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.
@article{JEDP_2018____A5_0, author = {Miot, Evelyne}, title = {On some coupled {PDE-ODE} systems in fluid dynamics}, journal = {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 = {http://www.numdam.org/articles/10.5802/jedp.665/} }
TY - JOUR AU - Miot, Evelyne TI - On some coupled PDE-ODE systems in fluid dynamics JO - Journées équations aux dérivées partielles N1 - talk:5 PY - 2018 SP - 1 EP - 13 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.665/ DO - 10.5802/jedp.665 LA - en ID - JEDP_2018____A5_0 ER -
Miot, Evelyne. On some coupled PDE-ODE systems in fluid dynamics. Journées équations aux dérivées partielles (2018), Talk no. 5, 13 p. doi : 10.5802/jedp.665. http://www.numdam.org/articles/10.5802/jedp.665/
[1] 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] Well posedness of ODEâs and continuity equations with nonsmooth vector fields, and applications (2017) (preprint)
[3] Estimates and regularity results for the DiPerna–Lions flow, J. Reine Angew. Math., Volume 616 (2008), pp. 15-46 | Zbl
[4] 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
[5] Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989) no. 3, pp. 511-547 | Zbl
[6] 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] 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
[8] Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Flüssigkeit, Math. Z., Volume 37 (1993), pp. 727-738 | Zbl
[9] Two dimensional incompressible ideal flow around a small obstacle, Commun. Partial Differ. Equations, Volume 28 (2003) no. 1-2, pp. 349-379 | Zbl
[10] 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] The vortex-wave system with gyroscopic effects (2019) (https://arxiv.org/abs/1903.01714)
[12] Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, 2002 | Zbl
[13] On the Euler equations with a singular external velocity field, Rend. Semin. Mat. Univ. Padova, Volume 84 (1990), pp. 61-69 | Zbl
[14] On the vortex-wave system, Mechanics, analysis, and geometry: 200 years after Lagrange (North-Holland Delta Series), North-Holland, 1991, pp. 79-95 | Zbl
[15] Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, 96, Springer, 1994 | Zbl
[16] 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] On a Vlasov–Euler system for 2D sprays with gyroscopic effects, Asymptotic Anal., Volume 81 (2013) no. 1, pp. 53-91 | Zbl
[18] Solvability of the problem of motion of concentrated vortices in an ideal fluid, Din. Splosh. Sredy, Volume 85 (1988), pp. 118-136
[19] 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] Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993 | Zbl
[21] 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] 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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