Nonlinear stability and instability of rotating Riesz star solutions of the compressible Euler-Riesz equations

Technical Summary: Nonlinear Stability and Instability of Rotating Riesz Star Solutions of the Compressible Euler-Riesz Equations

Problem Statement
This paper investigates the existence, nonlinear stability, and nonlinear instability of rotating steady-state solutions, termed "rotating Riesz stars," for the attractive three-dimensional compressible Euler–Riesz equations (CEREs). The system models compressible fluids with non-local attractive interactions governed by a Riesz potential $\Phi_\alpha(x) = -|x|^{-\alpha}/\alpha$ for $\alpha \in (0, 3)$. The study focuses on axisymmetric configurations where the fluid rotates about a fixed axis, characterized by a specific angular momentum profile. The analysis is divided into two regimes based on the polytropic exponent $\gamma$: the mass-subcritical regime ($\gamma > \frac{3+\alpha}{3}$) and the mass-supercritical regime ($\frac{6}{6-\alpha} < \gamma < \frac{3+\alpha}{3}$).

Methodology
The authors employ a variational framework combined with concentration compactness arguments and scaling analysis.

1. Variational Formulation:

   • Mass-Subcritical Regime: Rotating Riesz stars are constructed as minimizers of a free-energy functional $G(\varrho)$ defined over a set of axisymmetric densities with fixed mass $M$. This functional includes internal energy, rotational kinetic energy, and potential energy.
   • Mass-Supercritical Regime: Since the free-energy is unbounded from below in this regime, the authors introduce a constrained minimization problem. They define an admissible set $K$ consisting of densities satisfying a specific condition on the derivative of the energy under mass-preserving scalings (specifically $Q(\varrho)=0$ and $Q(\varrho_\lambda) > 0$ for $\lambda < 1$). Minimizers of a modified functional $S_\mu$ over $K$ are sought.

2. Concentration Compactness in Axisymmetry:
   A central technical challenge is adapting Lions' concentration compactness principle to the axisymmetric setting. Unlike the non-rotating case, translations in the radial direction destroy axisymmetry. The authors develop a refined framework to classify minimizing sequences, ruling out "vanishing" and "dichotomy." A novel difficulty arises from the rotational term: minimizing sequences may concentrate along rings with radii diverging to infinity. The authors prove that unless the mass remains tight around the origin, the potential energy vanishes in the limit, leading to a contradiction with the variational structure.

3. Scaling and Concavity Analysis:
   To address the mass-supercritical regime and instability, the paper analyzes the behavior of the energy functional under mass-preserving scalings $\varrho_\lambda(x) = \lambda^{3/2}\varrho(\lambda^{1/2}x)$.

   • In the stationary case, a unique critical scaling exists. In the rotating case, particularly for $\alpha \in (0, 2)$, the presence of rotational kinetic energy can lead to non-uniqueness of critical scalings (up to two distinct roots).
   • The authors introduce a quantity $\kappa(\varrho)$ to distinguish between these branches. They analyze the concavity of the energy along these scalings to identify the branch corresponding to local maxima (relevant for existence) and to establish instability mechanisms.

4. Stability and Instability Proofs:

   • Stability: For the subcritical regime, stability is proven by showing that solutions starting close to a minimizer remain close, utilizing the non-negativity of relative free-energy functionals and the compactness of minimizing sequences.
   • Instability: For the supercritical regime, instability is established by analyzing the concavity of the free-energy. The authors show that for initial data arbitrarily close to the steady state but lying in an invariant set where the energy is below the minimizer's value, the support of the solution grows indefinitely.

Key Contributions and Results

• Existence in Mass-Subcritical Regime: The paper proves the existence of rotating Riesz star solutions for general pressure laws under subhomogeneity assumptions on the angular momentum profile. The minimizers are shown to be continuous, axisymmetric, and possess compact support.
• Nonlinear Stability (Subcritical): It is established that these rotating Riesz stars are nonlinearly stable. Solutions starting sufficiently close to a steady state (in terms of relative free-energy and momentum deviation) remain close for all time, up to vertical translations.
• Existence in Mass-Supercritical Regime: The authors extend the theory to the mass-supercritical regime, proving the existence of rotating Riesz stars for polytropic fluids. This result holds beyond the small angular velocity regime, a significant extension even for the classical Euler-Poisson equations. The proof requires a superhomogeneity condition on the angular momentum profile ($L(am) \le a^\omega L(m)$ with $\omega > \omega^* + \bar{\omega}$) to ensure the admissible set is non-empty and minimizers correspond to physical solutions.
• Nonlinear Instability (Supercritical): The paper demonstrates that rotating Riesz stars in the mass-supercritical regime are nonlinearly unstable. Solutions starting arbitrarily close to the steady state exhibit unbounded growth of their support.
• Role of Rotation: The results highlight that rotation can have either a stabilizing or destabilizing effect depending on the singularity of the Riesz interaction ($\alpha$) and the regime (subcritical vs. supercritical). Specifically, in the supercritical regime, rotation introduces new difficulties regarding the uniqueness of critical scalings, which the authors resolve through a detailed analysis of the concavity of the energy functional.

Significance
The paper claims to provide the first comprehensive analysis of rotating Riesz stars in the mass-supercritical regime, extending variational methods to settings where the energy is unbounded from below. It addresses the specific analytical difficulties introduced by rotation, such as the loss of radial decreasing rearrangement properties and the emergence of multiple critical scalings. The work generalizes previous results on non-rotating stars and the classical Euler-Poisson equations to the non-local Euler-Riesz setting, covering a broad range of interaction singularities ($\alpha \in (0, 3)$). The findings suggest that the stability of stellar configurations is highly sensitive to the interplay between the interaction potential's singularity, the equation of state, and the angular momentum distribution.