Math Seminars
Alexander Shnirelman
TelAviv University and IAS
Large Weak Solutions of the Euler
Equations and their Possible Significance in the Turbulence Problem
The Euler equations describe the motion of an ideal incompressible fluid. Every real fluid, which is not the liquid helium, is viscous and compressible. But if the viscosity and compressibility effects are small, we may anticipate that the fluid behaves as if it were inviscid and incompressible. But this belief is not confirmed by observation of the real motions of nearly ideal fluids, which are called turbulent. The most striking fact is that the rate of the energy dissipation in such flows does not tend to zero, as the viscosity tends to zero!
The Euler equations may be recast as some integral identities, expressing the local mass and momentum balance. Every solution of the Euler equations satisfies these identities ( which may be called the weak Euler equations); but there may a priori exist very nonregular (merely square integrable) velocity fields, which satisfy the weak Euler equations. Such velocity fields are called weak solutions of the Euler equations.
Our main conjecture is that these weak solutions have something to do with the turbulent flows; namely, that the flow field for a turbulent flow of a fluid with very small viscosity and compressibility is asymptotically close to some weak solution, as viscosity and compressibility tend to zero. The aim of my lectures is to present some rigorous arguments in favor of this hypothesis.
There is a few sparce results concerning the weak solutions. The first nontrivial example was presented in 1993 by V. Scheffer. It was a vector field u(x, t) in L^2(R^2 x R), which vanishes outside the ball |x|^2+|t|^2<1, and thus violently breaks uniqueness, energy conservation, and even energy monotonicity.
I am going to explain the simplified example of such situation. This is a weak solution (in fact, an unbounded and everywhere discontinuous vector field) on a 2-dim torus u(x, t), which vanishes for |t|>1. These examples show that the formal definition of a weak solution is not complete, and we need some further restrictions. But such solutions are interesting by their own, for they demonstrate other interesting fluiddynamical phenomenon, namely the "inverse cascade."
Next, I shall explain the construction of a more realistic 3-dim weak solution, whose kinetic energy monotonically decreases in time. This solution is also everywhere discontinuous and unbounded, while has some realistic features. The construction starts from a simple mechanical system having the property that the kinetic energy decreases, while there is no explicit friction; it requires Generalized Flows, introduced recently by Y. Brenier, and some harmonic analysis.
At last, I am going to discuss the ways to the construction and theory of true, physically reasonable weak solutions of the Euler equations.