Vortex sheets - from new intuitions to crucial questions.

Project Information

VORTSHEET

Grant agreement ID: 706283

DOI

10.3030/706283

Project closed

EC signature date 14 April 2016

Start date 1 January 2017

End date 31 December 2018

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 195 454,80

EU contribution

€ 195 454,80

Coordinated by

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 1 - VORTSHEET (Vortex sheets - from new intuitions to crucial questions.)

Reporting period: 2017-01-01 to 2018-12-31

Summary of the context and overall objectives of the project

The project contains a series of problems in Partial Differential Equations arising from Fluid Mechanics.

Given the wide applicability of Fluid Mechanics in the Physical Sciences and Engineering it becomes therefore very relevant to understand fundamental properties of the equations that model the corresponding system.

In the project we focus in equations in vortex dynamics, where the vorticity is the curl of velocity. We also focus on the Camassa-Holm equation. For these systems we consider the following issues:

- Existence of solutions: Once a model has been posed, and described precisely by an partial differential equation, a fundamental question becomes understanding whether or not solutions actually exist, a result that might depend on the type of initial data prescribed.

- Uniqueness of solutions: ideally once a system has been posed it is higly desirable to have only solution for each initial condition. Mathematically this is know as uniqueness of solutions. A key result of this project is the uniqueness of solutions for the Camassa-Holm equation.

- Regulatiy of solutions: finally, once we have established the existence (and uniqueness) of a solution we want to understand mode qualitative properties of the solution, for example its regularity, and whether or not any singularities form.

All of these issues are very relevant in mathematical context, but also crutially important to validate the model and test its accurate as a predictor of the behaviour of solutions in real-worl problems.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project has addressed questions in 4 different directions, corresponding to as many work packages.

These are WP1: Existence, WP2: Singularities, WP3 Uniqueness and WP4: Other models. Without extending in technical aspects we summarise results in these packages:

- Existence of different classes of vortex-patch and vortex-sheet-like solutions have been considered

- For the Camass-Holm equation, and concerning singular solutions results established include: Holder continuity, Lipschitz continuity (into L^2, the energy space),...

- Uniqueness of dissipative solutions of the Camassa-Holm equation has been proven. This complete the global well-posedness theory of this celebrated model of shallow water, providing a mathematical rationale for the feasibility of the Camassa-Holm equation as a model of water waves encompassing both soliton interactions and wave-breaking.

- Other systems have been considered, including 2D Euler, SQG, and alpha models, obtaining results for symmetric vortex-patch solutions.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

This is a final report after the conclusion of the project. All progress beyond the state of the art have been reported in the space immediately above.

The effect of tranlations in the velocity