On the Landau law of decay of 3D shocks

Abstract: We will discuss stability and long term behavior of 3 dimensional compressible irrotational shocks arising in the compressible Euler equations. In 1945, Landau argued that spherically symmetric solutions which form weak shocks will settle to a profile with 2 shocks decaying at the rate proportionate to 1/{t\sqrt{\log t}}. We address this conjecture by first identifying the asymptotic profile which exhibit 2 shocks, as a self-similar solution of a related Burgers equation, and then proving its stability and the conjectured rate of decay for general (non-spherically symmetric) perturbations. This is joint work with D. Ginsberg.