Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data

1. Problem Statement

The paper addresses the long-standing open problem of the global existence of strong solutions for the compressible Navier-Stokes-Korteweg (NSK) system in two and three dimensions with arbitrarily large initial data.

The system models compressible fluids with capillarity effects (surface tension) and density-dependent viscosity. The governing equations are:
$$\begin{cases} \rho_t + \text{div}(\rho u) = 0, \\ (\rho u)_t + \text{div}(\rho u \otimes u) + \nabla P = \text{div}(2\mu(\rho)D u) + \nabla(\lambda(\rho)\text{div}u) + \text{div} K, \end{cases}$$
where:

• $\rho$ and $u$ are density and velocity.
• $P = \rho^\gamma$ is the pressure.
• $K$ is the Korteweg stress tensor representing capillarity.
• The viscosity coefficients $\mu(\rho)$ and $\lambda(\rho)$, and the capillarity coefficient $\kappa(\rho)$, are density-dependent.

Specific Assumptions:
The authors assume a specific algebraic relation between the coefficients (a generalized Bresch-Desjardins (BD) relation and a generalized Bohm identity):

• $\mu(\rho) = \nu \rho^\alpha$
• $\lambda(\rho) = 2\nu(\alpha - 1)\rho^\alpha$
• $\kappa(\rho) = \varepsilon^2 \alpha^2 \rho^{2\alpha - 3}$
• Regime: The analysis is conducted in the non-dispersive regime, where the viscosity constant $\nu$ is greater than or equal to the capillarity constant $\varepsilon$ ($\nu \ge \varepsilon > 0$).

The Challenge:
While global weak solutions exist for certain cases, proving the existence of strong (smooth) solutions for large initial data in multiple dimensions has been a major hurdle. Previous results were often restricted to small data, specific symmetries (spherical), or specific parameter ranges (e.g., $\alpha=1$).

2. Methodology

The authors employ a sophisticated combination of energy estimates, modified Nash-Moser iteration schemes, and elliptic regularity theory to overcome the nonlinearities introduced by $\alpha < 1$.

A. Effective Velocity Transformation

A crucial step is the introduction of an effective velocity $v$, defined as:
$$v = u + c \rho^{\alpha-2} \nabla \rho, \quad \text{where } c = \nu + \sqrt{\nu^2 - \varepsilon^2}.$$
This transformation reformulates the system into a parabolic system for $(\rho, v)$. The continuity equation becomes a fast diffusion equation with a convection term:
$$\rho_t + \text{div}(\rho v) - c \Delta \rho^\alpha = 0.$$
This structure allows the authors to exploit the dissipative properties of the diffusion term $\Delta \rho^\alpha$.

B. Modified Nash-Moser Iteration for Density Bounds

The core difficulty lies in establishing uniform upper and lower bounds for the density ($0 < \underline{\rho} \le \rho \le \overline{\rho} < \infty$) to prevent vacuum formation and concentration.

• Challenge: When $\alpha < 1$, the diffusion term acts on $\rho^\alpha$, making the equation nonlinear. Standard De Giorgi iteration (used for $\alpha=1$) fails.
• Solution: The authors develop a modified Nash-Moser iteration.
  • They derive Reverse Hölder inequalities for $\rho$ (for the upper bound) and $\tau = \rho^{-1}$ (for the lower bound).
  • The iteration relies on the integrability of the effective velocity $v$ in weighted $L^p$ spaces.
  • They carefully select integrability exponents ($\hat{q}_N$ and $\check{q}_N$) to ensure the iteration converges to $L^\infty$.
  • Key Insight: The constraints on the parameters $\alpha$ and $\beta$ (related to $\nu$ and $\varepsilon$) are derived specifically to ensure the existence of these integrability exponents.

C. Higher-Order Estimates and Coupling

Once density bounds are established, the authors derive higher-order Sobolev estimates ($H^3$ for density, $H^2$ for velocity).

