Hu, Dehua Wang|ArXiv.org|Apr 23, 2009

Navier-Stokes equation solutions19 references3 citations

TL;DR

This paper establishes the compactness of weak solutions to the three-dimensional full compressible magnetohydrodynamic (MHD) equations with density- and temperature-dependent viscosities and heat conductivity, even when these coefficients vanish in vacuum. By deriving a new entropy identity and using strong convergence of density and temperature, the authors prove that the limit of a sequence of weak solutions remains a weak solution, ensuring existence and stability of solutions under general physical conditions.

ABSTRACT

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space $\R^3$ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a new idea to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.

Motivation & Objective

• To establish the compactness of weak solutions for the full compressible MHD equations in three dimensions.
• To address the challenge of vanishing viscosities and heat conductivity in vacuum, which complicates the analysis of weak solutions.
• To provide an alternative approach to compactness that avoids relying on a positive lower bound for shear viscosity.
• To prove that the limit of a sequence of weak solutions remains a weak solution, ensuring existence and stability of solutions.
• To derive a new entropy identity that facilitates the analysis of energy and entropy dissipation in the system.

Proposed method

• Derives a new entropy identity for the compressible MHD system to control energy and entropy dissipation.
• Uses a sequence of approximate solutions with regularized viscosities and heat conductivity to handle the degeneracy at vacuum.
• Applies strong convergence of density and temperature sequences in $L^2$ spaces to pass to the limit in nonlinear terms.
• Employs weak convergence of velocity and magnetic field gradients, along with strong convergence of $\rho_n$, $\theta_n$, and $\mathbf{H}_n$, to pass to the limit in nonlinear terms.
• Establishes convergence of the viscous and magnetic terms via uniform bounds and weak-strong convergence arguments.
• Uses the induction equation and energy conservation in the sense of distributions to verify the limit satisfies the MHD equations.

Experimental results

Research questions

• RQ1 Can weak solutions to the 3D compressible MHD equations be shown to be compact when viscosities and heat conductivity depend on density and temperature and may vanish in vacuum?
• RQ2 Is it possible to prove the existence of global weak solutions without assuming a positive lower bound on the shear viscosity?
• RQ3 How can a new entropy identity be derived to control the behavior of solutions near vacuum and ensure compactness?
• RQ4 Can the limit of a sequence of weak solutions still satisfy the MHD equations in the weak sense when coefficients degenerate?
• RQ5 What convergence properties are required for nonlinear terms like $\mathbf{u} \times \mathbf{H} \times \mathbf{H}$ and $\nu(\rho,\theta) \mathbf{H} \times (\nabla \times \mathbf{H})$ to pass to the limit?

Key findings

• The limit of a sequence of weak solutions to the 3D compressible MHD equations remains a weak solution, even when viscosities and heat conductivity vanish in vacuum.
• A new entropy identity is derived that enables control of energy and entropy dissipation in the presence of degenerate coefficients.
• Strong convergence of $\rho_n$ and $\theta_n$ in $L^2$ spaces ensures the convergence of $\kappa(\rho_n,\theta_n)\nabla\theta_n$ to $\kappa(\rho,\theta)\nabla\theta$ in the sense of distributions.
• The viscous term $\Psi_n \mathbf{u}_n$ converges to $\Psi \mathbf{u}$ in the sense of distributions, relying on strong convergence of $\rho_n^{1/3}\mathbf{u}_n$ and $\rho_n^{-1/3}\sqrt{\mu(\rho_n)}$.
• The nonlinear term $({\bf u}_n \times {\bf H}_n) \times {\bf H}_n$ converges to $({\bf u} \times {\bf H}) \times {\bf H}$ in the sense of distributions due to weak convergence of $\mathbf{u}_n$ and strong convergence of $\mathbf{H}_n$.
• The induction equation and energy conservation hold in the limit in the sense of distributions, confirming the stability of the weak solution framework.