Eulerian and Lagrangian solutions to the continuity and Euler equations with L¹ vorticity

2017/05/17 by Crippa, Gianluca, Nobili, Camilla, Seis, Christian +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1705.06188

Abstract

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an L¹ function, extending the Lagrangian theory in \citeBouchutCrippa13. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \citeSeis16a and some tools from harmonic analysis introduced in \citeBouchutCrippa13. In the second part of the paper, we address a question that arose in \citeFilhoMazzucatoNussenzveig06, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~L¹, extending the proof for the L^p case in \citeCrippaSpirito15.

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• Dissipation concentration in two-dimensional fluids

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