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Degenerate singular parabolic problems with natural growth

El Ouardy Mounim, El Hadfi Youssef, Sbai Abdelaaziz

Opuscula Mathematica

2024

Vol. 44, no. 4

471--503

In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term [formula] where Ω is a bounded open subset of RN, N > 2, Q is the cylinder Ω × (0, T), T > 0, Γ the lateral surface ∂Ω×(0, T), 2 ≤ p < N, a(x, t) and b(x, t) are positive measurable bounded functions, q ≥ 0, 0 ≤ γ < 1, and ƒ non-negative function belongs to the Lebesgue space Lm(Q) with m > 1, and u0 ∈ L∞(Ω) such that ∀ω ⊂⊂ Ω∃Dω > 0 : u0 ≥ Dω in ω. More precisely, we study the interaction between the term uq (q > 0) and the singular lower order term d(x, t)|∇u|pu−γ (0 < γ < 1) in order to get a solution to the above problem. The regularizing effect of the term uq on the regularity of the solution and its gradient is also analyzed.

On convergence of explicit finite volume scheme for one-dimensional three-component two-phase flow model in porous media

Mostefai Mohamed Lamine, Choucha Abdelbaki, Cherif Bahri

Demonstratio Mathematica

2021

Vol. 54, nr 1

510--526

In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L1 . Results of numerical experiments are presented to validate the theoretical analysis.

Generation of analytic semi-groups in L² for a class of second order degenerate elliptic operators

Cannarsa P., Rocchetti D., Vancostenoble J.

Control and Cybernetics

2008

Vol. 37, no 4

831-878

We study the generation of analytic semigroups in the L² topology by second order elliptic operators in divergence form, that may degenerate at the boundary of the space domain. Our results, that hold in two space dimensions, guarantee that the solutions of the corresponding evolution problems support integration by parts. So, this paper provides the basis for deriving Carleman type estimates for degenerate parabolic operators.