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 . 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.
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Marco Bravin 1
@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 -
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] 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] 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] Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2002
[4] An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems, Springer, 2011 | Zbl
[5] 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] 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] The two-dimensional Euler equations on singular domains, Arch. Ration. Mech. Anal., Volume 209 (2013) no. 1, pp. 131-170 | DOI | MR | Zbl
[8] 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] Finite element methods for Navier–Stokes equations: theory and algorithms, Springer Series in Computational Mathematics, 5, Springer, 2012 | Zbl
[10] 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] Dynamics of several rigid bodies in a two-dimensional ideal fluid and convergence to vortex systems (2019) (https://arxiv.org/abs/1910.03158)
[12] 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] Two-dimensional incompressible viscous flow around a small obstacle, Math. Ann., Volume 336 (2006) no. 2, pp. 449-489 | DOI | MR | Zbl
[14] 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] Local and global unique solvability of the Navier–Stokes exterior problem with Cauchy data in the space , Houston J. Math., Volume 21 (1995) no. 4, pp. 755-799 | MR | Zbl
[16] 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] Recent developments in the Navier–Stokes problem, CRC Research Notes in Mathematics, 431, CRC Press, 2002 | MR | Zbl
[18] Local energy weak solutions for the Navier–Stokes equations in the half-space (2017) (https://arxiv.org/abs/1711.04486) | Zbl
[19] 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] 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
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