Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the <i>L</i> ^<i>p</i>-<i>L</i> ^<i>q</i> estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension <i>n</i>3, the

The nonlinear stability in the {L^p} -norm, p ≥ 1 , of stationary weak discrete shocks for the Lax-Friedrichs scheme approximating general m × m systems of nonlinear hyperbolic conservation laws is proved, provided that the summations of the initial perturbations equal zero. The result is proved by using both a weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.

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We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

We prove the existence of global solutions to a coupled system of NavierStokes, and reaction-diffusion equations (for temperature and mass fraction) with prescribed front data on an infinite vertical strip or tube. This system models a one-step exothermic chemical reaction. The heat release induced volume expansion is accounted for via the Boussinesq approximation. The solutions are time dependent moving fronts in the presence of fluid convection. In the general setting, the fronts are subject to intensive Rayleigh-Taylor and thermal-diffusive instabilities. Various physical quantities, such as fluid velocity, temperature, and front speed, can grow in time. We show that the growth is at most for large time <i>t</i> by constructing a nonlinear functional on the temperature and mass fraction components. These results hold for arbitrary order reactions in two space dimensions and for quadratic and cubic reactions in