Existence and stability of solutions to the compressible Euler equations with an outer force. (February 2016)

• Record Type: Journal Article
• Authors: Tsuge, Naoki
• Is Part Of: Nonlinear analysis. Volume 27(2016)
• Page Start: 203
• Page End: 220
• Publication Date: 2016-02
• Subjects: The compressible Euler equation -- Outer force -- The compensated compactness -- Difference scheme; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.07.017
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We study the compressible Euler equation with an outer force. The global existence theorem has been proved in many papers, provided that the outer force is bounded. However, the stability of their solutions has not yet been obtained until now. Our goal in this paper is to prove the existence of a global solution without such an assumption as boundedness. Moreover, we deduce a uniformly bounded estimate with respect to the time. This yields the stability of the solution. When we prove the global existence, the most difficult point is to obtain the bounded estimate for approximate solutions. To overcome this, we employ an invariant region, which depends on both space and time variables. To use the invariant region, we introduce a modified difference scheme. To prove their convergence, we apply the compensated compactness framework.