Weak Solutions with Decreasing Energy¶of Incompressible Euler Equations

Abstract:

Weak solution of the Euler equations is an L ²-vector field u(x, t), satisfying certain integral relations, which express incompressibility and the momentum balance. Our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations, as both viscosity and compressibility tend to zero; thus, we believe that weak solutions describe turbulent flows with very high Reynolds numbers.

Every physically meaningful weak solution should have kinetic energy decreasing in time. But the existence of such weak solutions have been unclear, and should be proven. In this work an example of weak solution with decreasing energy is constructed. To do this, we use generalized flows (GF), introduced by Y. Brenier. GF is a sort of a random walk in the flow domain, such that the mean kinetic energy of particles is finite, and the particle density is constant. We construct a GF such that fluid particles collide and stick; this sticking is a sink of energy. The GF which we have constructed is a GF with local interaction; this means that there are no external forces. The second important property is that the particle velocity depends only on its current position and time; thus we have some velocity field, and we prove that this field is a weak solution with decreasing energy of the Euler equations. The GF is constructed as a limit of multiphase flows (MF) with the mass exchange between phases.