Time-Periodic Linearized Solutions of the Compressible Euler Equations and a Problem of Small Divisors

Abstract

It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), pp. 341–363], we derive exact linearized solutions that exhibit the simplest possible periodic wave structure that can balance compression and rarefaction along characteristics in the nonlinear Euler problem. These linearized waves exhibit interesting phase and group velocities analogous to linear dispersive waves. Moreover, when the spatial period is incommensurate with the time period, the sound speed is incommensurate with the period, and a new periodic wave pattern is observed in which the sound waves move in a quasiperiodic trajectory though a periodic configuration of states. This establishes a new way in which nonlinear solutions that exist arbitrarily close to these linearized solutions can balance compression and rarefaction along characteristics in a quasiperiodic sense. We then rigorously establish the spectral properties of the linearized operators associated with these linearized solutions. In particular we show that the linearized operators are invertible on the complement of a one-dimensional kernel containing the periodic solutions only in the case when the wave speeds are incommensurate with the periods, but these invertible operators have small divisors, analogous to KAM theory. Almost everywhere algebraic decay rates for the small divisors are proven. In particular this provides a solid framework for the problem of perturbing these linearized solutions to exact nonlinear periodic solutions of the fully compressible Euler equations.