Nonlinear stability of 2-D Couette flow for the compressible Navier-Stokes equations at high Reynolds number

1. Problem Statement

The paper investigates the nonlinear stability of the two-dimensional Couette flow $(\bar{\rho}, \bar{u}) = (1, (y, 0)^T)$ for the compressible isentropic Navier-Stokes equations in the domain $\mathbb{T} \times \mathbb{R}$ at high Reynolds numbers ($Re$).

• The Challenge: Unlike the incompressible case, compressible flows exhibit a significantly stronger lift-up effect (a linear mechanism causing transient growth of disturbances) even in two dimensions. This effect, combined with the coupling between density and velocity, makes the stability analysis much more difficult.
• The Transition Threshold: The central question is determining the transition threshold exponent $\beta$ such that if the initial perturbation $\|( \rho_{in}, u_{in}) - (1, y, 0)\|_X \leq \epsilon Re^{-\beta}$, the solution remains globally stable. Previous works on incompressible flows established $\beta \leq 1/3$ (Sobolev) in 2D and $\beta \leq 1$ in 3D. For compressible flows, the best previous result (Huang, Li, Xu) was $\beta \leq 11/3$.
• Goal: To establish the optimal stability threshold for the 2D compressible case and prove global existence of solutions for perturbations of size $\epsilon \mu$ (where $\mu \sim Re^{-1}$).

2. Methodology

The authors employ a sophisticated combination of Fourier multiplier methods, coordinate transformations, and energy estimates tailored to the specific structure of the compressible equations.

A. Reformulation and "Good Unknowns"

To handle the strong coupling and the lift-up effect, the authors introduce a new set of variables:

1. Decomposition: They decompose the velocity into compressible (divergence) and incompressible (vorticity) parts: $d = \text{div } v$ and $\omega = \text{curl } v$.
2. New Variable $w$: Inspired by linear analysis, they define a "good unknown" $w = \omega + n - \mu d$.
   • Significance: The equation for $w$ contains no linear lift-up term involving the divergence $d$. This structural decoupling is crucial, as it prevents the unbounded growth associated with the lift-up mechanism in the vorticity equation.
3. Moving Frame: They transform to a moving frame coordinates $(x_1, y_1) = (x - ty, y)$ to eliminate the transport terms $y\partial_x$. This reveals the enhanced dissipation and inviscid damping mechanisms.

B. Fourier Multipliers

The core of the proof relies on constructing specific time-dependent Fourier multipliers to capture the interplay between dissipation and transient growth:

• $\phi(t, k, \xi)$: A weight designed to tame the lift-up effect. It grows in time near the critical layer ($t \approx \xi/k$) to control the transient growth, satisfying $1 \lesssim \phi \lesssim \mu^{-2/3}$.
• $m_1(t, k, \xi)$: Captures enhanced dissipation (decay rate $\sim e^{-\mu^{1/3}t}$).
• $m_2(t, k, \xi)$: Captures inviscid damping.
• Strategy: They apply these multipliers to the Fourier modes of the perturbations. The weight $\phi^{-1/4}$ is specifically used to control the lift-up term in the nonlinear estimates, reducing the loss of derivatives compared to previous approaches.

C. Energy Functional Construction

The authors construct distinct energy functionals for different components and derivative orders:

• Separation of Modes: They treat the zero mode (streamwise independent) and non-zero modes separately.
  • Zero Mode: Requires special handling because the $L^2$ estimate for the streamwise velocity component ($P_0 v_1$) is unavailable. They derive a separate $L^2$ estimate for $(P_0 n, P_0 v_2)$ and use the density equation to cancel problematic transport terms.
  • Non-zero Modes: Benefit from enhanced dissipation and inviscid damping.
• Good vs. Bad Derivatives: They distinguish between "good derivatives" ($\nabla$) and "bad derivatives" ($\tilde{\nabla} = (\partial_{x_1}, \partial_{y_1} - t\partial_{x_1})$). The energy functionals are designed to control cross-terms involving these derivatives, ensuring that the "bad" derivatives do not cause a loss of regularity.
• High-Order Estimates: They establish a hierarchy of energy estimates, controlling higher-order derivatives of the compressible part by lower-order derivatives of the incompressible part (and vice versa), utilizing the small viscosity parameter $\mu$ to compensate for derivative losses.

3. Key Contributions

1. Optimal Stability Threshold: The paper proves that the stability threshold is $\beta = 1$ (in terms of viscosity $\mu$, the condition is $\|(\rho_{in}, u_{in}) - (1, y, 0)\|_{H^4} \leq \epsilon \mu$). This improves significantly upon the previous best result of $\beta \leq 11/3$ for compressible flows.
2. Sharpness of the Threshold: Formal asymptotic analysis suggests this threshold is sharp within the class of Sobolev perturbations. The analysis shows that the lift-up effect in compressible flows is stronger than in incompressible flows, necessitating a higher regularity or smaller initial data size compared to the incompressible 2D case ($\beta=0$ in Gevrey, $\beta \leq 1/3$ in Sobolev).
3. Decoupling Mechanism: The introduction of the variable $w = \omega + n - \mu d$ successfully decouples the lift-up mechanism from the vorticity equation in the 2D compressible setting, a structural insight that was previously missing.
4. Handling Compressibility: The work rigorously addresses the unique difficulties of compressible flows, specifically the stronger lift-up effect and the lack of a divergence-free condition, which prevents the "circumvention" strategies used in 3D incompressible proofs.

4. Main Results

Theorem 1.1: There exists a small $\epsilon$ independent of the viscosity coefficients such that if the initial perturbation satisfies:
$$\|(n_{in}, v_{in})\|_{H^4(\mathbb{T} \times \mathbb{R})} \leq \epsilon \mu$$
then the system admits a unique global solution $(n, d, \omega)$ that remains close to the Couette flow for all time. Specifically:

• Zero Mode: The solution decays algebraically: $\|(P_0 \partial_y n, P_0 d, P_0 \partial_y \omega)\|_{L^2} \leq C(1+\mu t)^{-1/2}$.
• Non-Zero Modes: The solution exhibits enhanced dissipation and inviscid damping:
  $$\|(P_{\neq} n, P_{\neq} \Lambda_x^{-1} d)\|_{L^2} \leq C(1+t)^{1/2} e^{-\epsilon_1 \mu^{1/3} t}$$
  $$(1+t)\|P_{\neq} \partial_x \Delta_x^{-1} \omega\|_{L^2} + \|P_{\neq} \partial_y \Delta_x^{-1} \omega\|_{L^2} \leq C(1+t)^{-1/2} e^{-\epsilon_1 \mu^{1/3} t}$$

5. Significance

• Resolution of the Sommerfeld Paradox for Compressible Flows: This work provides a rigorous mathematical explanation for why Couette flow remains stable at high Reynolds numbers despite linear instability or transient growth, specifically for compressible fluids.
• Bridging the Gap: It closes the gap between numerical/physical observations and mathematical theory for compressible shear flows, which had lagged behind the incompressible case.
• Methodological Advancement: The techniques developed, particularly the "good unknown" $w$ and the specific Fourier multiplier weights for compressible modes, provide a new toolkit for analyzing stability in other compressible fluid problems where lift-up effects are dominant.
• Optimality: By establishing the threshold $\beta=1$ and arguing for its sharpness, the paper sets a definitive benchmark for the stability of 2D compressible Couette flow.