Partial suitable solutions for the micropolar equations and regularity properties

Diego Chamorro; David Llerena

doi:10.5802/ambp.428

Partial suitable solutions for the micropolar equations and regularity properties
[Solutions partiellement adaptées aux équations micrpolaires et leurs propriétés de régularité]

Diego Chamorro ¹ ; David Llerena ¹

¹ Université d’Évry Laboratoire de Mathématiques et Modélisation d’Évry 91 037 Évry Cedex (France)

Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 2, pp. 137-187.

Résumé
Abstract

Le système micropolaire incompressible est donné par deux équations couplées : la première équation donne l’évolution du champ de vitesse $\vec{u}$ tandis que la deuxième équation donne l’évolution du champ de microrotation $\vec{ω}$ . Dans cet article, nous étudierons la régularité des solutions faibles de ce système. Pour cela nous introduirons la nouvelle notion de solutions partiellement adaptées, qui impose des conditions uniquement pour le champ de vitesse $\vec{u}$ , ainsi, sous quelques hypothèses classiques sur la pression, nous obtiendrons un gain de régularité hölderienne pour les deux variables $\vec{u}$ et $\vec{ω}$ .

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field $\vec{u}$ while the second equation gives the evolution of the microrotation field $\vec{ω}$ . In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field $\vec{u}$ only, and under some classical hypotheses over the pressure, we will obtain a hölderian gain for both variables $\vec{u}$ and $\vec{ω}$ .

Publié le : 2025-02-18

DOI : 10.5802/ambp.428

Classification : 35B65, 35K55, 76D99
Keywords: Micropolar fluids equations, partial regularity
Mots-clés : équations des fluides micropolaires, régularité partielle

Affiliations des auteurs :

Diego Chamorro ¹ ; David Llerena ¹

¹ Université d’Évry Laboratoire de Mathématiques et Modélisation d’Évry 91 037 Évry Cedex (France)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Partial suitable solutions for the micropolar equations and regularity properties},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {137--187},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Diego Chamorro; David Llerena. Partial suitable solutions for the micropolar equations and regularity properties. Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 2, pp. 137-187. doi : 10.5802/ambp.428. https://ambp.centre-mersenne.org/articles/10.5802/ambp.428/

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