The global well-posedness of the multi-dimensional compressible Euler system with damping in the $L^p$ critical Besov spaces for $p<2$

1. Problem Statement

The paper investigates the Cauchy problem for the multi-dimensional compressible Euler system with linear damping, which models gas flow through a porous medium. The governing equations are:
$$\begin{cases} \partial_t \rho + \text{div}(\rho u) = 0, \\ \partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) + \nabla P(\rho) + \alpha \rho u = 0, \end{cases}$$
where $\rho$ is density, $u$ is velocity, $P(\rho)$ is the pressure function, and $\alpha > 0$ is the damping coefficient.

The Core Challenge:
While global well-posedness results for this system were previously established in $L^2$-type hybrid Besov spaces (where low frequencies are in $L^p$ and high frequencies in $L^2$) for $p \ge 2$, the case for $1 \le p < 2$ remained open.

• Physical Context: The case $p < 2$ corresponds to more singular and concentrated initial data (e.g., point sources, vortex filaments), which are physically relevant but mathematically more difficult to handle due to the lack of standard $L^2$ energy estimates for low frequencies in non-$L^2$ spaces.
• Mathematical Obstacle: Standard hyperbolic systems are often ill-posed in $L^p$ spaces for $p \neq 2$. The presence of damping is crucial, but extending the framework to $p < 2$ requires overcoming specific difficulties in estimating nonlinear terms in mixed $L^p-L^2$ spaces.

2. Methodology

The authors employ a hybrid Besov space framework that distinguishes between low and high frequencies based on a threshold $J_\varepsilon \approx -\log_2 \varepsilon$.

A. Reformulation

The system is transformed using the "enthalpy" variable $n(\rho) = \int_1^\rho \frac{P'(s)}{s} ds$ and a scaled velocity. The system is rewritten as:
$$\begin{cases} \partial_t n + v \cdot \nabla n + P'(\bar{\rho})\text{div}v + G(n)\text{div}v = 0, \\ \partial_t v + v \cdot \nabla v + \nabla n = -\frac{1}{\varepsilon}v, \end{cases}$$
where $\varepsilon$ is a scaling parameter related to the damping coefficient.

B. Functional Framework

The solution space $E^J_\varepsilon$ is defined with specific regularity requirements for low frequencies ($\ell, J_\varepsilon$) and high frequencies ($h, J_\varepsilon$):

• Low Frequencies: Measured in $L^p$-type critical Besov spaces $\dot{B}^{\frac{d}{p}}_{p,1}$.
• High Frequencies: Measured in $L^2$-type critical Besov spaces $\dot{B}^{\frac{d}{2}+1}_{2,1}$.
  This hybrid approach leverages the damping effect (which acts as a heat-like operator at low frequencies) while maintaining the energy structure at high frequencies.

C. Key Technical Innovation: New Product Estimate

The primary difficulty lies in handling the nonlinear terms (e.g., $v \cdot \nabla n$) in the $L^p$ ($p<2$) low-frequency regime. The authors develop a new product estimate (Lemma 2.1) in $L^p-L^2$ hybrid Besov spaces.

• The Estimate: For $1 \le p < 2$ and $0 < s_1 < d/2$:
  $$\|ab\|_{\dot{B}^{s_1}_{p,1}}^{\ell, J_\varepsilon} \lesssim \left( \|b\|_{\dot{B}^{s_1}_{p,1}}^{\ell, J_\varepsilon} + 2^{(s_1 - \frac{d}{p})J_\varepsilon} \|b\|_{\dot{B}^{\frac{d}{2}}_{2,1}}^{h, J_\varepsilon} \right) \|a\|_{\dot{B}^{\frac{d}{2}}_{2,1}}$$
• Significance: This inequality allows the control of products where one function is in the critical $L^p$ space and the other involves high-frequency components, effectively bridging the gap between $L^p$ and $L^2$ estimates using Bony's paraproduct decomposition.

D. A Priori Estimates

The proof relies on establishing global-in-time a priori estimates via a two-step process:

1. Low-Frequency Analysis ($L^p$ Framework):
   • Introduces an "effective velocity" $z = \varepsilon^{-1}v + \nabla n$.
   • The equation for $n$ behaves like a heat equation, and $z$ behaves like a damped equation.
   • The new product estimate (Lemma 2.1) is used to bound nonlinear terms like $v \cdot \nabla n$ and $G(n)\text{div}v$.
2. High-Frequency Analysis ($L^2$ Framework):
   • Uses a classical Lyapunov energy method (weighted energy estimates) suitable for dissipative systems.
   • Commutator estimates are bounded using the high-frequency $L^2$ structure.

3. Key Contributions

1. Extension to $p < 2$: The paper successfully extends the global well-posedness theory for the damped compressible Euler system from $p \ge 2$ to the critical range $1 \le p < 2$.
2. New Analytical Tool: The derivation of the specific product estimate (Lemma 2.1) for $L^p-L^2$ hybrid spaces is a significant methodological contribution. This tool is noted to be applicable to other systems, such as hyperbolic-parabolic chemotaxis systems and Navier-Stokes equations in similar frameworks.
3. Handling Singular Data: By working in $\dot{B}^{\frac{d}{p}}_{p,1}$ for $p < 2$, the authors demonstrate existence and uniqueness for initial data that are more singular than those allowed in the standard $L^2$ framework (since $\dot{B}^{\frac{d}{p}}_{p,1} \hookrightarrow \dot{B}^{\frac{d}{2}}_{2,1}$ when $p < 2$).

4. Main Results

Theorem 1.1 (Global Well-Posedness):
Assume $1 \le p < 2$, $d \ge 1$, and $P'(\bar{\rho}) > 0$. There exists a small constant $\delta_1 > 0$ such that if the initial data $(n_0, v_0)$ satisfies:
$$\|(n_0, v_0)\|_{\dot{B}^{\frac{d}{p}}_{p,1}}^{\ell, J_\varepsilon} + \varepsilon \|(n_0, v_0)\|_{\dot{B}^{\frac{d}{2}+1}_{2,1}}^{h, J_\varepsilon} \le \delta_1,$$
then the Cauchy problem admits a unique global solution in the space $E^J_\varefty$ for all $\varepsilon > 0$.

• Uniformity: The solution is uniform with respect to the relaxation parameter $\varepsilon$.
• Relaxation Limit: The paper notes that the relaxation limit (convergence to the porous medium equation) for the case $p < 2$ can be performed using the same procedure as in previous works for $p \ge 2$.

5. Significance

• Mathematical Rigor: This work resolves a natural question regarding the limits of the $L^p$ framework for damped hyperbolic systems, proving that the critical Besov space theory holds even when the integrability index $p$ drops below 2.
• Physical Relevance: The ability to handle $p < 2$ allows for the mathematical modeling of fluids with highly concentrated initial disturbances (singularities), which are common in physical scenarios like shock waves or point-source injections.
• Future Applications: The new product estimate developed in this paper provides a robust tool for analyzing other coupled hyperbolic-parabolic systems where mixed $L^p-L^2$ regularity is required, potentially impacting the study of chemotaxis and fluid dynamics models.

In summary, this paper bridges a critical gap in the theory of compressible fluids with damping by introducing a novel hybrid Besov estimate that enables the treatment of highly singular initial data ($p < 2$) while maintaining global existence and uniqueness.