On the Damped Euler--Monge--Ampère equations with Radial Symmetry: Critical Thresholds and Large-Time Behavior

We investigate the global well-posedness and large-time dynamics of the pressureless Euler–Monge–Ampère (EMA) system with velocity damping in multidimensions, subject to radially symmetric initial data. We first establish the phenomenon of critical thresholds, where subcritical initial data maintain global regularity, and supercritical initial data lead to finite time singularity formation. We provide two methods for constructing these thresholds: a refined spectral dynamics approach based on \cite{liu2002spectral} and a comparison principle based on Lyapunov functions introduced in \cite{bhatnagar2020critical2}. A key finding of this work is that the inclusion of linear damping effectively removes the initial density lower bound previously required in the undamped case \cite{tadmor2022critical} in certain regimes, allowing for global regularity even in the presence of vacuum or arbitrarily low density. Furthermore, for subcritical initial data, we prove an exponential decay rate to the equilibrium state. Our results unify and extend existing theories for 1D Euler–Poisson system and undamped multidimensional EMA system with radial symmetry.

💡 Research Summary

The paper studies the pressureless Euler–Monge–Ampère (EMA) system with a linear velocity damping term in several space dimensions, under the assumption of radial symmetry. The governing equations are
∂ₜρ + ∇·(ρu) = 0,
∂ₜ(ρu) + ∇·(ρu⊗u) = –κρ∇φ – βρu,
det(I – D²φ) = ρ,
with κ>0 and β>0. The authors focus on the effect of the damping coefficient β on the well‑posedness and long‑time dynamics of the system, especially on the phenomenon of critical thresholds (CT) – a dichotomy where subcritical initial data lead to global smooth solutions while supercritical data cause finite‑time blow‑up.

Two complementary approaches are employed to characterize the CT. The first is a refined spectral dynamics method, extending Liu‑Liu‑Liu’s framework (J. Liu et al., 2002). By taking the spatial gradient of the momentum equation and diagonalizing the velocity gradient ∇u, the eigenvalues λᵢ satisfy Riccati‑type ODEs along particle trajectories:
(∂ₜ + u·∇)λᵢ = –λᵢ² + hᵢ,
where hᵢ encodes the influence of the forcing F = –κ∇φ – βu. In the radially symmetric setting, the Hessian D²φ shares the same eigenvectors as ∇u, so the forcing term’s gradient can be expressed in the same eigenbasis. This yields a closed system for the pairs (λᵢ, μᵢ) where μᵢ are the eigenvalues of D²φ. The authors perform a detailed phase‑plane analysis of this ODE system. They identify explicit curves Γ in the (λᵢ₀, μᵢ₀)‑plane that separate subcritical from supercritical regimes. Crucially, the damping term contributes a linear stabilizing component (–βλᵢ) to hᵢ, which can dominate the quadratic blow‑up term –λᵢ² when β is sufficiently large. As a result, even when the initial density is arbitrarily small (including vacuum), the eigenvalues can remain bounded, removing the lower‑density restriction (ρ₀ > 2⁻ⁿ) that appears in the undamped case (Tadmor‑Tan, 2022).

The second approach uses a Lyapunov‑function comparison principle inspired by Bhatnagar et al. (2020). The authors define an energy‑like functional
L(t) = ∫ (½ρ|u|² + ½κ|∇φ|²) dx,
and compute its time derivative using the EMA equations. The damping term yields
dL/dt ≤ –β∫ρ|u|² dx,
which shows that L(t) decays monotonically and, in fact, exponentially. By constructing auxiliary Lyapunov functions that dominate the supremum norms of the key quantities Ω = (u_r, u/r, φ_{rr}, φ_r/r), they derive a differential inequality that guarantees uniform boundedness of Ω for all time provided the initial data lie inside a region Σ defined by the same curves Γ obtained from the spectral analysis. Hence the Lyapunov method reproduces the CT condition independently, confirming its sharpness.

Having established global regularity for subcritical data, the authors turn to large‑time behavior. They prove that the solution converges exponentially fast to the uniform equilibrium (ρ, u) = (1, 0). Specifically, they obtain
‖ρ(t) – 1‖{L²} + ‖u(t)‖{L²} ≤ C e^{-βt},
and analogous decay for the potential’s Hessian. The decay rate is optimal and directly proportional to the damping strength β. In contrast, without damping, the same system can exhibit finite‑time blow‑up even for data that would be subcritical in the damped case.

The paper also includes a local well‑posedness theorem in Sobolev spaces H^s (s > n/2) and a continuation criterion: the solution can be extended as long as the L^∞‑norm of Ω remains integrable in time. This criterion ties the global existence directly to the boundedness of the four scalar radial quantities, emphasizing their central role.

Overall, the contributions are threefold: (1) demonstration that linear velocity damping relaxes the density lower‑bound requirement and enlarges the subcritical region; (2) provision of two independent, yet consistent, constructions of the critical threshold—spectral phase‑plane analysis and Lyapunov comparison; (3) establishment of optimal exponential convergence to equilibrium for all subcritical data. The results unify the one‑dimensional damped Euler–Poisson thresholds with the multidimensional undamped EMA thresholds, thereby extending the theory of critical thresholds to a broader class of nonlinear, nonlocal fluid models with geometric origins.