CMSA General Relativity Seminar: Characteristic Initial Value Problem for the 3D Compressible Euler Equations

We present the first result for the characteristic initial value problem of the compressible Euler equations in three space dimensions without any symmetry assumption. We allow presence of vorticity and consider any equation of state. Compared to the standard Cauchy problem, where initial data can be freely prescribed on a constant-time hypersurface, we formulate the problem by distinguishing between the "free-component" and the "constrained-component" of the initial data. The latter is to be solved by the "free-component" utilizing the properties of the compressible Euler equations on the initial null hypersurfaces. Then, we establish a priori estimates, followed by a local well-posedness and a continuation criterion argument. Moreover, we prove a regularity theory in Sobolev norms. Our analysis critically relies on the vectorfield method due to the nature of the problem. This is a joint work with Jared Speck.