Spherically symmetric solutions to a model for phase transitions driven by configurational forces

We prove the global in time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. The problem models the behavior in time of materials in which martensitic phase transitions, driven by configurational forces, take place, and can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multidimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the reduced problem is one space dimensional.Comment: 25 page

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