Periodic Reporting for period 4 - 3DWATERWAVES (Mathematical aspects of three-dimensional water waves with vorticity)
Periodo di rendicontazione: 2020-09-01 al 2022-02-28
The goal of this project was to develop a mathematical theory for steady three-dimensional water waves with vorticity. This is important for modelling the interaction of surface waves with non-uniform currents. Such currents can for example be caused by wind or bottom friction and can have a profound effect on the surface waves.
The research is based on the Euler equations - a set of nonlinear partial differential equations which go back to the 18th century but are notorious for their complexity. Three-dimensional waves are waves whose surfaces vary in all horizontal directions. They may for example be doubly periodic, that is, periodic in two distinct directions - a situation which frequently occurs when waves travelling in different directions meet. In the two-dimensional case (that is, when the surface and fluid flow is independent of one horizontal direction), there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, which is a nice simplification of the Euler equations only available in the two-dimensional setting. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. Irrotational flow can be said to describe what happens when surface waves travel on still water or currents that are uniform in the vertical direction. In this project we seek to go beyond this restriction. In order to do so, we will both investigate particular situations when the Euler equations simplify (such as Beltrami flows) and try to work directly with the Euler equations. We will also consider new methods for studying flows with vorticity in fixed geometries as a natural part of the project.
The project has led to new results for three-dimensional water waves with vorticity in two different situations: in the setting of Beltrami flows and in the case when the vorticity is small. We have also developed a new method for constructing three-dimensional flows with vorticity in the case of fixed geometries, based on the dual stream function formulation. An interesting challenge for the future is to try to extend this to the free-boundary setting. Finally, we have obtained new existence results in the 2D and axisymmetric settings. Our approach has the advantage of allowing for both overhanging wave profiles, stagnation points and critical layers.
The research is based on the Euler equations - a set of nonlinear partial differential equations which go back to the 18th century but are notorious for their complexity. Three-dimensional waves are waves whose surfaces vary in all horizontal directions. They may for example be doubly periodic, that is, periodic in two distinct directions - a situation which frequently occurs when waves travelling in different directions meet. In the two-dimensional case (that is, when the surface and fluid flow is independent of one horizontal direction), there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, which is a nice simplification of the Euler equations only available in the two-dimensional setting. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. Irrotational flow can be said to describe what happens when surface waves travel on still water or currents that are uniform in the vertical direction. In this project we seek to go beyond this restriction. In order to do so, we will both investigate particular situations when the Euler equations simplify (such as Beltrami flows) and try to work directly with the Euler equations. We will also consider new methods for studying flows with vorticity in fixed geometries as a natural part of the project.
The project has led to new results for three-dimensional water waves with vorticity in two different situations: in the setting of Beltrami flows and in the case when the vorticity is small. We have also developed a new method for constructing three-dimensional flows with vorticity in the case of fixed geometries, based on the dual stream function formulation. An interesting challenge for the future is to try to extend this to the free-boundary setting. Finally, we have obtained new existence results in the 2D and axisymmetric settings. Our approach has the advantage of allowing for both overhanging wave profiles, stagnation points and critical layers.
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). 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). 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).
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). 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.
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). 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). 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).
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). 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.
Our results on water waves over Beltrami flows were the first existence results for three-dimensional travelling water waves with non-zero vorticity of any kind.
Moreover, our construction of steady rotational flows using two stream functions is the first time this approach has been used in a general existence theory.
Our new approach to the 2D water wave problem with vorticity is a breakthrough which has the potential to lead to further results on large-amplitude waves. There is strong numerical evidence for the existence of overhanging waves over flows, but before our paper there was no useful formulation that could capture such solutions except in the case of constant vorticity. It was surprising that we were able to find a new useful formulation in area which has been studied very actively in the last two decades.
The preprint "Symmetric doubly periodic gravity-capillary waves with small vorticity" draws on knowledge from the area of magnetohydrodynamics to construct three-dimensional doubly periodic waves on Beltrami flows. The equations for magnetohydrostatic equilibria are in fact equivalent to the steady Euler equations, and we were able to exploit this connection. It was surprising that we were able to construct such solutions without using Nash-Moser theory, even though the problem involves a loss of regularity.
Moreover, our construction of steady rotational flows using two stream functions is the first time this approach has been used in a general existence theory.
Our new approach to the 2D water wave problem with vorticity is a breakthrough which has the potential to lead to further results on large-amplitude waves. There is strong numerical evidence for the existence of overhanging waves over flows, but before our paper there was no useful formulation that could capture such solutions except in the case of constant vorticity. It was surprising that we were able to find a new useful formulation in area which has been studied very actively in the last two decades.
The preprint "Symmetric doubly periodic gravity-capillary waves with small vorticity" draws on knowledge from the area of magnetohydrodynamics to construct three-dimensional doubly periodic waves on Beltrami flows. The equations for magnetohydrostatic equilibria are in fact equivalent to the steady Euler equations, and we were able to exploit this connection. It was surprising that we were able to construct such solutions without using Nash-Moser theory, even though the problem involves a loss of regularity.