## Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves.(English)Zbl 0882.76074

Summary: Earlier 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” 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 deriving estimates for 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.

### MSC:

 76N15 Gas dynamics (general theory) 35Q30 Navier-Stokes equations 35L30 Initial value problems for higher-order hyperbolic equations 35K65 Degenerate parabolic equations
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