In this paper we study the distribution of scattering resonances for a multi-dimensional semi-classical Schrödinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier--Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the -norm of the initial perturbation) for some positive constant independent of . Here is the adiabatic gas constant. This is a Nishida--Smoller type global solvability result with large data.

For the class of graph-directed self-similar measures on R, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

We justify the smoothing effect for measure valued solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure valued initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.

We show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. To illustrate the advantages of this view, we provide a simple proof of the unconditional stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions.