On uniqueness techniques for degenerate convection-diffusion problems

Abstract : We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form $u_t-\div\, (a_0(\Grad w)+F(w))=f$, $u\in \beta(w)$ for a maximal monotone graph $\beta$, a Leray-Lions type nonlinearity $a_0$, a continuous convection flux $F$, and an initial condition $u|_{t=0}=u_0$. The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space. We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the ''parabolic-elliptic aspects''. We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of ''going to the boundary'' in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory.
Type de document :
Article dans une revue
International journal of dynamical systems and differential equations, 2012, 4 (1/2), pp. 3-34. 〈10.1504/IJDSDE.2012.045992〉
Liste complète des métadonnées

Littérature citée [59 références]  Voir  Masquer  Télécharger

Contributeur : Boris Andreianov <>
Soumis le : dimanche 9 janvier 2011 - 23:26:46
Dernière modification le : vendredi 6 juillet 2018 - 15:18:04
Document(s) archivé(s) le : dimanche 10 avril 2011 - 03:02:21


Fichiers produits par l'(les) auteur(s)


Distributed under a Creative Commons Paternité - Pas d'utilisation commerciale 4.0 International License




Boris Andreianov, Noureddine Igbida. On uniqueness techniques for degenerate convection-diffusion problems. International journal of dynamical systems and differential equations, 2012, 4 (1/2), pp. 3-34. 〈10.1504/IJDSDE.2012.045992〉. 〈hal-00553819〉



Consultations de la notice


Téléchargements de fichiers