# Preprint 2008-034

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

## Blake Temple and Robin Young

**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 [Preprint 2008-033], 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 spacial 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 quasi-periodic
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
quasi-periodic 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 nice starting framework for
the problem of perturbing these linearized solutions to exact
nonlinear periodic solutions of the full compressible Euler
equations.