It is well-known that one-dimensional isentropic gas dynamics has two elementary waves, i.e., shock wave and rarefaction wave. Among the two waves, only the rarefaction wave can be connected to a vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic NavierStokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained. The proof consists of a scaling argument and elementary energy analysis based on the underlying rarefaction wave structures.

This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions; the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium, to be precise. Together with the results of Parts I and II about the well-posedness of the Cauchy problem around the Maxwellian, we conclude this series with a satisfactory mathematical theory for the Boltzmann equation without angular cutoff.

A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.