For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the NavierStokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the NavierStokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate (1+ t) 1 4, it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of (1+ t) 1 4. Thus, this reciprocal order of decay rates for the time

We review the existence results of traveling wave solutions to the reaction-diffusion equations with periodic diffusion (convection) coefficients and combustion (bistable) nonlinearities. We prove that whenever traveling waves exist, the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions. In the case of bistable nonlinearity and one space dimension, we give an example of nonexistence of traveling waves which causes quenching (localization) of wavefront propagation. Quenching (localization) only occurs when the variations of the media from their constant mean values are large enough. Our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example. Finally, we comment on the role of the effective wave speeds in

For periodic initial data with density allowed to vanish initially, we establish the global existence of strong and weak solutions to the two-dimensional barotropic compressible Navier–Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is λ =ρ^β with β>4/3. These results generalize and improve the previous ones due to Vaigant–Kazhikhov [Sib. Math. J. 36 (1995) 1283–1316] who required β>3. Moreover, we also prove that the densities for both the strong and weak solutions remain bounded from above independently of time. As a consequence, it is shown that both the strong and weak solutions converge to the equilibrium state as time tends to infinity.

The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.