On the large-time behavior of strong solutions to the generalized compressible Navier-Stokes-Korteweg system in 2D and 3D for arbitrarily large initial data

Technical Summary: Global Existence and Large-Time Behavior of Strong Solutions to the Generalized Compressible Navier-Stokes-Korteweg System

Problem Statement
This paper investigates the global existence and large-time asymptotic behavior of strong solutions to the two- and three-dimensional periodic compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data $(\rho_0, u_0) \in H^3 \times H^2$. The system is characterized by density-dependent viscosity coefficients satisfying the Bresch-Desjardins (BD) relation, $\mu(\rho) = \nu\rho^\alpha$ and $\lambda(\rho) = 2\nu(\alpha-1)\rho^\alpha$, and a capillarity coefficient $\kappa(\rho) = \varepsilon^2\alpha^2\rho^{2\alpha-3}$.

The primary mathematical challenges addressed are:

1. Global Existence with Large Data: Establishing the existence of global strong solutions without smallness assumptions on the initial data, particularly in higher dimensions ($N=2,3$).
2. Uniform Density Bounds: Proving that the density remains uniformly bounded away from vacuum and concentration (i.e., $C^{-1} \le \rho(x,t) \le C$) for all time, a prerequisite for higher-order regularity.
3. Large-Time Behavior: Demonstrating the convergence of solutions to the equilibrium state as $t \to \infty$.

Previous works had established global existence for specific parameter regimes (e.g., Gu-Huang-Meng-Zhou [16] for $\alpha < 1$ and Huang-Meng-Zhang [25] for $\alpha=1$), but often with restricted adiabatic exponents $\gamma$ or without establishing uniform-in-time density bounds and large-time convergence for arbitrary large data.

Methodology
The authors employ a multi-faceted analytical strategy centered on the effective velocity transformation and a doubly parabolic structure of the reformulated system.

1. Effective Velocity Formulation: The system is reformulated using the effective velocity $v := u + c\rho^{\alpha-2}\nabla\rho$, where $c = \nu + \sqrt{\nu^2-\varepsilon^2}$. This transforms the original system into a coupled density-effective velocity system (Equation 11), which exhibits a doubly parabolic structure. This structure is crucial for deriving additional regularity and controlling the density.

2. Bootstrap Argument for Admissible Parameters:

   • For the case $\alpha < 1$, the authors improve the admissible range of the adiabatic exponent $\gamma$. They utilize a bootstrap argument that bridges the gap between the standard energy estimate (weighted $L^2$) and the critical estimate (weighted $L^3$). By iteratively improving the integrability of the effective velocity and the density, they extend the range of $\gamma$ beyond previous results (specifically, up to $\frac{111\alpha^2-122\alpha+33}{21\alpha-13}$ in 3D).

3. Time-Discretization Strategy for Upper Bounds:

   • To establish a uniform upper bound for the density, the authors adopt a time-discretization strategy. They partition the time axis $[0, \infty)$ into subintervals $[t_n, t_{n+1}]$ of uniformly bounded length.
   • Using the dissipation estimates, they identify points $t_n$ where the density and effective velocity satisfy uniform quantitative bounds.
   • On each subinterval, they apply a modified Nash-Moser iteration scheme (inspired by Gu-Huang-Meng-Zhou [16]) to derive a density upper bound that is independent of the specific subinterval, thus yielding a global uniform upper bound.

4. Inductive Bootstrap for Lower Bounds ($\alpha < 1$):

   • Establishing a uniform positive lower bound requires stronger integrability of the effective velocity ($L^p$ with $p > \frac{2}{1-\alpha}$).
   • The authors develop a novel inductive bootstrap procedure. Assuming $(k-1)$-th level estimates, they derive $k$-th level uniform and dissipation estimates. This process is repeated finitely many times until the required integrability is reached, allowing the application of the Nash-Moser iteration to the reciprocal variable $\tau = \rho^{-1}$ to obtain a uniform positive lower bound.
   • For the critical case $\alpha = 1$, the paper establishes a uniform upper bound but notes that the lower bound remains an open problem due to the inability to close the bootstrap at the $L^\infty$ level for the effective velocity.

5. Higher-Order Estimates and Decay:

   • With uniform density bounds established, the authors derive global higher-order estimates for the density and effective velocity.
   • They exploit the elliptic structure inherent in the effective velocity system to control higher-order derivatives.
   • These estimates lead to the proof of large-time behavior, showing that the $H^3$-norm of the density deviation and the $H^1$-norm of the velocity gradient decay to zero.

Key Contributions and Results

• Theorem 1.1 (Case $\alpha < 1$):

  • Establishes the global existence of unique strong solutions for arbitrarily large initial data in 2D and 3D.
  • Enlarged Parameter Range: Significantly expands the admissible range of the adiabatic exponent $\gamma$ compared to previous works (e.g., [16]). In 3D, $\gamma$ can now reach $\frac{111\alpha^2-122\alpha+33}{21\alpha-13}$.
  • Uniform Bounds: Proves that the density satisfies $C^{-1} \le \rho(x,t) \le C$ for all $t \ge 0$, ensuring the solution remains away from vacuum and concentration.
  • Large-Time Behavior: Proves that $\|\rho(t) - \rho_s\|_{H^3} + \|\nabla u(t)\|_{H^1} \to 0$ as $t \to \infty$, where $\rho_s$ is the average density. This holds without any smallness assumption on the initial data.

• Theorem 1.2 (Case $\alpha = 1$):

  • Establishes global existence for the critical case $\alpha = 1$ (which includes the viscous Saint-Venant model and quantum Navier-Stokes system under specific conditions).
  • Improved Parameter Range: Further enlarges the admissible range of $\gamma$ in 3D to $[1, 11/4)$, improving upon the $[1, 8/3)$ range in [22] and the $[1, 7/3)$ range in [25].
  • Uniform Upper Bound: Proves a uniform upper bound for the density ($\rho(x,t) \le C$).
  • Limitation: The paper explicitly states that establishing a uniform positive lower bound for the density in the critical case $\alpha=1$ remains an open challenge and is left for future work.

Significance
The paper provides an affirmative answer to the questions of whether global strong solutions with arbitrarily large initial data admit uniform-in-time density bounds and converge to equilibrium in the multidimensional periodic NSK system.

• Resolution of Open Problems: It resolves the open problem regarding the large-time behavior and uniform density bounds for the generalized NSK system with large data, which were previously unresolved even after the establishment of global existence in [16] and [25].
• Methodological Advancement: The development of the time-discretization strategy combined with the bootstrap argument for integrability represents a significant methodological advancement. It allows for the derivation of uniform bounds without relying on the restrictive intermediary regime ($\varepsilon = \nu$) often required in quantum Navier-Stokes analyses, thereby covering the broader parabolic regime ($\varepsilon < \nu$).
• Parameter Optimization: By refining the admissible ranges for $\gamma$, the results broaden the physical applicability of the global existence theory for compressible fluids with capillarity effects.

The authors emphasize that their results are derived without imposing smallness conditions on the initial data, making the findings robust for physically relevant scenarios involving large perturbations. The work leaves the extension to the Cauchy problem and the establishment of lower bounds for the critical case $\alpha=1$ as directions for future research.