Three-Scale Singular Limits of Evolutionary PDEs

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero

Convergence Rate Estimates for the Low Mach and Alfv\'en Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfv\'en numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors' previous paper, is much more complicated than for the classical two-scale singular limits

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier-Stokes-Poisson system converges strongly to the strong solution of the incompressible Navier-Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained.Comment: 21 page

Large-time behavior of non-symmetric Fokker-Planck type equations

Abstract. Large time asymptotics of the solutions to non-symmetric Fokker-Planck type equations are studied by extending the entropy method to this case. We present a modified Bakry-Emery criterion that yields covergence of the solution to the steady state in relative entropy with an explicit exponen-tial rate. In parallel it also implies a logarithmic Sobolev inequality w.r.t. the steady state measure. Explicit examples illustrate that skew-symmetric per-turbations in the Fokker Planck operator can “help ” to improve the constant in such a logarithmic Sobolev inequality. We dedicate this paper to Len Gross on the occasion of his 75th birthday. 1