A few remarks on the transport-Stokes system
[Quelques remarques sur le système de Transport-Stokes]
Mecherbet, Amina1 ; Sueur, Franck2
1Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité, 8 Place Aurélie Nemours, F75205 Paris Cedex 13 (France)
2Department of Mathematics Maison du nombre, 6 avenue de la Fonte, University of Luxembourg, L-4364 Esch-sur-Alzette (Luxembourg)
Annales Henri Lebesgue, Tome 7 (2024), pp. 1367-1408

We consider the so-called transport-Stokes system which describes sedimentation of inertialess suspensions in a viscous flow and couples a transport equation and the steady Stokes equations in the full three-dimensional space. First we present a global existence and uniqueness result for L 1 ∩L p initial densities where p≥3. Secondly, we prove that, in the case where p>3, the flow map which describes the trajectories of these solutions is analytic with respect to time. Finally we establish the small-time global exact controllability of the transport-Stokes system. These results extend to the transport-Stokes system some results obtained for the incompressible Euler system respectively by Yudovich in [Yud63], by Chemin in [Che92, Che95] and by Coron, and Glass, in [Cor96, Gla00].

Nous considérons le système de transport-Stokes décrivant la sédimentation de particules sans inertie dans un écoulement visqueux et couplant une équation de transport aux équations stationnaires de Stokes dans l’espace tridimensionnel complet. Nous présentons d’abord un résultat global d’existence et d’unicité pour des densités initiales L 1 ∩L p où p≥3. Deuxièmement, dans le cas où p>3, nous démontrons l’analyticité des trajectoires par rapport au temps. Enfin, nous établissons la contrôlabilité exacte globale à temps court du système de transport-Stokes. Ces résultats étendent au système transport-Stokes certains résultats obtenus pour le système d’Euler incompressible respectivement par Yudovich dans [Yud63], par Chemin dans [Che92, Che95] et par Coron et Glass, dans [Cor96, Gla00].

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DOI : 10.5802/ahl.222
Classification : 76D07, 35Q49, 76T20, 35A01, 35A20, 93B05
Keywords: Stokes flow, transport equation, Suspensions, global existence and uniqueness results for PDEs, Analyticity, controllability

Mecherbet, Amina  1   ; Sueur, Franck  2

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité, 8 Place Aurélie Nemours, F75205 Paris Cedex 13 (France)
2 Department of Mathematics Maison du nombre, 6 avenue de la Fonte, University of Luxembourg, L-4364 Esch-sur-Alzette (Luxembourg)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Mecherbet, Amina; Sueur, Franck. A few remarks on the transport-Stokes system. Annales Henri Lebesgue, Tome 7 (2024), pp. 1367-1408. doi: 10.5802/ahl.222

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