POINTWISE DECAY-ESTIMATES FOR MULTIDIMENSIONAL NAVIER-STOKES DIFFUSION WAVES

In [2], we determined a unique ''effective artificial viscosity'' syst em approximating the behavior of the compressible Navier-Stokes equati ons. Here, we derive a detailed, pointwise description of the Green's function for this system. This Green's function generalizes the notion of ''diffusion wave'' introduced by Liu in the one-dimensional case, being expressible as a nonstandard heat kernel convected by the hyperb olic solution operator of the linearized compressible Euler equations. It dominates the asymptotic behavior of solutions of the (nonlinear) compressible Navier-Stokes equations with localized initial data. The problem reduces to determining estimates on the wave equation, with in itial data consisting of various combinations of heat and Riesz kernel s; however, the calculations turn out to be surprisingly subtle, invol ving cancellation not captured by standard L-p estimates for the wave equation.