Energy conservation for 2D Euler with vorticity in $L(\log L{)}^{\alpha }$

Abstract

In these notes we discuss the conservation of the energy for weak solutions of the two-dimensional incompressible Euler equations. Weak solutions with vorticity in $L_t^{\infty }{L}_x^p$ with $p\ge 3/2$ are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class $L(\log L{)}^{\alpha }$ with $\alpha >1/2$.

arXiv commentarXiv admin note: text overlap with arXiv:1905.09720

References in corpus (3)

• On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity
• Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II
• Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy