Low Mach number limit and optimal time decay rates of the compressible Navier-Stokes-transport system in critical Besov spaces

Technical Summary: Low Mach Number Limit and Optimal Time Decay Rates of the Compressible Navier-Stokes-Transport System in Critical Besov Spaces

Problem Statement
This paper investigates the compressible Navier-Stokes-Transport (NST) system, a fluid model comprising a compressible Navier-Stokes system for density ($\rho$) and momentum ($\rho u$), coupled with a transport equation for the potential temperature ($\theta$). Unlike the well-known Navier-Stokes-Fourier (NSF) system, where temperature satisfies a parabolic equation providing dissipation for both temperature and density, the NST system features a transport equation for temperature. This structural difference precludes a dissipative mechanism for the density, leading to significantly different asymptotic behaviors.

The study addresses three primary mathematical challenges within the framework of critical Besov spaces ($\dot{B}^s_{p,r}$) on $\mathbb{R}^d$ ($d \ge 2$):

1. Global Well-posedness: Establishing the existence and uniqueness of strong solutions for small initial data in critical spaces, despite the lack of density dissipation.
2. Low Mach Number Limit: Rigorously proving the convergence of the compressible NST system to the incompressible inhomogeneous Navier-Stokes system as the Mach number $\varepsilon \to 0$, specifically for ill-prepared initial data.
3. Optimal Time Decay Rates: Deriving optimal decay rates for strong solutions in dimensions $d \ge 3$, addressing the difficulty that the density does not decay over time due to the absence of a dissipative structure.

Methodology
The authors employ a refined energy analysis within the Littlewood-Paley decomposition framework of Besov spaces. Key methodological innovations include:

• Reformulation and New Variables: To handle the strong nonlinearity of the pressure and the lack of density dissipation, the system is reformulated in terms of density perturbation $a = \rho - 1$, velocity $u$, and pressure perturbation $z = P - 1$. Crucially, the authors introduce a new derived quantity $\omega = \gamma a - z$. While $\omega$ itself lacks a dissipative structure, it provides a crucial link between $a$ and $z$, allowing for the derivation of uniform estimates for the density $a$ that would otherwise be unattainable due to the linear term $\text{div } u$ in the continuity equation.
• Frequency-Localized Analysis: The proof of global well-posedness utilizes a frequency-localized Lyapunov inequality. The system $(u, z)$ is treated as a hyperbolic-parabolic system where low frequencies exhibit heat-diffusive behavior and high frequencies exhibit damping. The authors estimate the incompressible part ($Pu$) and potential part ($Qu$) of the velocity separately in both low and high-frequency regimes, employing the effective velocity method for high frequencies.
• Low Mach Number Limit Strategy: For the singular limit $\varepsilon \to 0$, the authors utilize a rescaling argument. To overcome the lack of compactness for the density $a^\varepsilon$ (which does not belong to $L^2(\mathbb{R}^+; \dot{B}^{d/2}_{2,1})$), they analyze the equation for the new mode $\omega^\varepsilon = \gamma a^\varepsilon - z^\varepsilon$. This allows them to establish strong compactness for $\omega^\varepsilon$ and subsequently justify the convergence of the continuity equation in the sense of distributions. Dispersive estimates for $z^\varepsilon$ and $Qu^\varepsilon$ are derived to handle the convergence of the momentum equation.
• Time-Decay Analysis: For decay rates, the authors distinguish between bounded and smallness conditions on the initial data. They utilize time-weighted energy functionals and interpolation inequalities. A significant technical hurdle is the "derivative loss" caused by nonlinear terms involving the non-dissipative density $a$ (e.g., $f(a)\nabla z$). The authors overcome this by leveraging the coupling between $u$ and $z$ and the new mode $\omega$ to close energy estimates, avoiding the need for additional smallness assumptions on low-frequency initial data that are typical in isentropic studies.

Key Contributions and Results

1. Global Well-posedness (Theorem 1.2):
   The paper establishes the global existence and uniqueness of strong solutions for the compressible NST system in critical Besov spaces $\dot{B}^{d/2}_{2,1} \times \dot{B}^{d/2-1}_{2,1} \times \dot{B}^{d/2}_{2,1}$ for $d \ge 2$, provided the initial data is sufficiently small.

   • Significance: Unlike previous works (e.g., Zhai et al. [59]), this result provides optimal time decay rates and utilizes a different method that explicitly handles the lack of density dissipation via the new mode $\omega$.

2. Low Mach Number Limit (Theorem 1.3):
   The authors rigorously prove that as $\varepsilon \to 0$, the solutions of the compressible NST system converge to the global solution of the incompressible inhomogeneous Navier-Stokes system.

   • Significance: This result holds for ill-prepared initial data (where $\text{div } u_0 = O(1)$ and $\nabla z_0 = O(1)$), a case that is mathematically more challenging than the well-prepared case. The convergence is established in the sense of distributions, with explicit convergence rates for the acoustic modes.

3. Optimal Time Decay Rates (Theorems 1.4–1.6):
   For $d \ge 3$, the paper derives optimal time decay rates for the velocity $u$ and pressure/temperature perturbation $z$.

   • Upper Bounds: Under a boundedness condition on initial data (Theorem 1.4), decay rates of $(1+t)^{-\frac{1}{2}(\sigma - \sigma_0)}$ are obtained for $\sigma \in (\sigma_0, d/2 - 1]$. Under a smallness condition (Theorem 1.5), these rates extend to higher regularity $\sigma \in (\sigma_0, d/2 + 1]$.
   • Lower Bounds: Theorem 1.6 establishes matching lower bounds, confirming the optimality of the decay rates.
   • Distinctive Behavior: A key finding is that the density $a$ remains uniformly bounded in time but does not exhibit temporal decay. This contrasts sharply with the NSF system and isentropic Navier-Stokes equations, where the density possesses a dissipative structure and decays. The authors note that the lack of density dissipation prevents the nonlinear coupling terms from contributing additional decay rates to the system.

Significance and Claims
The paper claims to resolve significant technical challenges arising from the hyperbolic nature of the density and temperature transport in the NST system. By introducing the mode $\omega = \gamma a - z$, the authors successfully bypass the obstruction caused by the lack of a dissipative structure for the density, a limitation that prevents the application of classical perturbation theories used for isentropic or NSF systems.

The work extends the mathematical understanding of non-isentropic fluid dynamics, particularly in regimes where diabatic processes are neglected. The results on the low Mach number limit for ill-prepared data in critical Besov spaces fill a gap in the literature, providing a rigorous justification for the incompressible limit in a setting where standard compactness arguments fail. Furthermore, the derivation of optimal decay rates, while explicitly noting the non-decaying nature of the density, offers a precise characterization of the long-time behavior of this specific fluid model.