Sur la solution à support compact de l’equation d’Euler compressible

Abstract

Re~;u le 9 d› 1985The Cauchy problem for the compressible Euler equation is discussed with compactly supported initials. To establish the local existence of classical solutions by the aid of the theory of quasilinear symmetric hyperbolic systems, a new symmetrization is introduced which works for initials having compact support or vanishing at infinity. It is further shown that as far as the classical solution is concerned, its support does not change, and that the life span is finite for any solution except for the trivial zero solution.

Loss of regularity for supercritical nonlinear Schrödinger equations

“…As recalled above, if a 0 is compactly supported, then the ansatz that we consider remains supported in the same compact so long as the solution to (2.5) remains smooth; see [27], and also [34]. The justification of WKB analysis for short time shows that at least when σ = 1 ( [21]), or σ ∈ N and n 3 ([1]), we have, thanks to Borel lemma,…”

Existence of the unique strong solution for a class of non-Newtonian fluids with vacuum

“…This is somewhat surprising because in the context of the compressible Euler equations, there have been many works concerning the existence of a strong solution with compactly supported initial data. For these results, see [14,15,16]. …”

Geometric Optics and Boundary Layers for Nonlinear-Schrödinger Equations

“…Note that it does not seem possible to get a classical hyperbolic symmetric system (in the case ǫ = 0) involving only H and u as in the case of homogeneous pressure laws considered in [15].…”