Regularity and singularities of geometric evolution equations (RegSing)
Main funder
Funder's project number: 347550
Funds granted by main funder (€)
- 570 763,00
Funding program
Project timetable
Project start date: 01/09/2022
Project end date: 31/08/2026
Summary
The main challenge with geometric flows, such as the mean curvature flow, is that they develop singularities in finite time. Similarly when we study a related equations in fluid dynamics, such as the motion of a water drop in vacuum, we observe that it wobbles rather wildly and may develop singularities. It is therefore important to understand when and how these equations develop singularities and whether we may find a proper notion of solution. For some equations such as the mean curvature flow we may consider a weak solution which is defined for all times. In this case we are interested to understand how such a solution behave near the singularities.
In the proposed Academy project 'Regularity and singularities of geometric evolution equations' we study two evolution equations related to the isoperimetric problem; the mean curvature flow, which can be regarded as the gradient flow of the surface are, and the Euler equations with free boundary, which models the motion of the liquid drop. In the case of gradient flows our focus is on the weak solution which obtained via discrete minimizing movements method. Our aim is to study the problem of convergence, regularity and to show that it provides a well-defined solution of the associated equation. In the case of the Euler equations with free boundary, we will add an external electrostatic field and consider the motion of the charged liquid drop. Our aim is to show that the equations of motions are mathematically well-posed and study the appearance of a special type of singularity called Taylor-cone. The project is in the field of Calculus of Variations and nonlinear partial differential equations, but also intersects such areas differential geometry and fluid mechanics. The starting point is the PI's (Vesa Julin) ongoing research project on the stability of isoperimetric type problems.
The applied funding would be used to hire three postdoctoral researchers, for a total of 80 months over a period of four years, to work on the project with international collaboration. The budget contains travel funds for the postdoctoral researchers longer term research visits in the collaborating Universities.
In the proposed Academy project 'Regularity and singularities of geometric evolution equations' we study two evolution equations related to the isoperimetric problem; the mean curvature flow, which can be regarded as the gradient flow of the surface are, and the Euler equations with free boundary, which models the motion of the liquid drop. In the case of gradient flows our focus is on the weak solution which obtained via discrete minimizing movements method. Our aim is to study the problem of convergence, regularity and to show that it provides a well-defined solution of the associated equation. In the case of the Euler equations with free boundary, we will add an external electrostatic field and consider the motion of the charged liquid drop. Our aim is to show that the equations of motions are mathematically well-posed and study the appearance of a special type of singularity called Taylor-cone. The project is in the field of Calculus of Variations and nonlinear partial differential equations, but also intersects such areas differential geometry and fluid mechanics. The starting point is the PI's (Vesa Julin) ongoing research project on the stability of isoperimetric type problems.
The applied funding would be used to hire three postdoctoral researchers, for a total of 80 months over a period of four years, to work on the project with international collaboration. The budget contains travel funds for the postdoctoral researchers longer term research visits in the collaborating Universities.
