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

Technical Summary: $L^2 - L^\infty$ Decay Estimates and Inviscid Limits for Global Smooth Solutions to the Compressible Navier-Stokes-Riesz System

Problem Statement
This paper investigates the Cauchy problem for the compressible Navier-Stokes-Riesz system in $\mathbb{R}^3$, which models the motion of a charged compressible fluid with a self-consistent repulsive force. The system is governed by the continuity equation, the momentum equation with viscous terms, and a nonlocal potential equation involving the Riesz potential $(-\Delta)^{-s}$ with exponent $0 < s < 1$. The study focuses on two primary objectives:

1. Establishing the global existence, uniqueness, and time-decay rates of smooth solutions for small perturbations of a constant equilibrium state, specifically addressing the case where the initial solenoidal (divergence-free) velocity component is of order $O(\varepsilon)$.
2. Proving the global-in-time inviscid limit ($\varepsilon \to 0$) of these solutions to the corresponding irrotational solution of the compressible Euler-Riesz system, providing an explicit convergence rate.

The Riesz interaction term $\nabla(-\Delta)^{-s}(\rho - \bar{\rho})$ generalizes the Coulomb interaction (where $s=1$). While extensive literature exists for the Navier-Stokes-Poisson system ($s=1$) and the Euler-Riesz system, the range $0 < s < 1$ presents unique analytical challenges, particularly regarding the degeneracy of the phase function in the dispersive regime and the lack of existing global strong solution results for $s \in (0, 1/2)$.

Methodology
The authors employ a unified framework combining viscosity-adapted dispersive estimates, normal-form analysis, and nonlinear energy estimates. The methodology proceeds through the following key steps:

1. Decomposition and Approximation: The solution $(\rho^\varepsilon, u^\varepsilon)$ is decomposed into an approximate solution $(\rho, u)$ and a correction term $(n, v)$.

   • The approximate solution $(\rho, u)$ solves a viscous approximation of the Euler-Riesz system with irrotational initial data ($P^\perp u^\varepsilon_0$). This system preserves irrotationality and retains dissipative properties due to the artificial viscosity $\varepsilon L u$.
   • The correction system $(n, v)$ accounts for the difference between the full Navier-Stokes-Riesz system and the approximate irrotational system, driven by the initial solenoidal velocity component $P u^\varepsilon_0$.

2. Linear Analysis and Decay Estimates:

   • The linearized operator $A(D)$ is analyzed via its eigenvalues $\lambda_\pm(\xi) = -\varepsilon|\xi|^2 \pm i\sqrt{|\xi|^2 + |\xi|^{2-2s} - \varepsilon^2|\xi|^4}$.
   • The frequency domain is split into a dispersion-dominated region ($\Omega_L$, low frequencies where $\varepsilon|\xi|^2 \le 2\kappa_0$) and a dissipation-dominated region ($\Omega_H$, high frequencies).
   • In $\Omega_L$, the authors derive uniform-in-$\varepsilon$ $L^p$ decay estimates using stationary phase analysis. A critical distinction is made between $s \in (0, 1/2]$ (behaving like the Klein-Gordon equation) and $s \in (1/2, 1)$ (behaving more like the wave equation but with a degenerate phase function).
   • In $\Omega_H$, standard exponential decay estimates are derived.

3. Nonlinear Energy Estimates:

   • $H^N$ Estimates: The system is transformed into a symmetric hyperbolic-parabolic system using a change of variables. The authors derive high-order energy estimates, carefully handling the term $\int \nabla^k \phi \nabla^k \text{div} u \, dx$. For $s < 1/2$, this requires a modified energy approach involving negative Sobolev norms and commutator estimates.
   • $W^{k,p}$ Estimates: To capture optimal $L^\infty$ decay, the authors utilize the Duhamel formula. For the quadratic nonlinearity in the dispersive region, standard energy methods are insufficient. The authors apply Shatah's normal form method and the space-time resonance method. They construct a modified kernel to handle the singularity arising from the zero set of the phase function $\Phi_{jk}$, which is non-empty for the Riesz potential (unlike the Coulomb case). This allows control of the bilinear terms in negative Sobolev spaces.

4. Negative Sobolev Estimates and Iterative Decay:

   • To recover $L^2$ decay rates that reflect viscosity, the authors establish uniform estimates in negative Sobolev spaces $\dot{H}^{-\vartheta}$.
   • An iterative bootstrap procedure is used to improve decay rates step-by-step, leveraging the interplay between the $L^\infty$ decay (from dispersive estimates) and the $L^2$ dissipation.

5. Inviscid Limit:

   • The error system between the Navier-Stokes-Riesz solution and the Euler-Riesz solution is analyzed. Uniform energy estimates are derived to establish a convergence rate in $W^{N,p}$ norms.

Key Contributions and Results

• Global Existence for Full Range $s \in (0, 1)$: The paper proves the global existence and uniqueness of smooth solutions for the compressible Navier-Stokes-Riesz system for the entire range $0 < s < 1$. This extends previous results which were limited to $s \in (1/2, 1)$ or required specific conditions for $s=1$.
• Uniform Decay Estimates: The authors derive time-decay estimates in both $L^2$ and $L^\infty$ norms that are uniform with respect to the viscosity parameter $\varepsilon$.
  • For the density perturbation and velocity, the $L^\infty$ decay rate is shown to be $\min\{\varepsilon, (1+t)^{-\vartheta/2}\} + (1+t)^{-\beta_s}$, where $\beta_s$ depends on $s$ and the integrability index $p$.
  • The $L^2$ decay rates explicitly capture the viscosity-dependent dissipation, showing a rate of $(1+\varepsilon \Upsilon t)^{-s/2 - \vartheta/2}$ for density and $(1+\varepsilon \Upsilon t)^{-\vartheta/2}$ for velocity.
• Inviscid Limit with Explicit Rate: The paper establishes the global-in-time convergence of the Navier-Stokes-Riesz solution to the irrotational Euler-Riesz solution as $\varepsilon \to 0$. The convergence rate is explicitly quantified as $O(\varepsilon^{\lambda_s})$ in $W^{N,p}$ norms, where $\lambda_s$ depends on $s$ and $p$.
• Handling of Riesz Potential Degeneracy: A significant technical contribution is the analysis of the phase function for $s \in (1/2, 1)$, where the phase function exhibits degeneracy. The authors successfully adapt the normal form method to handle the non-empty time resonance sets and the resulting singularities in the kernel, a challenge not present in the $s=1$ (Coulomb) case.

Significance
The paper claims to provide a comprehensive analysis of the compressible Navier-Stokes-Riesz system, bridging the gap between the well-studied Navier-Stokes-Poisson system ($s=1$) and the Euler-Riesz system. By proving global existence for the full range $s \in (0, 1)$ and deriving uniform decay estimates, the work complements existing optimal decay results (such as those in Wang and Zhang [37] for $s > 1/2$) and extends them to the previously unresolved range $s \in (0, 1/2]$.

The authors note that while their derived $L^2$ decay rates are not necessarily sharp compared to the optimal rates for the Euler system, the analysis is crucial for understanding the interplay between dispersion and viscosity in the presence of nonlocal Riesz interactions. The explicit convergence rate for the inviscid limit provides a rigorous justification for using the Euler-Riesz system as an approximation for the Navier-Stokes-Riesz system in the low-viscosity regime, a result that was previously an open question for general $s \in (0, 1)$. The work relies heavily on the "dispersive" spectral condition for the initial velocity and the smallness of the initial data in specific Sobolev and $L^1$ spaces.