Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves

Abstract.

In [2], we determined a unique "effective artificial viscosity" system approximating the behavior of the compressible Navier-Stokes equations. 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 hyperbolic 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 initial data consisting of various combinations of heat and Riesz kernels; however, the calculations turn out to be surprisingly subtle, involving cancellation not captured by standard $L^p$ estimates for the wave equation.