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