Finite time singularities for the free boundary incompressible Euler equations
The 2D incompressible free boundary Euler problem is used to study the evolution of a fluid region and a vacuum region separated by a smooth (initial) interface. The fluid is irrotational and the surface tension is neglected. The main point is to describe the evolution of the vacuum-fluid interface \(WF\). Even if at the initial moment \(WF\) is smooth (but not necessary a graph), after a finite time a singularity appears -- the curve touches itself. Numerical results, based on the Beale-Hou-Lovengrub method, are used to motivate some conjectures. Then the authors explain what results can be proved. Interval arithmetic is used to produce a rigorous computer-assisted proof for the existence of an exact solution, close to the approximate numerical solution, which ends into a particular singularity. A local existence for analytic initial data in a transformed (``tilda'') domain is proved, by using the abstract form of the Cauchy-Kowalewski theorem.
scientific article; zbMATH DE number 6220729
scientific article