Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with $L^1$ Vorticity

Abstract

In the first part of this paper we establish a uniqueness result for continuity equations with a velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in [F. Bouchut and G. Crippa, J. Hyperbolic Differ. Equ., 10 (2013), pp. 235--282]. The proof is based on a combination of a stability estimate via optimal transport techniques developed in [C. Seis, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear] and some tools from harmonic analysis introduced in [F. Bouchut and G. Crippa, J. Hyperbolic Differ. Equ., 10 (2013), pp. 235--282]. In the second part of the paper, we address a question that arose in [M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes, Arch. Ration. Mech. Anal., 179 (2006), pp. 353--387], namely, whether 2 dimensional Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data have low integrability. We show that this is the case even when the initial vorticity is only in $L^1$, extending the proof for the $L^p$ case in [G. Crippa and S. Spirito, Comm. Math. Phys., 339 (2015), pp. 191--198].