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

• Published in: Journal of Differential Equations Vol. 346; pp. 254 - 276
• Main Authors: Liu, Lingjun, Wang, Danli, Xu, Lingda
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.02.2023
• Subjects: Composite wave; Navier-Stokes equations; Periodic perturbation; Composite wave; 35Q30; 76N30; 35L65; 76N10; Navier-Stokes equations; Periodic perturbation
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2022.11.040

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.