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  • Journées équations aux dérivées partielles
  • Année 2018
  • Exposé no. 5
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On some coupled PDE-ODE systems in fluid dynamics
Evelyne Miot1
1 CNRS-Université Grenoble Alpes Institut Fourier UMR 5582 100, rue des mathématiques 38610 Gières France
Journées équations aux dérivées partielles (2018), Exposé no. 5, 13 p.
  • Résumé

In this note we will present some existence and uniqueness issues for three coupled PDE-ODE systems. The common frame is that they arise as the asymptotical dynamics of a regular, incompressible two-dimensional flow interacting with:

  • points at which the vorticity is highly concentrated (point vortices);
  • an obstacle shrinking to a steady point;
  • rigid bodies contracting to moving massive particles.

We will mainly focus on the last situation, corresponding to the article [11], which is a joint work with Christophe Lacave.

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Publié le : 2019-06-19
DOI : 10.5802/jedp.665
Affiliations des auteurs :
Evelyne Miot 1

1 CNRS-Université Grenoble Alpes Institut Fourier UMR 5582 100, rue des mathématiques 38610 Gières France
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     author = {Evelyne Miot},
     title = {On some coupled {PDE-ODE} systems in fluid dynamics},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:5},
     pages = {1--13},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2018},
     doi = {10.5802/jedp.665},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.665/}
}
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Evelyne Miot. On some coupled PDE-ODE systems in fluid dynamics. Journées équations aux dérivées partielles (2018), Exposé no. 5, 13 p. doi : 10.5802/jedp.665. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.665/
  • Bibliographie
  • Cité par

[1] Luigi Ambrosio Transport equation and Cauchy problem for non-smooth vector fields, Calculus of variations and nonlinear partial differential equations (Lecture Notes in Mathematics), Volume 1927, Springer, 2008, pp. 1-42 | Zbl

[2] Luigi Ambrosio Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications (2017) (preprint)

[3] Gianluca Crippa; Camillo De Lellis Estimates and regularity results for the DiPerna–Lions flow, J. Reine Angew. Math., Volume 616 (2008), pp. 15-46 | Zbl

[4] Gianluca Crippa; Lopes Milton C. Filho; Evelyne Miot; Helena J. Nussenzveig Lopes Flows of vector fields with point singularities and the vortex-wave system, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 5, pp. 2405-2417 | Zbl

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[6] Olivier Glass; Christophe Lacave; Franck Sueur On the motion of a small body immersed in a two dimensional incompressible perfect fluid, Bull. Soc. Math. Fr., Volume 142 (2014) no. 3, pp. 489-536 | Zbl

[7] Olivier Glass; Christophe Lacave; Franck Sueur On the motion of a small light body immersed in a two dimensional incompressible perfect fluid with vorticity, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 1015-1065 | Zbl

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[9] Dragoş Iftimie; Milton C. Lopes Filho; Helena J. Nussenzveig Lopes Two dimensional incompressible ideal flow around a small obstacle, Commun. Partial Differ. Equations, Volume 28 (2003) no. 1-2, pp. 349-379 | Zbl

[10] Christophe Lacave; Evelyne Miot Uniqueness for the vortex-wave system when the vorticity is initially constant near the point vortex, SIAM J. Math. Anal., Volume 41 (2009) no. 3, pp. 1138-1163 | Zbl

[11] Christophe Lacave; Evelyne Miot The vortex-wave system with gyroscopic effects (2019) (https://arxiv.org/abs/1903.01714)

[12] Andrew J. Majda; Andrea L. Bertozzi Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, 2002 | Zbl

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[15] Carlo Marchioro; Mario Pulvirenti Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96, Springer, 1994 | Zbl

[16] Evelyne Miot Quelques problèmes relatifs à la dynamique des points vortex dans les équations d’Euler et de Ginzburg-Landau complexe, Université Pierre et Marie Curie - Paris VI (France) (2009) (Ph. D. Thesis)

[17] Ayman Moussa; Franck Sueur On a Vlasov–Euler system for 2D sprays with gyroscopic effects, Asymptotic Anal., Volume 81 (2013) no. 1, pp. 53-91 | Zbl

[18] Victor N. Starovoǐtov Solvability of the problem of motion of concentrated vortices in an ideal fluid, Din. Splosh. Sredy, Volume 85 (1988), pp. 118-136

[19] Victor N. Starovoǐtov Uniqueness of a solution to the problem of evolution of a point vortex, Sib. Math. J., Volume 35 (1994) no. 3, pp. 625-630 | Zbl

[20] Elias M. Stein Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993 | Zbl

[21] Witold Wolibner Un théorème sur l’existence du mouvement plan d’un fluide parfait homogène, incompressible, pendant un temps infiniment long, Math. Z., Volume 37 (1933) no. 1, pp. 698-726 | Zbl

[22] Viktor I. Yudovich Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., Volume 3 (1963) no. 6, pp. 1032-1066 English translation in USSR Comput. Math. Math. Phys. 3 (1963), no. 6, p. 1407-1456

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