Analytic rates of solutions to the Euler equations

Abstract

The Cauchy problem of the Euler equations is considered with initial data with possibly less regularity. The time-local existence and the uniqueness of strong solutions were established by Pak-Park, when the initial velocity is in the Besov space $B^1_{\infty, 1}$. By treating non-decaying initial data, we are able to discuss the propagation of almost periodicity. It is also proved that if the initial data are real analytic, then the solutions become necessarily real analytic in space variables with an explicit convergence rate of the radius in Taylor's expansion. This result comes from the calculation of higher order derivatives, inductively.