Time-asymptotic expansion with pointwise remainder estimates for 1D viscous compressible flow

Abstract

We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier--Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are newly introduced family of waves which we call \textit{higher-order diffusion waves}. In particular, these provide accurate description of the power-law asymptotics of the solution around the origin $x=0$ where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the $L^p(\mathbb{R})$-norm for all $1\le p\le \infty$. The proof is based on pointwise estimates of Green's function.

30 pages

References in corpus (1)

• Refined pointwise estimates for solutions to the 1D barotropic compressible Navier--Stokes equations: An application to the long-time behavior of a point mass