No abstract

We consider a general class of degenerate elliptic parabolic problems associated with the equation b(v) t =div a(v, Dv)+f. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in L 1 for renormalized solutions.1999 Academic Press

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.

No abstract