Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

Abstract

19 pagesWe consider the general degenerate hyperbolic-parabolic equation: \begin{equation}\label{E}\tag{E} u_t+\div f(u)-\Delta\phi(u)=0 \mbox{ in } Q = (0,T)\times\Omega,\;\;\;\; T>0,\;\;\;\Omega\subset\mathbb R^N ; \end{equation} with initial condition and the zero flux boundary condition. Here $\phi$ is a continuous non decreasing function. Following [B\"{u}rger, Frid and Karlsen, J. Math. Anal. Appl, 2007], we assume that $f$ is compactly supported (this is the case in several applications) and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension $N\geq 1$ under a partial genuine nonlinearity assumption on $f$. Uniqueness is shown for the case $N=1$, using the idea of [Andreianov and Bouhsiss, J. Evol. Equ., 2004], nonlinear semigroup theory and a specific regularity result for one dimension

Convergence of Finite Volume Scheme for Degenerate Parabolic Problem with Zero Flux Boundary Condition

“…(the latter means the space of measurable on Ω functions with values in [0, u max ]). In the work [4], inspired by [7] we proposed a new entropy formulation of (P) saying that u ∈ L ∞ (Q; [0, u max ]) is an entropy solution of (P) if u ∈ C([0, T ]; L 1 (Ω )) with u(0) = u 0 , φ (u) ∈ L 2 (0, T ; H 1 (Ω )) and ∀k ∈ [0, u max ] (1) in D ((0, T ) × Ω ), where η is the exterior unit normal vector to the boundary Σ = (0, T ) × ∂ Ω and the last term is taken with respect to the Hausdorff measure H −1 on Σ . Contrary to the Dirichlet case (cf.…”

“…But in order to prove uniqueness, one faces a serious difficulty (not relevant in the case u c = u max , [7]) related to the lack of regularity of the flux F [u] := f (u) − ∇φ (u) and specifically, to the weak sense in which the normal component F [u].η of the flux annulates on Σ . Techniques of nonlinear semigroup theory (see, e.g., [6,5]) can be used to circumvent this regularity problem in some cases (see [3,4]) and to prove well-posedness for (P) in the sense (1). Let us present the key arguments: indeed, they are also important for study of convergence of the Finite Volume scheme for (P), which is the goal of this note.…”

“…Let us stress that the question of uniqueness for (P) with u c / ∈ {0, u max } and > 1 remains open. The one-dimensional hyperbolic-parabolic case ( = 1, Ω = (a, b) with arbitrary u c ∈ [0, u max ]) has been treated by the authors in [4], using the above abstract approach along with the elementary observation that yields trace-regularity:…”

“…Hence, we found it useful to define the new notion of integral-process solution in the framework of abstract problem (3). Following the pattern of the uniqueness proofs in [3,4], we compare an entropy-process solution of (P) and a trace regular solution of (S), then we prove that an entropy-process solution of (P) is an integral-process solution of (3) defined for an appropriate m-accretive operator A f ,φ . The convergence result holds due to the fact that the integral-process solution coincides with the unique integral solution of (3); and the latter one coincides with the unique entropy solution of (P) in the sense (1).…”

The Semigroup Approach to Conservation Laws with Discontinuous Flux

“…Then the notion of integral solution can be exploited, following [12,14], as it was done in [16,3,6] in various contexts. Indeed, u is an integral solution of (AbEv) if the comparison inequality in D ′ (0, T ) holds:…”