In this paper, we investigate the initial-boundary value problem to the heat conductive compressible Navier-Stokes equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial density $\rho_0$, velocity $u_0$, and temperature $\theta_0$ satisfy $\rho_0\in W^{1,q}$, with $q\in(3,6)$, $u_0\in H^1$, and $\sqrt{\rho_0}\theta_0\in L^2$. The initial density is assumed to be only nonnegative and thus the initial vacuum is allowed. In addition to the necessary regularity assumptions, we do not require any initial compatibility conditions such as those proposed in (Y. Cho and H. Kim, \emph{Existence results for viscous polytropic fluids with vacuum}, J. Differential Equations {\bf 228} (2006), no.~2, 377--411.), which although are widely used in many previous works but put some inconvenient constraints on the initial data. Due to the weaker regularities of the initial data and the absence of the initial compatibility conditions, leading to weaker regularities of the solutions compared with those in the previous works, the uniqueness of solutions obtained in the current paper does not follow from the arguments used in the existing literatures. Our proof of the uniqueness of solutions is based on the following new idea of two-stages argument: (i) showing that the difference of two solutions (or part of their components) with the same initial data is controlled by some power function of the time variable; (ii) carrying out some singular-in-time weighted energy differential inequalities fulfilling the structure of the Gr\"onwall inequality. The existence is established in the Euler coordinates, while the uniqueness is proved in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

In this paper, we first show that the regular solutions of compressible Euler equations in<i>R</i>³with damping will not be global if the initial density function has compact support. This implies that<i>c</i>²cannot be smooth across the boundary<i></i>separating the gas and the vacuum after a finite time, where<i>c</i>is the speed of sound. Then we study the local existence of solutions for isentropic gas flow in<i>R</i>when<i>c</i>, 0&lt;<i></i>1, is smooth across<i></i>, using the energy method and the characteristics method.

Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].