Nonlinear Stability and Instability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with General Pressure Laws

Technical Summary: Nonlinear Stability and Instability of Finite-Energy Solutions of the Compressible Euler-Riesz Equations with General Pressure Laws

Problem Statement
This paper investigates the nonlinear stability and instability of steady states for the multidimensional compressible Euler-Riesz equations (CEREs) in $\mathbb{R}^n$ ($n \ge 2$). The system models physical phenomena such as stellar dynamics, plasma physics, and swarming, governed by the conservation of mass and momentum coupled with a nonlocal Riesz potential interaction. The equations are given by:
$$\begin{cases} \partial_t \rho + \text{div} M = 0, \\ \partial_t M + \text{div}\left(\frac{M \otimes M}{\rho}\right) + \nabla p + \kappa \rho \nabla W_\alpha = 0, \end{cases}$$
where $\rho$ is the density, $M$ is the momentum, $p=p(\rho)$ is a general pressure law, and $W_\alpha$ is the Riesz potential (or logarithmic potential for $\alpha=0$). The parameter $\kappa$ determines the nature of the interaction (attractive for $\kappa > 0$, repulsive for $\kappa < 0$).

The study focuses on the existence of global finite-energy weak solutions and the dynamical behavior of perturbations around steady states. A central challenge is the competition between internal energy and potential energy, which depends critically on the scaling exponent $\gamma$ in the polytropic case $p(\rho) \sim \rho^\gamma$. The authors distinguish between the mass-critical exponent $\gamma = \frac{n+\alpha}{n}$, the mass-supercritical regime $\gamma < \frac{n+\alpha}{n}$, and the mass-subcritical regime $\gamma > \frac{n+\alpha}{n}$.

Methodology
The authors employ a combination of variational methods, relative entropy techniques, and compensated compactness arguments to address the difficulties posed by the nonlocal potential, vacuum states, and the hyperbolic nature of the system.

1. Variational Approach and Concentration-Compactness:
   To establish the existence of steady states (minimizers of the free energy functional), the authors utilize Lions' concentration-compactness principle. They analyze the behavior of minimizing sequences to rule out vanishing and dichotomy, ensuring the existence of global minimizers for general pressure laws under specific asymptotic conditions on $p(\rho)$ near zero and infinity.

2. Relative Entropy and Stability Estimates:
   For the stability of steady states in the subcritical regime ($\gamma > \frac{n+\alpha}{n}$), the authors develop a quantitative relative entropy method. A key innovation is the construction of a modified relative internal energy that incorporates the Lagrange multiplier associated with the steady state's Euler-Lagrange equation. This modification allows for the control of density perturbations in vacuum regions (where the steady state vanishes) without requiring uniform positive lower bounds on the steady density, a restriction present in prior works.

3. Instability Analysis via Scaling and Concavity:
   In the mass-supercritical regime ($\frac{2n}{2n-\alpha} < \gamma < \frac{n+\alpha}{n}$), the authors prove nonlinear instability. They analyze the concavity of the free energy along mass-preserving dilations. By exploiting the scaling properties of the energy functional, they demonstrate that steady states are unstable, leading to the growth of the support of solutions.

4. Compensated Compactness for Global Existence:
   To prove the global existence of finite-energy weak solutions with spherical symmetry, the authors use a vanishing physical viscosity approach (approximating CEREs with compressible Navier-Stokes-Riesz equations). They derive uniform energy estimates, Bresch-Desjardins (BD) entropy estimates, and higher integrability estimates for the density and velocity. The convergence of the approximate solutions is established using the compensated compactness framework, specifically analyzing the entropy-entropy flux kernels and their singularities to handle the nonlocal Riesz potential term.

Key Contributions and Results

• Nonlinear Instability in the Mass-Supercritical Regime:
  The paper establishes the nonlinear instability of steady states for attractive CEREs when $\frac{2n}{2n-\alpha} < \gamma < \frac{n+\alpha}{n}$. This is achieved by showing that the free energy is concave along mass-preserving dilations. The authors prove that solutions starting arbitrarily close to a steady state can develop growing support, with the radius of the support growing linearly in time for solutions with positive energy. This result extends previous instability findings for the Euler-Poisson equations to the broader Riesz setting and general pressure laws.

• Nonlinear Stability in the Mass-Subcritical Regime:
  For general pressure laws satisfying specific asymptotic behaviors (analogous to $\gamma > \frac{n+\alpha}{n}$), the authors prove the existence of energy minimizers and their nonlinear orbital stability. They establish that solutions starting sufficiently close to a steady state remain close to it for all time, modulo translations.

• Quantitative Stability via Modified Relative Entropy:
  A significant technical contribution is the derivation of a relative entropy bound for finite-energy solutions without assuming uniform pointwise upper and lower bounds on the density. The authors introduce a modified relative entropy functional that accounts for the vacuum degeneracy. This allows them to quantify the stability of compactly supported steady states with vacuum free boundaries, overcoming limitations of previous methods that relied on $L^\infty$ bounds or strictly positive densities.

• Global Existence of Finite-Energy Weak Solutions:
  The paper proves the global existence of spherically symmetric finite-energy weak solutions for CEREs with general pressure laws and $\alpha \in (-1, n-1)$. This is achieved through the compensated compactness method, handling the nonlocal interaction term by establishing strong convergence of the density and the potential term. This result yields unconditional stability around steady states within the class of weak solutions for the subcritical case.

• Growth of Support for Positive Energy Solutions:
  The authors derive lower bounds on the asymptotic growth rate of the support radius for global classical solutions with positive energy. They address the case $\alpha > 2$, where standard virial arguments fail due to sign changes in the potential energy term, by showing that for sufficiently small mass, the internal energy dominates, restoring a positive lower bound on the second derivative of the second moment.

Significance and Claims
The authors claim that their approach provides a unified framework for analyzing the stability and instability of CEREs with general pressure laws, extending beyond the classical polytropic and Newtonian cases. The development of the modified relative entropy method is highlighted as a novel tool for handling vacuum states in nonlocal interaction models, removing restrictive assumptions on the steady state density found in previous literature. Furthermore, the global existence result for general pressure laws in the presence of Riesz interactions represents a significant advancement in the theory of hyperbolic systems with nonlocal terms. The paper suggests that the developed techniques, particularly the handling of nonlocal terms via concentration-compactness and compensated compactness, are applicable to other nonlinear partial differential equations with similar structural difficulties.