A characteristic initial boundary value problem for a symmetric positive system

Abstract

We study the simplest maximal positive boundary value problem for symmetric positive systems in a bounded open set for which the boundary matrix is not of constant rank. To be precise, the boundary matrix changes the definiteness simply crossing an embedded manifold in the boundary which is the intersection of the boundary with a non-characteristic hypersurface. Assuming that the flow passing the hypersurface compensates for the degeneracy of the boundary matrix on the embedded manifold, we discuss the existence of regular solutions to the boundary value problem.

Inflow-outflow problems for Euler equations in a rectangular cylinder

“…Then, for all (π, w) ∈ S T there exists a unique (ρ, v) ∈ K T solving (25)-(4)- (6). Moreover, there exist two constants…”

“…where ∂ t = ∂/∂t, M is a matrix which depends on the particular inflow-outflow problem considered, see (6), (9), and ν denotes the unit outward normal to ∂Ω, when it exists; the density ρ = ρ(x, t), the velocity field v = v(t, x) = (v 1 , v 2 , v 3 ) and the pressure p = p(t, x) are unknown functions of time t ∈ (0, T ) and space variable x ∈ Ω. In (1) the density ρ is assumed to be positive for physical reasons; moreover, ρ and p are related by the equation of state p = p(ρ) where p : R + → R + is a given smooth function such that p (s) > 0 for all s > 0.…”

“…In next section we write (1) as a symmetric hyperbolic system [4]; with our assumptions it becomes partly characteristic and partly noncharacteristic: more precisely, the boundary Γ 1 is noncharacteristic while the boundary Γ 0 is characteristic of rank two. To our knowledge no general theory is available for these problems even if some partial results may be found in [6,7,8,11,13].…”

Couches limites semilinéaires

“…La proposition suivante fait le lien entre U [21], [27], [26], [20] s'intéressent au cas où l'Hypothèse 1.1 est violée. L'auteur ignore dans quelle mesure la théorie de couche limite développée précédemment peut se maintenir.…”

“…-Enoncer un théorème analogue dans un cadre de ré-gularité finie est loin d'êtreévident. Une condition suffisante aété donnée dans [9] et une condition nécessaire et suffisante dans [20].…”