Abstract
In this paper, we consider two systems modelling the evolution of a rigid body in an incompressible fluid in a bounded domain of the plane. The first system corresponds to an inviscid fluid driven by the Euler equation whereas the other one corresponds to a viscous fluid driven by the Navier–Stokes system. In both cases we investigate the uniqueness of weak solutions, à la Yudovich for the Euler case, à la Leray for the Navier–Stokes case, as long as no collision occurs.
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Conca C., San Martin J.A., Tucsnak M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Commun. Partial Differ. Equ. 25(5–6), 1019–1042 (2000)
Dashti M., Robinson J.C.: The motion of a fluid–rigid disc system at the zero limit of the rigid disc radius. Arch. Ration. Mech. Anal. 200(1), 285–312 (2011)
Desjardins B., Esteban M.: On weak solutions for fluid–rigid structure interaction: compressible and incompressible models. Commun. Partial Differ. Equ. 25(7–8), 1399–1413 (2000)
Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146(1), 59–71 (1999)
DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Feireisl E.: On the motion of rigid bodies in a viscous incompressible fluid. Dedicated to Philippe Bénilan. J. Evol. Equ. 3(3), 419–441 (2003)
Feireisl, E.: On the motion of rigid bodies in a viscous fluid. Mathematical theory in fluid mechanics (Paseky, 2001). Appl. Math. 47(6), 463–484 (2002)
Feireisl E.: On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167(4), 281–308 (2003)
Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. Handbook of Mathematical Fluid Dynamics, Vol. I. North-Holland, Amsterdam, 653–791, 2002
Geissert M., Götze K., Hieber M.: Lp-theory for strong solutions to fluid–rigid body interaction in Newtonian and generalized Newtonian fluids. Trans. Am. Math. Soc. 365(3), 1393–1439 (2013)
Gérard-Varet D., Hillairet M.: Regularity issues in the problem of fluid–structure interaction. Arch. Ration. Mech. Anal. 195(2), 375–407 (2010)
Gilbarg, D., Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer, Berlin, 1983
Glass O., Lacave C., Sueur F.: On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Bull. Soc. Math. Fr. 142(3), 489–536 (2014)
Glass O., Sueur F.: The movement of a solid in an incompressible perfect fluid as a geodesic flow. Proc. Am. Math. Soc. 140, 2155–2168 (2012)
Glass O., Sueur F., Takahashi T.: Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid. Ann. Sci. E. N. S. 45(1), 1–51 (2012)
Glass O., Sueur F.: On the motion of a rigid body in a two-dimensional irregular ideal flow, SIAM J. Math. Anal. 44(5), 3101–3126 (2012)
Grandmont C., Maday Y.: Existence for an unsteady fluid–structure interaction problem. M2AN Math. Model. Numer. Anal. 34(3), 609–636 (2000)
Gunzburger M., Lee H.-C., Seregin G.A.: Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2(3), 219–266 (2000)
Hesla, T.I.: Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation. PhD thesis, University of Minnesota, revised version (2005)
Hillairet M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ. 32(7–9), 1345–1371 (2007)
Hoffmann K.-H., Starovoitov V.N.: On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9(2), 633–648 (1999)
Inoue A., Wakimoto M.: On existence of solutions of the Navier–Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(2), 303–319 (1977)
Judakov, N.V.: The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid (in Russian). Dinamika Splošn. Sredy 18, 249–253 (1974)
Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1933)
Leray J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes de l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)
Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, Vol. 3, 1996
San Martin J.A., Starovoitov V., Tucsnak M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161(2), 113–147 (2002)
Serre D.: Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Jpn. J. Appl. Math. 4(1), 99–110 (1987)
Starovoitov V.N.: Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid. J. Math. Sci. 130(4), 4893–4898 (2005)
Takahashi T.: Analysis of strong solutions for the equations modeling the motion of a rigid–fluid system in a bounded domain. Adv. Differ. Equ. 8(12), 1499–1532 (2003)
Takahashi T., Tucsnak M.: Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6(1), 53–77 (2004)
Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963) (in Russian). [English translation in USSR Comput. Math. Math. Phys. 3, 1407–1456 (1963)]
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Glass, O., Sueur, F. Uniqueness Results for Weak Solutions of Two-Dimensional Fluid–Solid Systems. Arch Rational Mech Anal 218, 907–944 (2015). https://doi.org/10.1007/s00205-015-0876-8
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DOI: https://doi.org/10.1007/s00205-015-0876-8