Article

Journal of Dynamics and Differential Equations

, Volume 21, Issue 4, pp 631-661

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • James P. KelliherAffiliated withDepartment of Mathematics, University of California Email author 

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.

Keywords

Statistical solutions Navier–Stokes equations Euler equations

Mathematics Subject Classification (2000)

76D06 76D05