Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions
Jasor, Marie-Josée ; Lévi, Laurent
Annales mathématiques Blaise Pascal, Tome 10 (2003), p. 269-296 / Harvested from Numdam
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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.

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     author = {Jasor, Marie-Jos\'ee and L\'evi, Laurent},
     title = {Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     year = {2003},
     pages = {269-296},
     doi = {10.5802/ambp.177},
     zbl = {1065.35158},
     mrnumber = {2031272},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2003__10_2_269_0}
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Jasor, Marie-Josée; Lévi, Laurent. Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions. Annales mathématiques Blaise Pascal, Tome 10 (2003) pp. 269-296. doi : 10.5802/ambp.177. http://gdmltest.u-ga.fr/item/AMBP_2003__10_2_269_0/

[1] Ball, J.M. A Version of the Fundamental Theorem for Young Measures, PDEs and Continuum Model of Phase Transition, Springer-Verlag, Berlin (1995), pp. 241-259 | MR 1036070 | Zbl 0991.49500

[2] Bamberger, A. Etude d’une équation doublement non linéaire, J. Func. Anal., Tome 24 (1977), pp. 148-155 | Article | MR 470490 | Zbl 0345.35059

[3] Bardos, C.; Leroux, A.Y.; Nedelec, J.C. First-Order Quasilinear Equations with Boundary Conditions, Commun. in Partial Differential Equations, Tome 4 (1979) no. 9, pp. 1017-1034 | Article | MR 542510 | Zbl 0418.35024

[4] Burgers, R.; Frid, H.; Karlsen, K.H. 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] Carrillo, J. Entropy Solution for Nonlinear Degenerate Problems, Arch. Rat. Mech. Anal., Tome 147 (1999) no. 2, pp. 269-361 | Article | MR 1709116 | Zbl 0935.35056

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

[7] Diperna, R. J. Measure-Valued Solutions to Conservation Laws, Arch. Rat. Mech. Anal., Tome 88 (1985) no. 3, pp. 223-270 | Article | MR 775191 | Zbl 0616.35055

[8] Eymard, R.; Gallouet, T.; Herbin, R. Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation, Chin. Ann. of Math., Tome 16B (1995) no. 1, pp. 1-14 | MR 1338923 | Zbl 0830.35077

[9] Eymard, R.; Michel, A.; Gallouet, T; Herbin, R Convergence of a Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations, Numer. Math., Tome 92 (2002) no. 1, pp. 41-82 | Article | MR 1917365 | Zbl 1005.65099

[10] Gagneux, G.; Madaune-Tort, M. Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Springer-Verlag, Berlin, Mathématiques & Applications - SMAI, Tome 22 (1996) | MR 1616513 | Zbl 0842.35126

[11] Gagneux, G.; Rouvre, E. Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, Ann. Fac. Sci. Toulouse, Tome X (2001) no. 1, pp. 163-183 | Numdam | MR 1928992 | Zbl 1027.35062

[12] Jasor, M. J. Behaviour of a Class of Nonlinear Diffusion-Convection Equations, Adv. in Math. Sci. and Appl., Tome 5 (1995) no. 2, pp. 631-638 | MR 1361008 | Zbl 0844.35049

[13] Jasor, M. J. Perturbations singulières de problèmes aux limites, non linéaires paraboliques dégénérés-hyperboliques, Ann. Fac. Sci. Toulouse, Tome VIII (1998) no. 2, pp. 267-291 | Article | Numdam | MR 1656170 | Zbl 0919.35013

[14] Kruskov, S. N. First-Order Quasilinear Equations in Several Independent Variables, Math. USSR Sb., Tome 10 (1970) no. 2, pp. 217-243 | Article | Zbl 0215.16203

[15] Lévi, L. Singular Perturbations of Unilateral Problems Arising from the Theory of Flows through Porous Media, Adv. in Math. Sci. and Appl., Tome 9 (1999) no. 2, pp. 597-620 | MR 1725675 | Zbl 0965.35099

[16] Lévi, L. Strong Variational Formulations for Bilateral Obstacle Problems for Parabolic Degenerate Equations and Singular Perturbations Properties, Université de Pau et des Pays de l’Adour, Laboratoire de Mathématiques Appliquées ERS 2055 - CNRS (2001) no. 26 (Technical report)

[17] Madaune-Tort, M. Un résultat de perturbations singulières pour des inéquations variationnelles dégénérées, Annali di Matematica pura et applicata, Tome IV (1982) no. CXXXI, pp. 117-143 | Article | MR 681560 | Zbl 0523.35068

[18] Malek, J.; Necas, J.; Rokyta, M.; Ruzicka, M. 2, Weak and Measure-Valued Solutions to Evolutionary PDE’s, Chapman and Hall (Applied Mathematics and Mathematical Computation) Tome 4 (1996) | MR 1409366 | Zbl 0851.35002

[19] Mascia, C.; Porreta, A.; Terracina, A. Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations, Arch. Rat. Mech. Anal., Tome 163 (2002) no. 2, pp. 87-124 | Article | MR 1911095 | Zbl 1027.35081

[20] Mignot, F.; Puel, J.P. Un résultat de perturbations singulières dans les inéquations variationnelles, Lecture Notes in Mathematics, Singular Perturbations and Boundary Layer Theory, Springer-Verlag (1977) | MR 463670 | Zbl 0446.35009

[21] Tartar, L.; Knops, R. J. Compensated Compactness and Applications to Partial Differential Equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman Advanced Publishing Program (1979) | MR 584398 | Zbl 0437.35004