Low Mach number limit for non-isentropic magnetohydrodynamic equations with ill-prepared data and zero magnetic diffusivity in bounded domains

In this article, we verify the low Mach number limit of strong solutions to the non-isentropic compressible magnetohydrodynamic equations with zero magnetic diffusivity and ill-prepared initial data in three-dimensional bounded domains, when the density and the temperature vary around constant states. Invoking a new weighted energy functional, we establish the uniform estimates with respect to the Mach number, especially for the spatial derivatives of high order. Due to the vorticity-slip boundary condition of the velocity, we decompose the uniform estimates into the part for the fast variables and the other one for the slow variables. In particular, the weighted estimates of highest-order spatial derivatives of the fast variables are crucial for the uniform bounds. Finally, the low Mach number limit is justified by the strong convergence of the density and the temperature, the divergence-free component of the velocity, and the weak convergence of other variables. The methods in this paper can be applied to singular limits of general hydrodynamic equations of hyperbolic-parabolic type, including the full Navier-Stokes equations

Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow

We justify the incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow in the framework of classical solutions. We first derive the uniform energy estimates on the Mach number \eps for both the compressible system and its differential system with respect to time under uniformly in \eps small initial data. Then, based on these uniform estimates, we pass to the limit \eps \rightarrow 0 in the compressible system, so that we establish the global classical solution of the incompressible system by the compactness arguments. Moreover, we also obtain the convergence rates associated with $L^2$-norm in the case of well-prepared initial data.Comment: 49 page