MULTIDIMENSIONAL DIFFUSION WAVES FOR THE NAVIER-STOKES EQUATIONS OF COMPRESSIBLE FLOW

We derive a detailed, pointwise description of the asymptotic behavior of solutions of the Cauchy problem for the Navier-Stokes equation of compressible flow in several space dimensions, with initial data in L( 1) boolean AND H-k(n). We show that, asymptotically, the solution deco mposes into the sum of two terms, one of which dominates in L(p) for p >2, the other for p<2. The dominant term for p>2 has constant density and divergence-free momentum field, decaying at the rate of a heat ker nel. Thus, as measured in L(p) for p>2, all smooth, small-amplitude so lutions of the Navier-Stokes equations are asymptotically incompressib le. When p<2, the dominant term reflects instead the spreading effect of convection, and decays more slowly than a heat kernel; in fact, the solution may grow without bound in L(p) for p near 1. These features of the solution do not arise in one dimensional flow, nor are they app arent from previously known L(2) decay rates. An alternative interpret ation of our estimates shows that the solution is asymptotically given by a diffusion about the origin whose mass is that of the initial dat a, convected by the fundamental solution of the linearized Euler equat ions. This result thus defines the correct notion of ''diffusion wave' ' in the context of compressible, Navier-Stokes flows. Our analysis in volves a number of interesting issues in Fourier multiplier and Paley- Wiener theory, required for the derivation of pointwise bounds from re presentations of the Fourier transforms of Green's matrices.