$L^{2}-L^{\infty}$ decay estimates and inviscid limits for Global smooth solutions to the compressible Navier-Stokes-Riesz system

Abstract

We study the Cauchy problem in $\mathbb{R}^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-Δ)^{-s}(ρ-\barρ)$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.