Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

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.