Asymptotic stability of the combination of a viscous contact wave with two rarefaction waves for 1-D Navier-Stokes equations under periodic perturbations

Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. Perturbations of this kind oscillate in the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation as that of the initial data, but due to the degeneration of contact discontinuity, the construction is more subtle. A novel method for constructing robust ansatzes is presented. It allows the same weight function to be used on different variables and wave patterns while maintaining control over the errors. As a result, it is possible to apply this construction to contact discontinuities and composite waves. Lastly, we demonstrate that the Cauchy problem admits a unique global-in-time solution and the composite wave remains stable under space-periodic perturbations through the energy method. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we consider the time-asymptotic stability of a composite wave consisting of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations with space-periodic perturbations. The key is to construct a suitable ansatz that carries the same oscillation as the initial data, but the construction is more subtle due to the degeneration of the contact discontinuity. We propose a novel method for constructing robust ansatzes that allows for the same weight function to be used on different variables and wave patterns while maintaining control over the errors. Furthermore, we demonstrate the unique global-in-time solution and the stability of the composite wave under space-periodic perturbations through the energy method.