Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions

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

Annales mathématiques Blaise Pascal (2003)

  • Volume: 10, Issue: 2, page 269-296
  • ISSN: 1259-1734

Abstract

top
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 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.

How to cite

top

Jasor, Marie-Josée, and Lévi, Laurent. "Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions." Annales mathématiques Blaise Pascal 10.2 (2003): 269-296. <http://eudml.org/doc/10490>.

@article{Jasor2003,
abstract = {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 $\mathbb\{R\}^p$, $1 \le p &lt;+\infty $. 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^\infty $ together with a weak formulation of boundary conditions for scalar conservation laws.},
affiliation = {Université Blaise Pascal Laboratoire de Mathématiques Appliquées UMR 6620 CNRS 24 avenue des Landais 63117 Aubiere Cedex FRANCE; Université de Pau Laboratoire de Mathématiques Appliquées ERS 2055 CNRS BP 1155 64013 Pau Cedex FRANCE},
author = {Jasor, Marie-Josée, Lévi, Laurent},
journal = {Annales mathématiques Blaise Pascal},
keywords = {existence and uniqueness theorem; first-order hyperbolic equation; entropy solution},
language = {eng},
month = {7},
number = {2},
pages = {269-296},
publisher = {Annales mathématiques Blaise Pascal},
title = {Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions},
url = {http://eudml.org/doc/10490},
volume = {10},
year = {2003},
}

TY - JOUR
AU - Jasor, Marie-Josée
AU - Lévi, Laurent
TI - Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions
JO - Annales mathématiques Blaise Pascal
DA - 2003/7//
PB - Annales mathématiques Blaise Pascal
VL - 10
IS - 2
SP - 269
EP - 296
AB - 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 $\mathbb{R}^p$, $1 \le p &lt;+\infty $. 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^\infty $ together with a weak formulation of boundary conditions for scalar conservation laws.
LA - eng
KW - existence and uniqueness theorem; first-order hyperbolic equation; entropy solution
UR - http://eudml.org/doc/10490
ER -

References

top
  1. J.M. Ball, A Version of the Fundamental Theorem for Young Measures, PDEs and Continuum Model of Phase Transition (1995), 241-259, Springer-Verlag, Berlin Zbl0991.49500MR1036070
  2. A. Bamberger, Etude d’une équation doublement non linéaire, J. Func. Anal. 24 (1977), 148-155 Zbl0345.35059MR470490
  3. C. Bardos, A.Y. LeRoux, J.C. Nedelec, First-Order Quasilinear Equations with Boundary Conditions, Commun. in Partial Differential Equations 4 (1979), 1017-1034 Zbl0418.35024MR542510
  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. 147 (1999), 269-361 Zbl0935.35056MR1709116
  6. G. Chavent, J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation, (1986), North Holland, Amsterdam Zbl0603.76101
  7. R. J. Diperna, Measure-Valued Solutions to Conservation Laws, Arch. Rat. Mech. Anal. 88 (1985), 223-270 Zbl0616.35055MR775191
  8. R. Eymard, T. Gallouet, R. Herbin, Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation, Chin. Ann. of Math. 16B (1995), 1-14 Zbl0830.35077MR1338923
  9. R. Eymard, A. Michel, T Gallouet, R Herbin, Convergence of a Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations, Numer. Math. 92 (2002), 41-82 Zbl1005.65099MR1917365
  10. G. Gagneux, M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, 22 (1996), Springer-Verlag, Berlin Zbl0842.35126MR1616513
  11. G. Gagneux, E. Rouvre, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, Ann. Fac. Sci. Toulouse X (2001), 163-183 Zbl1027.35062MR1928992
  12. M. J. Jasor, Behaviour of a Class of Nonlinear Diffusion-Convection Equations, Adv. in Math. Sci. and Appl. 5 (1995), 631-638 Zbl0844.35049MR1361008
  13. M. J. Jasor, Perturbations singulières de problèmes aux limites, non linéaires paraboliques dégénérés-hyperboliques, Ann. Fac. Sci. Toulouse VIII (1998), 267-291 Zbl0919.35013MR1656170
  14. S. N. Kruskov, First-Order Quasilinear Equations in Several Independent Variables, Math. USSR Sb. 10 (1970), 217-243 Zbl0215.16203
  15. L. Lévi, Singular Perturbations of Unilateral Problems Arising from the Theory of Flows through Porous Media, Adv. in Math. Sci. and Appl. 9 (1999), 597-620 Zbl0965.35099MR1725675
  16. L. Lévi, Strong Variational Formulations for Bilateral Obstacle Problems for Parabolic Degenerate Equations and Singular Perturbations Properties, (2001) 
  17. M. Madaune-Tort, Un résultat de perturbations singulières pour des inéquations variationnelles dégénérées, Annali di Matematica pura et applicata IV (1982), 117-143 Zbl0523.35068MR681560
  18. J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDE’s, 4 (1996), Chapman and Hall Zbl0851.35002MR1409366
  19. C. Mascia, A. Porreta, A. Terracina, Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations, Arch. Rat. Mech. Anal. 163 (2002), 87-124 Zbl1027.35081MR1911095
  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 (1977), Springer-Verlag Zbl0446.35009MR463670
  21. L. Tartar, Compensated Compactness and Applications to Partial Differential Equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (1979), KnopsR. J.R. J. Zbl0437.35004MR584398

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.