Comptes Rendus
Mathematical Analysis, Ordinary Differential Equations
On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy
Comptes Rendus. Mathématique, Volume 358 (2020) no. 3, pp. 303-319.

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body’s boundary. The whole “viscous incompressible fluid + rigid body” system is assumed to occupy the full plane 2 . We prove the existence of global-in-time weak solutions with constant non-zero circulation at infinity.

Dans cet article, nous considérons le mouvement d’un corps rigide dans un fluide visqueux incompressible avec des conditions de glissement avec friction de Navier à l’interface. Le système “fluide+corps rigide” est supposé occuper le plan tout entier. Nous prouvons l’existence de solutions globales en temps avec une circulation constante non nulle à l’infini.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.36
Classification: 35Q30, 70E15, 76D05, 76D03

Marco Bravin 1

1 Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2020__358_3_303_0,
     author = {Marco Bravin},
     title = {On the {2D} {\textquotedblleft}viscous incompressible fluid + rigid body{\textquotedblright} system with {Navier} conditions and unbounded energy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {303--319},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.36},
     language = {en},
}
TY  - JOUR
AU  - Marco Bravin
TI  - On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 303
EP  - 319
VL  - 358
IS  - 3
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.36
LA  - en
ID  - CRMATH_2020__358_3_303_0
ER  - 
%0 Journal Article
%A Marco Bravin
%T On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy
%J Comptes Rendus. Mathématique
%D 2020
%P 303-319
%V 358
%N 3
%I Académie des sciences, Paris
%R 10.5802/crmath.36
%G en
%F CRMATH_2020__358_3_303_0
Marco Bravin. On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy. Comptes Rendus. Mathématique, Volume 358 (2020) no. 3, pp. 303-319. doi : 10.5802/crmath.36. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.36/

[1] Paul Acevedo; Chérif Amrouche; Carlos Conca; Amrita Ghosh Stokes and Navier-–Stokes equations with Navier boundary condition, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 2, pp. 115-119 | DOI | MR | Zbl

[2] Marco Bravin Energy equality and uniqueness of weak solutions of a “viscous incompressible fluid + rigid body” system with Navier slip-with-friction conditions in a 2D bounded domain, J. Math. Fluid Mech., Volume 21 (2019) no. 2, 23, 31 pages | MR | Zbl

[3] Lawrence C. Evans Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2002

[4] Giovanni P. Galdi An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems, Springer, 2011 | Zbl

[5] Isabelle Gallagher; Thierry Gallay Uniqueness for the two-dimensional Navier-–Stokes equation with a measure as initial vorticity, Math. Ann., Volume 332 (2005) no. 2, pp. 287-327 | DOI | MR | Zbl

[6] David Gérard-Varet; Matthieu Hillairet Existence of weak solutions up to collision for viscous fluid-solid systems with slip, Commun. Pure Appl. Math., Volume 67 (2014) no. 12, pp. 2022-2076 | DOI | MR | Zbl

[7] David Gérard-Varet; Christophe Lacave The two-dimensional Euler equations on singular domains, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 131-170 | DOI | MR | Zbl

[8] Yoshikazu Giga; Tetsuro Miyakawa; Hirofumi Osada Two-dimensional Navier–Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal., Volume 104 (1988) no. 3, pp. 223-250 | DOI | MR | Zbl

[9] Vivette Girault; Pierre-Arnaud Raviart Finite element methods for Navier–Stokes equations: theory and algorithms, Springer Series in Computational Mathematics, 5, Springer, 2012 | Zbl

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

[11] Olivier Glass; Franck Sueur Dynamics of several rigid bodies in a two-dimensional ideal fluid and convergence to vortex systems (2019) (https://arxiv.org/abs/1910.03158)

[12] 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 | DOI | MR | Zbl

[13] Dragoş Iftimie; Milton C. Lopes Filho; Helena J. Nussenzveig Lopes Two-dimensional incompressible viscous flow around a small obstacle, Math. Ann., Volume 336 (2006) no. 2, pp. 449-489 | DOI | MR | Zbl

[14] Keisuke Kikuchi Exterior problem for the two-dimensional Euler equation, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 30 (1983) no. 1, pp. 63-92 | MR | Zbl

[15] Hideo Kozono; Masao Yamazaki Local and global unique solvability of the Navier–Stokes exterior problem with Cauchy data in the space L n, , Houston J. Math., Volume 21 (1995) no. 4, pp. 755-799 | MR | Zbl

[16] Christophe Lacave; Takéo Takahashi Small moving rigid body into a viscous incompressible fluid, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 3, pp. 1307-1335 | DOI | MR | Zbl

[17] Pierre G. Lemarié-Rieusset Recent developments in the Navier–Stokes problem, CRC Research Notes in Mathematics, 431, CRC Press, 2002 | MR | Zbl

[18] Yasunori Maekawa; Hideyuki Miura; Christophe Prange Local energy weak solutions for the Navier–Stokes equations in the half-space (2017) (https://arxiv.org/abs/1711.04486) | Zbl

[19] Jaime Ortega; Lionel Rosier; Takéo Takahashi On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 24 (2007) no. 1, pp. 139-165 | DOI | Numdam | MR | Zbl

[20] Gabriela Planas; Franck Sueur On the “viscous incompressible fluid + rigid body” system with Navier conditions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 1, pp. 55-80 | DOI | Numdam | MR | Zbl

Cited by Sources:

Comments - Policy