Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy

Abstract

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\le p\le \infty$. Moreover, if $p\ge 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.

28 pages

References in corpus (2)

• Convex integration solutions to the transport equation with full dimensional concentration
• A Posteriori Error Estimates for Self-Similar Solutions to the Euler Equations

Cited by in corpus (6)

• Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit
• On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity
• A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation
• Energy balance for forced two-dimensional incompressible ideal fluid flow
• $L^2$-critical nonuniqueness for the 2D Navier-Stokes equations
• Energy conservation for 2D Euler with vorticity in $L(\log L{)}^{\alpha }$