B. Buffoni and the PI ("Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration", Analysis & PDE, 2019) constructed steady 3D rotational ideal flows near a given parallel flow in a fixed geometry by using a formulation based on two stream functions. Instead of prescribing the relationship between the vorticity and the stream function (as in 2D), we prescribe the relation between the Bernoulli function and the stream functions.
E. Lokharu and the PI ("A variational principle for three-dimensional water waves over Beltrami flows", Nonlinear Analysis, 2019) derived a variational principle for doubly periodic 3D travelling water waves over Beltrami flows. We are currently using this principle to construct fully localised solitary waves.
E. Lokharu, D. Svensson Seth and the PI ("An existence theory for small-amplitude doubly periodic water waves with vorticity", Archive for Rational Mechanics and Analysis, 2020) developed an existence theory for small-amplitude doubly periodic three-dimensional travelling waters on Beltrami flows. These bifurcate from flows with a flat surface in which the velocity is constant at each depth but the direction of the velocity field is depth dependent (see attached figure). The theory is based on a multi-parameter bifurcation approach.
D. Svensson Seth constructed 3D steady ideal flows in fixed domains with edges ("Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges", Journal of Differential Equation, 2021). This can be used to model flow in a pipe with inflow at one end and outflow at the other. The result is based on a method by Alber, who considered smooth domains. The extension to domains with edges is natural, but challenging since the edges affect the regularity of the solutions.
In a series of works, we have constructed large-amplitude, 2D waves with vorticity. J. Weber and the PI constructed large-amplitude periodic capillary-gravity waves using a conformal change of variables ("Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity",
https://arxiv.org/abs/2109.06070(si apre in una nuova finestra)). This was recently extended to pure gravity waves ("Large-amplitude steady gravity water waves with general vorticity and critical layers",
https://arxiv.org/abs/2204.13093(si apre in una nuova finestra)). Finally, in the axisymmetric setting, we have together with A. Erhardt constructed periodic capillary waves using a similar approach, but with a different change of variables ("Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl",
https://arxiv.org/abs/2202.01754(si apre in una nuova finestra)).
D. Svensson Seth, K. Varholm and the PI constructed doubly periodic waves with small vorticity bifurcating from uniform flows ("Symmetric doubly periodic gravity-capillary waves with small vorticity",
https://arxiv.org/abs/2204.13093(si apre in una nuova finestra)). The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a representation of the vorticity as the cross product of two gradients, and on prescribing a relation between the Bernoulli function and the orbital period of the water particles. The free surface introduces new challenges. In particular, the free boundary problem is not elliptic, and the involved maps incur a loss of regularity under Fréchet differentiation.
The Lund Workshop on Fluid Dynamics and Dispersive Equations was organised in June, 2018. It gathered many renowned experts on mathematical aspects of fluid dynamics and waves. The support from the ERC allowed a number of junior researchers to participate and present posters.
