Time-asymptotic stability of composite wave for the one-dimensional compressible fluid of Kortwewg type
Sungho Han, Jeongho Kim
TL;DR
The authors address the time-asymptotic stability of a composite wave consisting of a 1-rarefaction and a 2-viscous-dispersive shock for the one-dimensional Navier-Stokes-Korteweg system in Lagrangian coordinates. They extend the a-contraction with shift method to NSK by introducing an augmented system with an auxiliary variable, and construct a composite wave with a Lipschitz shift X(t). Through a meticulous weighted relative-entropy analysis and wave-interaction estimates, they prove global existence of a perturbed solution and show convergence to the composite wave in appropriate norms, with X′(t)→0. The results cover a broad class of capillarity and viscosity coefficients and provide a rigorous framework for nonlinear stability of NSK composite waves, advancing understanding of capillarity-influenced fluid dynamics. The work has potential implications for modeling phase-transition phenomena and quantum-fluid dynamics where capillary effects are essential.
Abstract
We study the asymptotic stability of a composition of rarefaction and shock waves for the one-dimensional barotropic compressible fluid of Korteweg type, called the Navier-Stokes-Korteweg(NSK) system. Precisely, we show that the solution to the NSK system asymptotically converges to the composition of the rarefaction wave and shifted viscous-dispersive shock wave, under certain smallness assumption on the initial perturbation and strength of the waves. Our method is based on the method of $a$-contraction with shift developed by Kang and Vasseur \cite{KV16}, successfully applied to obtain contraction or stability of nonlinear waves for hyperbolic systems.