| Abstract: |
| We prove existence of small, symmetric, doubly periodic and rotational gravity-capillary waves in water of infinite depth in three space dimensions.
Following Lortz` ansatz from magnetohydrodynamics [1], we write the vorticity as the cross-product of two gradients and we assume the Bernoulli function to depend on the orbital period of the particles. This gives rise to a coupled elliptic-hyperbolic formulation which we cast as a fixed-point problem. Here we follow the strategy in [2], where the case of finite depth is treated. In our case the domain is not compact, which requires a more careful treatment of both the elliptic and the hyperbolic parts. Finally, we prove existence of small-amplitude solutions to the free-boundary problem with small vorticity using local bifurcation theory.
This is ongoing work.
[1] D. Lortz, \Uber die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation, Journal of Applied Mathematics and Physics (ZAMP), 1970
[2] D. S. Seth, K. Varholm, and E. Wahl\`en, Symmetric doubly periodic gravity-capillary waves with small vorticity, Advances in Mathematics, 2024 |
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