Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation

Abstract

We establish sharp estimates for distributional solutions to the Euler–Poisson–Darboux equation posed in a periodic domain. These equations are highly singular, and setting the Cauchy problem requires a precise understanding of the nature of the singularities that may arise in weak solutions. We consider initial data in a space of functions with fractional derivatives such that weak solutions are solely integrable, and we derive sharp continuous dependence estimates for solutions to the initial-value problem. Our results strongly depend on a key parameter arising in the Euler–Poisson–Darboux equation.