Existence and stability of steady compressible Navier–Stokes solutions on a finite interval with noncharacteristic boundary conditions
Melinand, Benjamin; Zumbrun, Kevin (2019), Existence and stability of steady compressible Navier–Stokes solutions on a finite interval with noncharacteristic boundary conditions, Physica D: Nonlinear Phenomena, 394, 1, p. 16-25. 10.1016/j.physd.2019.01.006
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01625910v1
Journal namePhysica D: Nonlinear Phenomena
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Department of mathematics [Bloomington]
Abstract (EN)We study existence and stability of steady solutions of the isentropic compressible Navier–Stokes equations on a finite interval with noncharacteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possesses (i) a spectral gap between neutral and growing/decaying modes, and (ii) an even number of nonstable eigenvalues (with a nonnegative real part). In the case that there are no nonstable eigenvalues, i.e., of spectral stability, we show this solution to be nonlinearly exponentially stable in . Using “Goodman-type” weighted energy estimates, we establish spectral stability for small-amplitude data. For large-amplitude data, we obtain high-frequency stability, reducing stability investigations to a bounded frequency regime. On this remaining, bounded-frequency regime, we carry out a numerical Evans function study, with results again indicating universal stability of solutions.
Subjects / KeywordsStability of steady waves; Bounded domains; Isentropic gas; Evans function
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