| Abstract: |
| The motion of a two-dimensional incompressible fluid is governed by the 2D Euler equations
\begin{equation}
\begin{cases}
\partial_tv+\left(v\cdot\nabla\right)v+\nabla p=0,\
\mathrm{div}\,v=0,\
v(0,\cdot)=v_0,
\end{cases}
\label{eq:eu}
\end{equation}
where $v:[0,T]\times\mathbb{R}^2\to\mathbb{R}^2$ is the velocity field, $p:[0,T]\times\mathbb{R}^2\to\mathbb{R}$ is the scalar pressure and $v_0:\mathbb{R}^2\to\mathbb{R}^2$ is a given divergence-free initial velocity.
Smooth solutions, whose global existence is well-known, enjoy two very natural properties: the first one is that they are \emph{Lagrangian}, namely the vorticity is advected by the flow of the velocity; the second property is that smooth solutions conserve the \emph{kinetic energy}.
When we consider solutions in weaker classes, precisely when the initial vorticity is in $L^p$ with very low $p$, the existence of Lagrangian solutions and the conservation of the energy may depend in general on the approximation scheme. In this talk, we prove the existence of solutions that enjoy the above properties constructed via different methods. Specifically, we will consider the vanishing viscosity limit and the vortex-blob approximation, which are important from a physical and numerical point of view. |
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