Abstract
We develop the concept of an infinite-energy statistical solution to the Navier–Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier–Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by making a vanishing viscosity argument.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ben-Artzi M.: Global solutions of two-dimensional Navier-Stokes and Euler equations. Arch. Ration. Mech. Anal. 128(4), 329–358 (1994)
Chemin J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differ. Equat. 21(11–12), 1771–1779 (1996)
Chemin, J.-Y.: Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1998) (Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie)
Constantin P., Ramos F.: Inviscid limit for damped and driven incompressible Navier-Stokes equations in \({\mathbb{R}^{2}}\) . Comm. Math. Phys. 275(2), 529–551 (2007)
Foiaş C.: Statistical study of Navier-Stokes equations. I, II. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972)
Foiaş C.: Statistical study of Navier-Stokes equations. I, II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973)
Foias C., Manley O., Rosa R., Temam R.: Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)
Iftimie D., Kelliher J.P.: Remarks on the vanishing obstacle limit for a 3d viscous incompressible fluid. Proc. Am. Math. Soc. 137(2), 685–694 (2009)
Kelliher J.P.: The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity. Math. Res. Lett. 11(4), 519–528 (2004)
Kelliher J.P.: Expanding domain limit for incompressible fluids in the plane. Comm. Math. Phys. 278(3), 753–773 (2008)
Kelliher, J.P., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Vanishing viscosity limit for an expanding domain in space. Ann. Inst. Henri Poincaré (C). doi:10.1016/j.anihpc.2009.07.007
Kim, N.: Large friction limit and the inviscid limit of 2-D Navier-Stokes equations under Navier friction condition. SIAM Math. Anal., to appear
Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI (2001) (Reprint of the 1984 edn)
Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3:1032–1066 (1963) (Russian)
Yudovich V.I.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2(1), 27–38 (1995)
Acknowledgments
The author was supported in part by NSF grant DMS-0705586 during the period of this work.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Kelliher, J.P. Infinite-Energy 2D Statistical Solutions to the Equations of Incompressible Fluids. J Dyn Diff Equat 21, 631–661 (2009). https://doi.org/10.1007/s10884-009-9151-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-009-9151-8