Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations. (December 2022)

• Record Type: Journal Article
• Authors: Yin, Rong; Li, Yeping
• Is Part Of: Nonlinear analysis. Volume 68(2022)
• Publication Date: 2022-12
• Subjects: Planar rarefaction wave -- Compressible Navier–Stokes–Korteweg equations -- Zero-viscosity-capillarity limit -- Energy estimate; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103685
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.