an:01057384
Zbl 0882.76074
Hoff, David; Zumbrun, Kevin
Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves
EN
Z. Angew. Math. Phys. 48, No. 4, 597-614 (1997).
00041456
1997
j
76N15 35Q30 35L30 35K65
effective artificial viscosity; hyperbolic operator; Green's function; heat kernel; linearized compressible Euler equations; wave equation; Riesz kernels
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.