Annales Mathématiques
Blaise Pascal
no. 2
Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions
Marie-Josée Jasor1 ; Laurent Lévi2
1 Université Blaise Pascal Laboratoire de Mathématiques Appliquées UMR 6620 CNRS 24 avenue des Landais 63117 Aubiere Cedex FRANCE
2 Université de Pau Laboratoire de Mathématiques Appliquées ERS 2055 CNRS BP 1155 64013 Pau Cedex FRANCE
Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 269-296.

We establish a singular perturbation property for a class of quasilinear parabolic degenerate equations associated with a mixed Dirichlet-Neumann boundary condition in a bounded domain of ℝ p , 1≤p<+∞. In order to prove the L 1 -convergence of viscous solutions toward the entropy solution of the corresponding first-order hyperbolic problem, we refer to some properties of bounded sequences in L ∞ together with a weak formulation of boundary conditions for scalar conservation laws.

DOI : 10.5802/ambp.177
Marie-Josée Jasor 1 ; Laurent Lévi 2

1 Université Blaise Pascal Laboratoire de Mathématiques Appliquées UMR 6620 CNRS 24 avenue des Landais 63117 Aubiere Cedex FRANCE
2 Université de Pau Laboratoire de Mathématiques Appliquées ERS 2055 CNRS BP 1155 64013 Pau Cedex FRANCE 
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Marie-Josée Jasor; Laurent Lévi. Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions. Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 269-296. doi : 10.5802/ambp.177. https://ambp.centre-mersenne.org/articles/10.5802/ambp.177/

[1] J.M. Ball A Version of the Fundamental Theorem for Young Measures, PDEs and Continuum Model of Phase Transition, Springer-Verlag, Berlin, 1995, pp. 241-259 | MR | Zbl

[2] A. Bamberger Etude d’une équation doublement non linéaire, J. Func. Anal., Volume 24 (1977), pp. 148-155 | DOI | MR | Zbl

[3] C. Bardos; A.Y. LeRoux; J.C. Nedelec First-Order Quasilinear Equations with Boundary Conditions, Commun. in Partial Differential Equations, Volume 4 (1979) no. 9, pp. 1017-1034 | DOI | MR | Zbl

[4] R. Burgers; H. Frid; K.H. Karlsen On a Free Boundary Problem for a Strongly Degenerate Quasilinear Parabolic Equation with an Application to a Model of Pressure Filtration, Web Site Conservation Laws http://www.math.ntnu.no/conservation/, 2002

[5] J. Carrillo Entropy Solution for Nonlinear Degenerate Problems, Arch. Rat. Mech. Anal., Volume 147 (1999) no. 2, pp. 269-361 | DOI | MR | Zbl

[6] G. Chavent; J. Jaffré Mathematical Models and Finite Elements for Reservoir Simulation, North Holland, Amsterdam, 1986

[7] R. J. Diperna Measure-Valued Solutions to Conservation Laws, Arch. Rat. Mech. Anal., Volume 88 (1985) no. 3, pp. 223-270 | DOI | MR | Zbl

[8] R. Eymard; T. Gallouet; R. Herbin Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation, Chin. Ann. of Math., Volume 16B (1995) no. 1, pp. 1-14 | MR | Zbl

[9] R. Eymard; A. Michel; T Gallouet; R Herbin Convergence of a Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations, Numer. Math., Volume 92 (2002) no. 1, pp. 41-82 | DOI | MR | Zbl

[10] G. Gagneux; M. Madaune-Tort Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques & Applications - SMAI, 22, Springer-Verlag, Berlin, 1996 | MR | Zbl

[11] G. Gagneux; E. Rouvre Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, Ann. Fac. Sci. Toulouse, Volume X (2001) no. 1, pp. 163-183 | Numdam | MR | Zbl

[12] M. J. Jasor Behaviour of a Class of Nonlinear Diffusion-Convection Equations, Adv. in Math. Sci. and Appl., Volume 5 (1995) no. 2, pp. 631-638 | MR | Zbl

[13] M. J. Jasor Perturbations singulières de problèmes aux limites, non linéaires paraboliques dégénérés-hyperboliques, Ann. Fac. Sci. Toulouse, Volume VIII (1998) no. 2, pp. 267-291 | DOI | Numdam | MR | Zbl

[14] S. N. Kruskov First-Order Quasilinear Equations in Several Independent Variables, Math. USSR Sb., Volume 10 (1970) no. 2, pp. 217-243 | DOI | Zbl

[15] L. Lévi Singular Perturbations of Unilateral Problems Arising from the Theory of Flows through Porous Media, Adv. in Math. Sci. and Appl., Volume 9 (1999) no. 2, pp. 597-620 | MR | Zbl

[16] L. Lévi Strong Variational Formulations for Bilateral Obstacle Problems for Parabolic Degenerate Equations and Singular Perturbations Properties (2001) no. 26 (Technical report)

[17] M. Madaune-Tort Un résultat de perturbations singulières pour des inéquations variationnelles dégénérées, Annali di Matematica pura et applicata, Volume IV (1982) no. CXXXI, pp. 117-143 | DOI | MR | Zbl

[18] J. Malek; J. Necas; M. Rokyta; M. Ruzicka 2, Weak and Measure-Valued Solutions to Evolutionary PDE’s (Applied Mathematics and Mathematical Computation), Volume 4 (1996) | MR | Zbl

[19] C. Mascia; A. Porreta; A. Terracina Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations, Arch. Rat. Mech. Anal., Volume 163 (2002) no. 2, pp. 87-124 | DOI | MR | Zbl

[20] F. Mignot; J.P. Puel Un résultat de perturbations singulières dans les inéquations variationnelles, Lecture Notes in Mathematics, Singular Perturbations and Boundary Layer Theory, Springer-Verlag, 1977 | MR | Zbl

[21] L. Tartar; R. J. Knops Compensated Compactness and Applications to Partial Differential Equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman Advanced Publishing Program, 1979 | MR | Zbl

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