# MIRaW – Mathematical Interdisciplinary Research at Warwick

### Weak solutions of the 3d Euler equations

Monday 8 June 2009

Chair person: José Rodrigo

Organiser: Xinyu He

##### All talks will be in B3.02, Mathematics Institute, Zeeman Building

The meeting will focus on the development of weak solutions of the incompressible Euler equations, and will promote informal discussion of weak solutions, energy conservation and turbulence. ** Programme ** as follows.

Coffee 11:00 am in the Maths Common Room

- 11:30 - 12:30
**Keith Moffatt**(DAMTP, Cambridge):*Relaxation Routes to Steady Euler Flows of Complex Topology.*( PDF of talk)The well-known analogy between vorticity dynamics and the induction equation of magnetohydrodynamics provides a natural relaxation route for the determination of steady Euler flows of arbitrarily prescibed streamline topology. Such relaxation processes converge in L^2 norm. Alternative relaxation procedures that conserve vorticity topology are available only for 2D or axisymmetric conditions; they involve 'artificial' dynamics that drive the system to a topologically accessible state of maximal energy. These relaxation routes will be reviewed, and open problems identified.

Lunch 12:30 - 1:30 in the Maths Common Room (sandwiches will be provided).

- 1:30 - 2:30
**Camillo De Lellis**(Universität Zürich):*h - Principle and Fluid Dynamics.**( PDF of reading material)**( PDF of reading material)*In the early nineties Scheffer produced a complicated example of a nontrivial weak solution to the incompressible Euler equations, having compact support in space and time. Subsequent papers by Shnirelman produced other examples of quite irregular solutions by different, yet complicated, methods.

In a recent joint work with László Székelyhidi we have used a suitable "

*h*-principle'' to produce solutions with the same behavior in a relatively simple way. Our approach answers to further questions left open by the works of Scheffer and Shnirelman and might be relevant in understanding a long-standing conjecture of Onsager. The same kind of analysis has supripising applications also to the theory of hyperbolic systems of conservation laws and shares some striking similarities with the theory of fully developped turbulence. - 2:40 - 3:40
**Alexander Shnirelman**(Concordia University):*Two Basic Problems on the Weak Solutions of 3-d Euler Equations.**( PDF of talk)*There are, roughly speaking, four classes of weak solutions of 3-d Euler equations: (1) Classical regular solutions; (2) Moderately irregular solutions, which are Hölder continuous in space-time with the Hölder exponent more than 1/3; (3) Hölder continuous solutions with the Hölder exponent less or equal than 1/3; (4) Everywhere discontinuous and, possibly, unbounded, square integrable weak solutions.

Solutions of the class 2 enjoy the energy conservation, and, possibly, even the uniqueness property (with fixed initial velocity). So, they are too regular to describe the developed turbulence. Solutions of the class 4 are too flexible (highly nonunique), and have other nonphysical features (like the absence of pressure), which makes them poor candidates for realistic description of turbulence. As for solutions of the class 3, they appear to be the most suitable to describe the developed turbulence. However, they are the least studied. There are two basic problems about these solutions:

1. Construct an example of such solution with monotonically decreasing energy and nontrivial pressure;

2. Given an initial, Hölder continuous velocity field, prove that there exists a global in time weak solution with decreasing energy.

In the talk I'll tell what I know about these (and some related) problems.

Tea 3:40 pm in the Maths Common Room

- 4:00 - 5:00
**James Robinson**(Warwick):*Almost Everywhere Uniqueness of Particle Trajectories for Suitable Weak Solutions of the 3D Navier-Stokes Equations.( PDF of talk)*Given an initial condition u_0 in H^{1/2}, it is known that the 3D Navier-Stokes equations have a unique solution u(x,t) on some finite time interval [0,T). For t >= T there exists at least one weak solution.

In this talk I will first give a simple proof that for every initial particle position, the equation dX/dt= u(X,t) has a unique solution on [0,T) (this is joint work with Masoumeh Dashti, Warwick).

After time T , if one takes u(x,t) to be any fixed suitable weak solution (in the sense of Caffarelli-Kohn-Nirenberg), almost every initial particle position gives rise to a unique solution of dX/dt = u(X,t) on [0,\infty ) (this is joint work with Witold Sadowski, Warsaw).

Reception 5:05 pm in the Maths Common Room

** Participants **: Mitchell Berger (Exeter), Geoffrey Burton (Bath), Camillo De Lellis (Zürich), Andrew Gilbert (Exeter), Robert Kerr (Warwick), Keith Moffatt (Cambridge), Sergey Nazarenko (Warwick), Koji Ohkitani (Sheffield), James Robinson (Warwick), Jonathan Robbins (Bristol), José Rodrigo (Warwick), Alexander Shnirelman (Concordia)