Finite time singularities for the free boundary incompressible Euler equations. (English) Zbl 1291.35199

Castro, Angel; Córdoba, Diego; Fefferman, Charles; Gancedo, Francisco; Gómez-Serrano, Javier

Ann. Math. (2) 178, No. 3, 1061-1134 (2013).

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.

Reviewer: Gelu Paşa (Bucureşti)

MSC:

35Q31	Euler equations
76D05	Navier-Stokes equations for incompressible viscous fluids
76D27	Other free boundary flows; Hele-Shaw flows
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B44	Blow-up in context of PDEs

Keywords:

incompressible Euler equations; potential theory; water waves problem

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