Zero dissipation and time decay rates to the planar rarefaction wave for 3-D compressible Euler equation in the whole space

Abstract

We investigate the large-time behavior and inviscid limit of solutions to a class of three-dimensional (3D) viscous hyperbolic systems, focusing on their convergence toward a planar rarefaction wave of the 3D isentropic compressible Euler equation. We prove that the planar rarefaction wave is time-asymptotically stable and the solutions converge to the Euler planar rarefaction wave. Notably, decay rates that depends explicitly on both time and the viscosity coefficient are obtained. Moreover, the strength of the rarefaction wave can be large, and the initial perturbation is in the whole space. To the best of our knowledge, this is the first result to derive both the time-dependent and viscosity-dependent dissipation rates for rarefaction waves. Further, it is also the first result which successfully obtained the time decay rate around the planar rarefaction wave in the case of systems.

To address the error terms arising from approximating the viscous hyperbolic system with the inviscid planar rarefaction wave, we construct a new smooth viscous profile using Riemann invariants. This profile satisfies a one-dimensional (1D) parabolic system and serves as an effective approximation to the planar rarefaction wave. By applying a scaling transformation and deriving delicate time-weighted $L^2$-energy estimates, we establish a uniform convergence rate to the planar rarefaction wave. Moreover, the result is in the whole space, which means we don't need any periodic assumptions on any spatial directions.