Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity

Abstract This article shows time-asymptotic nonlinear stability of rarefaction wave to the Cauchy problem for the one-dimensional relaxed compressible Navier-Stokes equations with density-dependent viscosity. We prove that the solution to this typical system tends time-asymptotically to the rarefaction wave. For this, we technically construct the correction function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mo>ˆ</m:mo> </m:mrow> </m:mover> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \hat{S}\left(x,t) , which means that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mo>±</m:mo> </m:mrow> </m:msub> </m:math> {S}_{\pm } can be non-zero. The proof is accomplished by virtue of energy estimates.