Singularities and non-uniqueness for the 2-dimensional Euler equations

Singularities and non-uniqueness for the 2-dimensional Euler equations

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Alberto Bressan, Penn State University
Fine Hall 322

In connection with the 2-dimensional incompressible Euler equations, we study initial data where the vorticity is supported on two wedges, symmetric w.r.t. the origin, with density $O(r^{-\alpha})$. Numerical simulations by Wen Shen (Oxford, 2017) have shown that, by approximating this same initial data with smooth functions in two different ways, one obtains two distinct limit solutions.  One contains a single spiraling vortex, while the other solution contains two vortices.

The talk will report recent work aimed at a rigorous validation of these numerical results.The  main ingredients are: (i) an analytic construction of the solution in the exterior of a disc, (ii)  a posteriori error estimates for the numerically computed solution, valid on a bounded domain where the solution is smooth, and (iii)  an analytic construction of the solution near the spirals' centers,where singularities occur. This last step is largely based on techniques introduced by Volker Elling (2013).