The global Cauchy problem for the Euler–Riesz equations

Abstract

We completely resolve the global Cauchy problem for the multi-dimensional Euler–Riesz equations, where the interaction forcing is given by $\nabla {(-\Delta )}^{-\sigma /2}\rho$ for some $\sigma \in (0,2)$. We construct the global-in-time unique solution to the Euler–Riesz system in a $H^s$ Sobolev space under a smallness assumption on the initial density and a dispersive spectral condition on the initial velocity. Moreover, we investigate the algebraic time decay of convergences for the constructed solutions. Our results cover the both attractive and repulsive cases as well as the whole regime $\sigma \in (0,2)$.