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Vort3DEuler Report Summary

Project ID: 616797
Funded under: FP7-IDEAS-ERC
Country: United Kingdom

Mid-Term Report Summary - VORT3DEULER (3D Euler, Vortex Dynamics and PDE)

The main objects of study in the project are the three-dimensional Euler equations, and various related models, with particular emphasis on the surface quasi-geostrophic equations (SQG). The equations of Magneto-hydrodynamics have also been considered, in connection with the idea of magnetic relaxation to obtain physically meaningful solutions of Euler.

The first stage of the proposal focused on the model equations to gain insight on the Euler equations. The main objects of study are almost-sharp fronts. These objects are solutions of SQG with arbitrarily large gradient. The main goals of the project are the study of their evolution and the construction of families for which the time.

Concerning the construction of almost-sharp fronts, the analytic case was completed with Fefferman (appeared recently in the Archives for Rational Mechanics and Analysis). The main ingredient s in the proof involved the understanding of the Prandtl-like limit structure of the equations and development of a version of the Cauchy-Kowaleskaya theorem that allows for the presence of nonlinear operators of degree higher than one.

The study in the smooth case is also very advanced. Given the Prandtl-like structure of the limit equation it seems reasonable to restrict the approach to monotonic solutions. In this case, the PI, together with C. Fefferman has developed an approach (using as a main tool weighted norms depending on the solution itself) that makes it possible to construct families of almost-sharp fronts under monotonicity assumptions (and some other additional technical conditions). In addition to removing these conditions, the non-monotonic case remains of interest, and partial results have been obtained.

An unexpected recent development involves the development of the theory of sharp fronts for more singular scenarios (in fact to allow a fractional derivative of order less than 1). This was completely unexpected as SQG already provides the dimensionally correct singularity. We are now focusing on the construction of the analogue of almost-sharp fronts for these more singular models.

Concerning Euler (to be developed in the second half of the project), initial progress has been made in method to desingularise the velocity using fractional derivatives. This approach yields the evolution of a vortex line by the binormal, while using objects that are weak solutions of the new equation. While early stages this approach looks very promising.

In the development of the various techniques used for the problems described above progress has been possible in other related projects, such as magnetic relaxation, blow up rates for solutions, and the development of optimal conditions for energy conservation for Euler without the use of Besov spaces (including the case of some domains with boundary).

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United Kingdom
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