Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3D compressible Navier-Stokes equations

Abstract

In this paper, we construct a family of global in time solutions of the 3D compressible Navier-Stokes (N-S) equations with temperature-dependent viscosity and heat-conductivity, and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of \(L^{\infty}(\R^3)\). Different from \cite{LWW2020,LWW2022}, which study the vanishing viscocity limit to rarefaction wave away from vaccum in the domain \(\Omega=\mathbb{R} \times \mathbb{T}^2\) with \(\mathbb{T}^2=\) \((\mathbb{R} / \mathbb{Z})^2\) denoting a two-dimensional unit flat torus with periodic boundary condition, the Cauchy problem in \(\R^3\) is considered in this paper. Since planar rarefaction waves are not exact solutions of compressible N-S equations, we consider perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. Different from \cite{HuangXuYuan2022}, because the viscosity is temperature-dependent and degeneracies are caused by vaccum, the a priori assumptions and two Gargliardo-Nirenberg type inequalities is essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also exponential time decay rate for the non-zero mode.