Low Mach number limit for the Quantum-Hydrodynamics system

In this paper we deal with the low Mach number limit for the system of quantum-hydrodynamics, far from the vortex nucleation regime. More precisely, in the framework of a periodic domain and ill-prepared initial data we prove strong convergence of the solutions towards regular solutions of the incompressible Euler system. In particular we will perform a detailed analysis of the time oscillations and of the relative entropy functional related to the system.Comment: To appear in Research in the Mathematical Science

Low Mach number limit of the full Navier-Stokes equations

The low Mach number limit for classical solutions to the full Navier Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are accounted. In particular we consider general initial data. The equations leads to a singular problem, depending on a small scaling parameter, whose linearized is not uniformly well-posed. Yet, it is proved that the solutions exist and are uniformly bounded for a time interval which is independent of the Mach number Ma in (0,1], the Reynolds number Re in [1,+\infty] and the Peclet number Pe in [1,+\infty]. Based on uniform estimates in Sobolev spaces, and using a Theorem of G. Metivier and S. Schochet, we next prove that the large terms converge locally strongly to zero. It allows us to rigorously justify the well-known formal computations described in the introduction of the book of P.-L. Lions.Comment: 69 page

Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.Comment: 37page

Low Mach number limit of viscous polytropic fluid flows

This paper studies the singular limit of the non-isentropic Navier-Stokes equations with zero thermal coefficient in a two-dimensional bounded domain as the Mach number goes to zero. A uniform existence result is obtained in a time interval independent of the Mach number, provided that the initial data satisfy the "bounded derivative conditions", that is, the time derivatives up to order two are bounded initially, and Navier's slip boundary condition is imposed

All speed scheme for the low mach number limit of the Isentropic Euler equation

An all speed scheme for the Isentropic Euler equation is presented in this paper. When the Mach number tends to zero, the compressible Euler equation converges to its incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, nonphysical oscillations can be suppressed. We develop this semi-implicit time discretization in the framework of a first order local Lax-Friedrich (LLF) scheme and numerical tests are displayed to demonstrate its performances