A BGK approximation to nonlinear parabolic initial‐boundary value problems

Article type: Research Article

Authors: Guarguaglini, F.R. | Terracina, A.

Affiliations: Dipartimento di Matematica Pura e Applicata, Universitá degli Studi di L'Aquila, Via Vetoio, I‐67100 Coppito (L'Aquila), Italy E‐mail: guarguag@univaq.it | Dipartimento di Matematica Universitá di Roma “La Sapienza”, Piazzale Aldo Moro n. 5, I‐00185 Roma, Italy E‐mail: terracin@mat.uniroma1.it

Abstract: We present a relaxation semilinear system of conservation laws with source which approximates nonlinear parabolic equations with initial and boundary conditions. The system can be interpreted as a BGK (Bhatnagar, Gross, Krook) model with a finite number of velocities. We prove the well‐posedness of the model, a priori estimates and we obtain the convergence towards the solution of the parabolic problem. Moreover we prove a similar result for a weakly degenerate problem.

Keywords: parabolic nonlinear conservation laws, degenerate parabolic equations, singular perturbation problems, BGK models

Journal: Asymptotic Analysis, vol. 28, no. 1, pp. 75-89, 2001