Dirichlet problems with skew-symmetric drift terms

Lucio Boccardo; Juan Casado-Diaz; Luigi Orsina

doi:10.5802/crmath.564

Research article - Partial differential equations

Dirichlet problems with skew-symmetric drift terms

Lucio Boccardo ¹; Juan Casado-Diaz ²; Luigi Orsina ³

¹ Istituto Lombardo & Sapienza Università di Roma, Italy
² Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain
³ Dipartimento di Matematica, Sapienza Università di Roma, Italy

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 301-306.

Abstract
Résumé

We prove existence of finite energy solutions for a linear Dirichlet problem with a drift and a convection term of the form $A E (x) \nabla u + div (u E (x))$ , with $A > 0$ and $E$ in ${(L^{r} (Ω))}^{N}$ . The result is obtained using a nonlinear function of $u$ as test function, in order to “cancel” this term.

Nous prouvons l’existence de solutions d’énergie finie pour un problème de Dirichlet linéaire avec un terme de la forme $A E (x) \nabla u + div (u E (x))$ , où $A > 0$ et $E$ est dans ${(L^{r} (Ω))}^{N}$ . Le résultat est obtenu en utilisant une fonction non linéaire de $u$ comme fonction test, afin d’“annuler” ce terme.

Received: 2023-08-04
Revised: 2023-09-04
Accepted: 2023-09-06
Published online: 2024-05-02

DOI: 10.5802/crmath.564

Mots-clés : Singular drift, Dirichlet problems, nonlinear test functions

Author's affiliations:

Lucio Boccardo ¹; Juan Casado-Diaz ²; Luigi Orsina ³

¹ Istituto Lombardo & Sapienza Università di Roma, Italy
² Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain
³ Dipartimento di Matematica, Sapienza Università di Roma, Italy

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Lucio Boccardo and Juan Casado-Diaz and Luigi Orsina},
     title = {Dirichlet problems with skew-symmetric drift terms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {301--306},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.564},
     language = {en},
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Lucio Boccardo; Juan Casado-Diaz; Luigi Orsina. Dirichlet problems with skew-symmetric drift terms. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 301-306. doi : 10.5802/crmath.564. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.564/

References
Cited by

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