Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations

Abstract

While the asymptotic stability of rarefaction waves for the compressible Navier--Stokes equations has been well established since the pioneering works of Matsumura--Nishihara \cite{MatsumuraNishihara1986} and Liu--Xin \cite{LiuXin1988}, the classical $L^2$ energy method yields only qualitative convergence in $L^\infty(\R)$ as $t\to\infty$ without any explicit decay rate. In this paper, we establish for the first time a standalone \(L^4\) energy estimate for perturbations of a weak rarefaction wave in the one-dimensional isentropic compressible Navier--Stokes system, thereby obtaining the first explicit algebraic time-decay rates for this model. Specifically, for small initial perturbations in $H^2(\R)$ and small wave strength $δ$, we prove that $(v-V,u-U)$ decays in $L^4(\R)$ at the rate \[ \|(v-V,u-U)(t)\|_{L^4}\le C\bigl(\|(v_0-V(0),u_0-U(0))\|_{H^2}+\sqrtδ\bigr)(1+t)^{-1/8}, \] where \((V,U)\) denotes the smooth rarefaction wave, while the relative Riemann invariants \(z_\pm\) of the perturbed system decay in \(L^\infty(\R)\) at the rate \((1+t)^{-1/6}\). The proof relies on a new local quartic energy estimate for \(z_\pm\) and \(z_\pm-Q\), where \(Q=μ(v)v_x-μ(V)V_x\), as well as a nonlocal half-space wave-interaction functional for corrected characteristic modes and a weighted higher-order derivative energy estimate coupled at the scale of the one-dimensional Nash inequality.