The stability of irrotational shocks and the Landau law of decay

We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.