Coupling of Multidimensional Parabolic and Hyperbolic Equations

Abstract

This paper deals with the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set, while in the complementary Ωh ≡ Ω\Ωp, only advection-reaction terms are taken into account. To begin we provide a definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh contains outward characteristics for the first-order operator in Ωh, the uniqueness proof starts by considering first the hyperbolic zone and then the parabolic one. The existence property uses the vanishing viscosity method and to pass to the limit on the hyperbolic zone, we refer to the notion of process solution.

Entropy formulations for a class of scalar conservations laws with space-discontinuous flux functions in a bounded domain

“…So we refer to [2,3] to propose a weak formulation through a global entropy inequality on the whole Q. That is why it will be said that:…”

“…But due to (3) and to the monotonicity of K h , the first integral in the right-hand side is non-positive. We deduce that, if u is a measurable and bounded function on Q satisfying (5) and (6), then for any κ in R and any ϕ of…”

Obstacle problems for a coupling of quasilinear hyperbolic-parabolic equations

“…We give the definition of a weak solution to (1)-(6) by first keeping in mind that it has to involve an entropy criterion on Q h and secondly by taking into account the obstacle condition for u. That is why, by noting that (1)-(6) can be viewed as an obstacle problem for a quasilinear parabolic evolution equation that strongly degenerates on a fixed subdomain, we make use of related work ( [1], [8]) to propose a weak formulation through a global entropy inequality on the whole Q, the latter giving rise to a variational inequality on the parabolic domain, and to an entropy inequality on the hyperbolic one so as to ensure uniqueness.…”

“…We derive from (13) and (14) an entropy inequality on the hyperbolic domain that will be the starting point to establish a Lipschitzian time-dependence in L 1 (Ω h ) of a weak solution to (1)-(6) with respect to the corresponding initial data. To do so we need a lemma proved as in [1]:…”