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Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations
Author(s) -
Da Veiga H. Beirão
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160460206
Subject(s) - barotropic fluid , mathematics , inviscid flow , bounded function , hadamard transform , euler equations , nonlinear system , mathematical analysis , domain (mathematical analysis) , compressibility , perturbation (astronomy) , pure mathematics , classical mechanics , physics , quantum mechanics , mechanics , thermodynamics
The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well‐posed problem (in the Hadamard classical sense) in spaces H k (Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if ( a n , ϕ n ) → ( a , ϕ) in H k × H k and if f n → f in ℒ 2 (0, T ; H k ), then ( v n , g n ) → ( v, g ) in (0, T ; H k × H k ). The method followed here (see also [8]) also applies to the non‐barotropic case p = p (ρ, s ) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural‐stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.