Global smooth solutions to the irrotational Euler-Riesz system in three dimensions

TL;DR

This paper proves the existence of global smooth solutions for the three-dimensional irrotational Euler-Riesz system with small initial data, demonstrating decay over time and extending understanding of such nonlinear PDEs.

Contribution

It establishes the global well-posedness and decay rates for the Euler-Riesz system in three dimensions under small irrotational initial conditions.

Findings

Global smooth solutions exist for small initial perturbations.

Solutions decay over time at a rate depending on .

The system generalizes classical fluid interactions with Riesz potentials.

Abstract

This paper investigates the global dynamics of the Euler--Riesz system in three dimensions, focusing on the well-posedness and large-time behavior of solutions near equilibrium. The system generalizes classical interactions by incorporating the Riesz interactions $\nabla (-\Delta)^{-\sigma/2}(\rho - 1)$. We show that the system admits a global smooth solution for small irrotational initial perturbations. Specifically, we establish that if the initial data is sufficiently small, the solution remains regular globally in time and decays over time at a rate dependent on $\sigma$.