Periodic Solutions of Compressible Euler

We have been working on a long term program to construct the first time-periodic solutions of the compressible Euler equations. Our program from the start has been to construct the simplest periodic wave structure, and then tailor the analysis in a rigorous existence proof to the explicit structure. In [26] have found the periodic structure, in [27, 28] we showed that linearized solutions with this structure are isolated it in the kernel of its linearized operator, expressing all of this terms of a new dimensionless formulation of the problem which converts linearized evolution into rotation of the representations of the Fourier modes. But the linearized operator has resonances on a measure zero set of periods, and small divisors at non-resonant periods. In [29] we have shown the consistency of a LiapunovSchmidt decomposition and proven convergence subject to a finite Fourier mode cutoff. We are now trying to close the final proof by implementing a framework for period extraction into a Nash-Moser argument set out in [30]. Here is an accounting of our progress so far.