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For the viscous and heat-conductive fluids governed by the compressible NavierStokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the L^p - L^q estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L²-norm are obtained when the L¹-norm of the perturbation is bounded.

We study a biodegradation model for the time evolution of concentrations of contaminant, nutrient, and bacteria. The bacteria has a natural concentration which will increase when the nutrient reaches the substrate (contaminant). The growth utilizes nutrients and degrades the substrate. Eventually, such a process removes all the substrate and can be described by traveling wave solutions. The model consists of advection-reaction-diffusion equations for the substrate and nutrient concentrations and a rate equation (ODE) for the bacteria concentration. We first show the existence of approximate traveling wave solutions to an elliptically regularized system posed on a finite domain using degree theory and the elliptic maximum principle. To prove that the approximate solutions do not converge to trivial solutions, we construct comparison functions for each component and employ integral identities of the governing

We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de- veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well- posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

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.