On the interior regularity of weak solutions to the 2-D incompressible Euler equations
Siljander, J., & Urbano, J. M. (2017). On the interior regularity of weak solutions to the 2-D incompressible Euler equations. Calculus of Variations and Partial Differential Equations, 56(5), Article 126. https://doi.org/10.1007/s00526-017-1231-8
© Springer-Verlag GmbH Germany 2017. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi-Nash-Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result u ∈ L 2+ε loc (ΩT ) =⇒ local regularity for weak solutions in the energy space L∞t L2 x, satisfying appropriate vorticity estimates. We also obtain improved integrability for the vorticity – which is to be compared with the DiPerna-Lions assumptions. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier-Stokes equations in any dimension. ...