Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations
Abstract
We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a $C^{2,\alpha}$ smooth curve that intersects itself at one point, and the vorticity density on the interface is of class $C^\alpha$. The proof consists in perturbing Crapper's family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2019
- DOI:
- 10.48550/arXiv.1906.02612
- arXiv:
- arXiv:1906.02612
- Bibcode:
- 2019arXiv190602612C
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 31 pages