Abstract: | 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 Hk(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (an, ?n) → (a, ?) in Hk × Hk and if fn → f in ??2(0, T;Hk), then (vn, gn) → (v, g) in ?? (0, T; Hk × Hk). 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. |