Abstract
We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).
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References
Borsuk, M., Kondratiev, V.: Elliptic boundary value problems of second order in piecewise smooth domains. North-Holland Mathematical Library, Vol. 69. Elsevier Science B.V., Amsterdam, 2006
Bucur D., Feireisl E., Necasova S., Wolf J.: On the asymptotic limit of the Navier–Stokes system with rough boundaries. J. Differ. Equ. 244(11), 2890–2908 (2008)
Casado-Diaz J., Fernandez-Cara E., Simon J.: Why viscous fluids adhere to rugose walls: a mathematical explanation. J. Differ. Equ. 189(2), 26–537 (2003)
Delort J.-M.: Existence de nappes de tourbillon en dimension deux. (French) [Existence of vortex sheets in dimension two]. J. Am. Math. Soc. 4(3), 553–586 (1991)
DiPerna R.J., Majda A.J.: Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Math. 40(3), 301–345 (1987)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, 2nd edn. Springer Monographs in Mathematics. Springer, New York, 2011
Henrot, A., Pierre, M.: Variation et optimisation de formes. Une analyse géométrique (French) [Shape variation and optimization. A geometric analysis], Mathématiques & Applications, Vol. 48. Springer, Berlin, 2005
Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J.: Two dimensional incompressible ideal flow around a small obstacle. Commun. Partial Diff. Equ. 28(1–2), 349–379 (2003)
Kikuchi, K.: Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 30(1), 63–92 (1983)
Kozlov, V.A., Mazya, V.G., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs, Vol. 85. American Mathematical Society, Providence, RI, 2001
Lacave C.: Two dimensional incompressible ideal flow around a thin obstacle tending to a curve. Annales de l’IHP, Anl 26, 1121–1148 (2009)
Lacave C.: Two-dimensional incompressible ideal flow around a small curve. Commun. Partial Diff. Equ. 37(4), 690–731 (2012)
Lacave, C.: Uniqueness for two dimensional incompressible ideal flow on singular domains. submitted, preprint 2011. arXiv:1109.1153
Lopes Filho M.C.: Vortex dynamics in a two dimensional domain with holes and the small obstacle limit. SIAM J. Math. Anal. 39(2), 422–436 (2007)
Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, 2002
McGrath F.J.: Nonstationary plane flow of viscous and ideal fluids. Arch. Ration. Mech. Anal. 27, 329–348 (1967)
Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen, 1975
Simon J.: Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)
Sverák V.: On optimal shape design. J. Math. Pures Appl. (9) 72(6), 537–551 (1993)
Taylor, M.: Incompressible fluid flows on rough domains. Semigroups of Operators: Theory and Applications (Newport Beach, CA, 1998). In: Progress in Nonlinear Differential Equations and their Applications, Vol. 42. Birkhauser, Basel, 320–334, 2000
Wolibner, W.: Un théorème sur l’existence du mouvement plan d’un fluide parfait homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726, 1933
Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z. Vycisl. 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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Gérard-Varet, D., Lacave, C. The Two-Dimensional Euler Equations on Singular Domains. Arch Rational Mech Anal 209, 131–170 (2013). https://doi.org/10.1007/s00205-013-0617-9
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DOI: https://doi.org/10.1007/s00205-013-0617-9