Inflow-outflow Problems For Euler Equations In A Rectangular Cylinder

Inflow-outflow problems for Euler equations in a rectangular cylinder FILIPPO GAZZOLA Dipartimento di Scienze T.A. - via Cavour 84, 15100 Alessandria, Italy PAOLO SECCHI Dipartimento di Matematica - via Valotti 9, 25133 Brescia, Italy Abstract We prove that some inflow-outflow problems for the Euler equations in a (nonsmooth) bounded cylinder admit a regular solution. The problems considered are symmetric hyperbolic systems with partly characteristic and partly noncharacteristic boundary; for such problems, no general theory is available. Therefore, we introduce particular spaces of functions satisfying suitable additional boundary conditions which allow to determine a regular solution by means of a “reflection technique”. 1 Introduction Let Ω be an open bounded cylinder in IR3 having a rectangular section: more precisely, let Ω be the cartesian product of an open rectangle R with a bounded interval (a, b) (Ω = R × (a, b)) and let Γ1 = R × {a, b} and Γ0 = ∂R × [a, S b] so that the piecewise smooth boundary ∂Ω may be characterized by ∂Ω = Γ1 Γ0 . In the cylinder Ω we consider the following initial-boundary value problem for the Euler equations for a barotropic inviscid compressible fluid ∂t ρ + ∇ · (ρv) = 0 in [0, T ] × Ω ρ(∂t v + (v · ∇)v − f ) + ∇p = 0 in [0, T ] × Ω M (ρ, v) = G on (0, T ) × Γ1 (1) v·ν =0 on (0, T ) × Γ0 ρ(0, x) = ρ0 (x) in Ω v(0, x) = v (x) in Ω , 0 1 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) = (v1 , v2 , v3 ) 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