[Paper Review] Incompressible limit of the compressible non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity

Song Jiang, Qiangchang Ju+1 more|arXiv (Cornell University)|Nov 12, 2011

Navier-Stokes equation solutions18 references3 citations

TL;DR

This paper establishes the incompressible limit of the compressible non-isentropic magnetohydrodynamic (MHD) equations with zero magnetic diffusivity in $[\mathbb{R}^d$ ($d=2,3$) for general initial data. By deriving uniform a priori estimates independent of the Mach number $\em>\epsilon$ and proving existence of solutions on a time interval uniform in $\em>\epsilon$, the authors show convergence to the incompressible MHD system as $\em>\epsilon \to 0$, including the limit system's structure and the role of entropy and magnetic fields.

ABSTRACT

We study the incompressible limit of the compressible non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity and general initial data in the whole space $\mathbb{R}^d$ $(d=2,3)$. We first establish the existence of classic solutions on a time interval independent of the Mach number. Then, by deriving uniform a priori estimates, we obtain the convergence of the solution to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero.

Motivation & Objective

• To analyze the incompressible limit of compressible non-isentropic MHD equations with zero magnetic diffusivity in $[\mathbb{R}^d$ ($d=2,3$).
• To establish the existence of classical solutions on a time interval independent of the Mach number $\em>\epsilon$.
• To derive uniform a priori estimates in Sobolev spaces that are independent of $\em>\epsilon$.
• To prove convergence of the compressible MHD solution to the incompressible MHD solution as $\em>\epsilon \to 0$ under general initial data conditions.
• To characterize the limit system, including the role of entropy, magnetic field, and velocity in the incompressible regime.

Proposed method

• Introduce a dimensionless Mach number $\em>\epsilon$ to scale the compressible MHD equations and study the limit as $\em>\epsilon \to 0$.
• Use the entropy $S$ and pressure $p$ as state variables, with equations of state $\rho = R(S,p)$, $\theta = \Theta(S,p)$, and assume $\partial R/\partial p > 0$.
• Rewrite the system in terms of $S$, $p$, velocity $\mathbf{u}$, and magnetic field $\mathbf{H}$, with the internal energy equation (1.8) and entropy equation (1.10) under $\kappa = 0$.
• Apply a change of variables to the momentum and energy equations to express them in terms of $q = p - p_0$ and $\mathbf{u}$, with $p_0$ being a reference pressure.
• Derive uniform $H^s$ estimates for the solution $(S^\epsilon, q^\epsilon, \mathbf{u}^\epsilon, \mathbf{H}^\epsilon)$ on a time interval $[0,T]$ independent of $\em>\epsilon$, using the magnetic diffusion term to control $\mathbf{H}^\epsilon$ terms.
• Use weak and strong convergence in $L^\infty(0,T;H^s)$ and $L^2(0,T;H^{s'})$ to identify the limit $(\bar{S}, 0, \mathbf{v}, \bar{\mathbf{H}})$ as a solution to the incompressible MHD system (5.14)–(5.17).

Experimental results

Research questions

• RQ1 Does the solution of the compressible non-isentropic MHD equations with zero magnetic diffusivity converge to that of the incompressible MHD system as the Mach number tends to zero?
• RQ2 Can classical solutions be constructed on a time interval independent of the Mach number $\em>\epsilon$ for general initial data in $[\mathbb{R}^d$?
• RQ3 What are the uniform a priori estimates for the solution in Sobolev spaces that are independent of $\em>\epsilon$?
• RQ4 How does the entropy $S$ and magnetic field $\mathbf{H}$ behave in the incompressible limit, and what is the role of the initial data's decay properties?
• RQ5 What is the precise structure of the limit system, and how is the velocity field $\mathbf{v}$ determined from the initial data in the limit?

Key findings

• For any $M_0 > 0$, there exists a time $T = T(M_0)$ such that the Cauchy problem for the compressible MHD system has a unique solution in $C^0([0,T], H^s(\mathbb{R}^d))$ with $\|\cdot\|_{H^s} \leq N(M_0)$ for all $\epsilon \in (0,1]$.
• The solution $(S^\epsilon, q^\epsilon, \mathbf{u}^\epsilon, \mathbf{H}^\epsilon)$ converges weakly in $L^\infty(0,T; H^s(\mathbb{R}^d))$ and strongly in $L^2(0,T; H^{s'}_{\text{loc}}(\mathbb{R}^d))$ to $(\bar{S}, 0, \mathbf{v}, \bar{\mathbf{H}})$ for all $s' < s$.
• The limit $(\bar{S}, \mathbf{v}, \bar{\mathbf{H}})$ solves the incompressible MHD system (5.14)–(5.17), with $\text{div}\,\mathbf{v} = 0$, $\text{div}\,\bar{\mathbf{H}} = 0$, and $\partial_t \bar{S} + (\mathbf{v} \cdot \nabla)\bar{S} = 0$.
• The velocity $\mathbf{v}$ is determined by the initial data $\mathbf{v}_0$ through the condition $\text{div}\,\mathbf{w}_0 = 0$ and $\text{curl}(r(S_0,0)\mathbf{w}_0) = \text{curl}(r(S_0,0)\mathbf{v}_0)$, where $r(S_0,0) = \lim_{\epsilon \to 0} r^\epsilon(S^\epsilon_0, 0)$.
• The pressure gradient $\nabla \hat{\pi}$ is in $C([0,T], H^{s-1}(\mathbb{R}^d))$, ensuring regularity of the limit pressure.