Projektdetails
Abstract
1) Wider research context
This project concerns the theoretical analysis of steady (travelling) water waves, which are ubiquitous in nature and whose investigation has heavily influenced mathematical research for centuries. The project is within the scope of the full water wave model, that is, governed by Euler's equations with a free boundary. We are in particular interested in water waves with overhanging surfaces, that is, waves with height ceasing to be single-valued. The project does not focus on numerics.
2) Research objectives
We are concerned with the mathematical understanding and construction of overhanging water waves, connected by a bifurcation branch to simpler, explicitly known waves. In the first part, devoted to the twodimensional situation, we construct large pure gravity water waves with a submerged vortex patch and study in detail their structural properties. In the second part, devoted to the axisymmetric situation, we investigate recently constructed capillary waves in more detail, while extending the set of waves that can be captured by rigorous analysis. In the third part, devoted to the three-dimensional situation, by a dimension-breaking bifurcation analysis we shall for the first time rigorously construct fully three-dimensional, overhanging water waves.
3) Methods
The overall strategy is to employ bifurcation methods. A substantial part of the required analysis is to write the equations in a form that is amenable to an abstract bifurcation theorem, while allowing for as many as possible physical configurations. Since the focus of this project is on overhanging waves, we shall often make use of conformal changes of variables, mapping the unknown fluid domain to a fixed domain. Thus, tools from complex analysis will be helpful. Other tools are a precise nodal analysis through sharp maximum principles, a careful study of the spectrum of operators corresponding to certain linearisations and tools from spatial dynamics.
4) Level of originality
In the first part, we will for the first time construct water waves with a vortex patch in the pure gravity case while allowing for overhanging profiles. Our nodal analysis will be completely novel in the vortex patch setting.
In the second part, we will provide a mathematical formulation that for the first time can capture the formation of bubbles and is not restricted to the irrotational, swirl-free case. Also, we will study in more detail solution branches of waves with vorticity and swirl; regarding their visualisation nothing at all is available in the literature so far. To date there is a complete lack of theory in the large regime for three-dimensional water waves. The third part of this project aims to make substantial progress in this direction and prove the existence of overhanging 3D waves as dimension-breaking perturbations of the classical 2D Crapper waves.
5) Primary researchers involved
Principal investigator: Jörg Weber. Mentor: Adrian Constantin.
This project concerns the theoretical analysis of steady (travelling) water waves, which are ubiquitous in nature and whose investigation has heavily influenced mathematical research for centuries. The project is within the scope of the full water wave model, that is, governed by Euler's equations with a free boundary. We are in particular interested in water waves with overhanging surfaces, that is, waves with height ceasing to be single-valued. The project does not focus on numerics.
2) Research objectives
We are concerned with the mathematical understanding and construction of overhanging water waves, connected by a bifurcation branch to simpler, explicitly known waves. In the first part, devoted to the twodimensional situation, we construct large pure gravity water waves with a submerged vortex patch and study in detail their structural properties. In the second part, devoted to the axisymmetric situation, we investigate recently constructed capillary waves in more detail, while extending the set of waves that can be captured by rigorous analysis. In the third part, devoted to the three-dimensional situation, by a dimension-breaking bifurcation analysis we shall for the first time rigorously construct fully three-dimensional, overhanging water waves.
3) Methods
The overall strategy is to employ bifurcation methods. A substantial part of the required analysis is to write the equations in a form that is amenable to an abstract bifurcation theorem, while allowing for as many as possible physical configurations. Since the focus of this project is on overhanging waves, we shall often make use of conformal changes of variables, mapping the unknown fluid domain to a fixed domain. Thus, tools from complex analysis will be helpful. Other tools are a precise nodal analysis through sharp maximum principles, a careful study of the spectrum of operators corresponding to certain linearisations and tools from spatial dynamics.
4) Level of originality
In the first part, we will for the first time construct water waves with a vortex patch in the pure gravity case while allowing for overhanging profiles. Our nodal analysis will be completely novel in the vortex patch setting.
In the second part, we will provide a mathematical formulation that for the first time can capture the formation of bubbles and is not restricted to the irrotational, swirl-free case. Also, we will study in more detail solution branches of waves with vorticity and swirl; regarding their visualisation nothing at all is available in the literature so far. To date there is a complete lack of theory in the large regime for three-dimensional water waves. The third part of this project aims to make substantial progress in this direction and prove the existence of overhanging 3D waves as dimension-breaking perturbations of the classical 2D Crapper waves.
5) Primary researchers involved
Principal investigator: Jörg Weber. Mentor: Adrian Constantin.
| Kurztitel | Überhängende Wasserwellen |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/01/25 → 31/12/27 |