Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping

Technical Summary: Global Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping

Problem Statement
The paper addresses the longstanding open problem of the global well-posedness of the multi-dimensional compressible Euler equations with damping. While the global existence of smooth solutions for small initial data in the isentropic regime ($\gamma > 1$) has been partially established in previous works (e.g., [29, 31]), the case allowing for partially large initial data in three dimensions remains challenging. Specifically, the authors investigate the Cauchy problem for the three-dimensional damped Euler equations in both the isentropic regime ($\gamma > 1$) and the isothermal regime ($\gamma = 1$). The primary difficulty lies in the fact that without damping, solutions typically develop singularities (shocks) in finite time regardless of the smoothness of initial data. While damping improves regularity for small data, it does not prevent singularity formation for large data in general. This work seeks to establish global existence and optimal decay rates for initial data where the $L^2$-norm is allowed to be large, provided the third-order Sobolev norm is sufficiently small.

Methodology
The authors employ a combination of energy methods, Green function analysis, and compactness arguments.

1. Transformation to Symmetric Hyperbolic System: For the isentropic case ($\gamma > 1$), the authors introduce a new variable $\phi = \frac{2}{\gamma-1}(\sigma(\rho) - \sigma(\rho^*))$, where $\sigma(\rho) = \sqrt{P'(\rho)}$ is the sound speed. This transforms the original system into a symmetric hyperbolic system with partial damping. This reformulation is crucial for deriving a priori estimates.
2. Regularity Criterion: A key step is establishing a regularity criterion (Theorem 3.1). The authors prove that if the integral of the $L^\infty$ norms of the gradients of $\phi$ and $u$ over time remains sufficiently small, the local solution can be extended globally.
3. Green Function and Decay Analysis: The linearized system is analyzed via its Green function $G$. By decomposing the solution into low-frequency ($G^\ell$) and high-frequency ($G^h$) parts, the authors derive precise decay rates.
   • The low-frequency part is shown to satisfy optimal algebraic decay rates similar to the heat equation, derived from the eigenvalues of the linearized operator.
   • The high-frequency part is controlled using energy estimates and the damping structure, which provides exponential decay.
4. Handling Large Initial Data: The framework allows for large $L^2$ initial data ($\|( \rho_0 - \rho^*, u_0) \|_{L^2} \le M_0$) by assuming the higher-order norm is small ($\|D^3(\rho_0 - \rho^*, u_0)\|_{L^2} \le \delta_0$). The proof relies on the smallness of $\delta_0$ to close the a priori estimates, utilizing interpolation inequalities (Gagliardo-Nirenberg) to bound the $L^\infty$ norms of gradients by the small high-order norms and the potentially large low-order norms.
5. Isothermal Limit: To address the case $\gamma = 1$, where the transformation $\phi$ becomes singular, the authors utilize a compactness argument. They first establish uniform estimates for solutions with respect to $\gamma \in (1, 3]$. Then, using the Aubin-Lions lemma, they prove the strong convergence of the solution sequence as $\gamma \to 1$, thereby rigorously justifying the isothermal limit.

Key Contributions and Results

• Global Well-Posedness with Partially Large Data: The paper establishes the existence of unique global classical solutions for the 3D damped Euler equations in both isentropic ($\gamma > 1$) and isothermal ($\gamma = 1$) regimes. The initial data is permitted to have a large $L^2$-norm, provided the third-order Sobolev norm is small. This extends previous results which typically required smallness in the full Sobolev norm.
• Optimal Algebraic Decay Rates: The authors derive optimal decay rates for the global solutions. Specifically, for $k=0, 1, 2, 3$:
  • $\|D^k(\rho - \rho^*)(t)\|_{L^2} \le C(1+t)^{-k/2}$
  • $\|D^k u(t)\|_{L^2} \le C(1+t)^{-(k-1)/2}$
    These rates match those of the corresponding linearized system, confirming their optimality.
• Isothermal Limit: The paper rigorously proves that as $\gamma \to 1$, the solutions of the isentropic damped Euler system strongly converge to the solution of the isothermal damped Euler system. This result simultaneously establishes the global well-posedness and asymptotic decay for the isothermal case ($\gamma = 1$).
• New Analytical Framework: For the isentropic case, a new framework is developed where all required a priori estimates are derived under the condition that $\int_0^T (\|\nabla \rho\|_{L^\infty} + \|\nabla u\|_{L^\infty}) dt$ remains small. This condition is verified using the derived optimal decay rates.

Significance
The paper claims to resolve a significant gap in the theory of compressible fluid dynamics by handling "partially large" initial data in three dimensions. By decoupling the size of the $L^2$ norm from the higher-order regularity requirements, the authors demonstrate that global smooth solutions exist even when the initial perturbation in density and velocity is large in the $L^2$ sense, as long as the high-frequency components are small. Furthermore, the unified treatment of the isothermal limit provides a rigorous mathematical foundation for the transition between isentropic and isothermal models in the presence of damping, a transition that was previously difficult to justify due to the singularity in the standard variable transformation at $\gamma = 1$. The results are presented as a direct consequence of the developed analytical techniques, without invoking experimental proposals or unverified future applications.