Young Researchers Workshop: Current trends in kinetic theory
An Onsager singularity theorem for the compressible Euler equations
We prove that bounded weak solutions of the compressible Euler
equations will conserve thermodynamic entropy, unless the solution fields have
sufficiently low space-time Besov regularity. A quantity measuring kinetic energy
cascade will also vanish for such Euler solutions, unless the same singularity
conditions are satisfied. It is shown furthermore that strong limits of solutions
of compressible Navier-Stokes equations that are bounded and exhibit anomalous
dissipation are weak Euler solutions. These inviscid limit solutions have
non-negative anomalous entropy production and kinetic energy dissipation, with
both vanishing when solutions are above the critical degree of Besov regularity.
Stationary, planar shocks in Euclidean space with an ideal-gas equation of state
provide simple examples that satisfy the conditions of our theorems and which
demonstrate sharpness of our L^3-based conditions. This talk is based on joint
work with G. Eyink.