Infinite-Energy 2D Statistical Solutions to the Equations of Incompressible Fluids

Open Access


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.


Statistical solutions Navier–Stokes equations Euler equations 

Mathematics Subject Classification (2000)

76D06 76D05 



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.


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© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA

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