We consider a general class of degenerate elliptic-parabolic problems associated with the equation b(v)t = div a(v; Dv) + f . Existence of renormalized solutions is established for general L 1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: u t + div Φ(u) = f on Q = (0, T ) × Ω, u(0, ·) = u 0 on Ω and "u = a on some part of the boundary (0, T ) × ∂Ω." Existence and uniqueness of the entropy solution is established for any Φ ∈ C(R; R N ), u 0 ∈ L ∞ (Ω), f ∈ L ∞ (Q), a ∈ L ∞ ((0, T ) × ∂Ω). In the L 1 -setting, a corresponding result is proved for the more general notion of renormalised entropy solution.

We are interested in parabolic problems with L 1 data of the typeHere, is an open bounded subset of R N with regular boundary ∂ and a: × R N → R N is a Caratheodory function satisfying the classical Leray-Lions conditions and β is a monotone graph in R 2 with closed domain and such that 0 ∈ β(0).We study these evolution problems from the point of view of semi-group theory, then we identify the generalized solution of the associated Cauchy problem with the entropy solution of (P i,j (φ, ψ, β)) in the usual sense introduced in [5].

No abstract

No abstract