• Nonlinear Obstructions: Differentiating the continuity equation generates difficult nonlinear terms involving $|\nabla \rho|^6$ and $|\nabla \rho|^2 |\nabla^2 \rho|^2$.
• 2D Case: These terms are controlled using Gagliardo-Nirenberg inequalities and the fact that the fast diffusion structure allows rewriting terms in terms of $z = \rho^\alpha$.
• 3D Case (The Major Difficulty): The nonlinear terms cannot be treated perturbatively using only standard dissipation.
  • The authors derive an auxiliary evolution estimate for $\|\nabla \rho\|_{L^4}^4$.
  • This estimate generates a crucial positive quantity $A(t) = \int \rho^{\alpha-1} |\nabla \rho|^2 |\nabla^2 \rho|^2 dx$.
  • They prove that $A(t)$ controls the critical cubic terms (like $\int \rho^{\alpha-3} |\nabla \rho|^6 dx$) via sharp algebraic estimates.
  • This mechanism closes the energy estimates only if $\alpha$ satisfies a strict lower bound (specifically $\alpha > \frac{9\sqrt{3}-4\sqrt{2}}{9\sqrt{3}-2\sqrt{2}} \approx 0.778$ in 3D).

D. Elliptic Regularity

The momentum equation is treated as a Lamé-type elliptic system with variable coefficients. The uniform bounds on $\rho$ allow the authors to recover second-order derivatives of velocity ($\nabla^2 v$) from the time derivative $\rho v_t$ and lower-order terms, preventing a loss of derivatives in the coupling between continuity and momentum equations.

3. Key Contributions

1. Resolution of a Longstanding Open Problem: The paper provides the first proof of global-in-time strong solutions for the 3D generalized NSK system with arbitrarily large initial data in the non-dispersive regime ($\nu \ge \varepsilon$).
2. Generalization of Previous Results: It extends previous results (which were limited to $\alpha=1$ or symmetric data) to a general range of $\alpha \in (0, 1)$ and general (non-symmetric) initial data.
3. Novel Iteration Scheme: The development of a modified Nash-Moser iteration tailored for fast diffusion equations with convection terms ($\alpha < 1$) is a significant methodological advancement.
4. 3D Closure Mechanism: The identification of the auxiliary $L^4$-estimate for $\nabla \rho$ and the quantity $A(t)$ to control critical nonlinearities in 3D is a novel technical breakthrough.
5. Precise Parameter Ranges: The paper explicitly defines the admissible ranges for the exponents $\alpha$, $\gamma$, and the ratio $\beta = \sqrt{1 - \varepsilon^2/\nu^2}$ required for global existence.

4. Main Results

Theorem 1.1:
For $N \in \{2, 3\}$, under the assumptions that the viscosity and capillarity coefficients satisfy the specified algebraic relations and $\nu \ge \varepsilon$, if the initial data $(\rho_0, u_0)$ are sufficiently regular ($H^3$ for density, $H^2$ for velocity) and satisfy $0 < \underline{\rho} \le \rho_0 \le \overline{\rho}$, then the system admits a unique global strong solution $(\rho, u)$.

The solution satisfies:

• Uniform Bounds: $0 < C(T)^{-1} \le \rho(x, t) \le C(T) < \infty$.
• Regularity:
  • $\rho \in C([0, T]; H^3) \cap L^2(0, T; H^4)$
  • $u \in C([0, T]; H^2) \cap L^2(0, T; H^3)$
• Constraints:
  • 2D: $\alpha \in (\frac{\sqrt{5}-1}{2}, 1)$, $\beta \in [0, \beta_2^-(\alpha))$.
  • 3D: $\alpha \in (\approx 0.778, 1)$, $\beta \in [0, \beta_3^-(\alpha))$, and $\gamma$ is bounded above.

5. Significance

• Mathematical Physics: This work bridges the gap between the theory of compressible Navier-Stokes equations and Korteweg fluids, confirming that capillarity and density-dependent viscosity do not prevent global regularity even for large data, provided the viscosity dominates capillarity.
• Methodological Impact: The techniques developed, particularly the handling of fast diffusion nonlinearities in 3D via auxiliary $L^4$ estimates, offer a new toolkit for analyzing other systems with similar dispersive-dissipative structures.
• Future Directions: The paper leaves open the case where $\nu < \varepsilon$ (dispersive regime) and the endpoint cases where $\alpha$ falls outside the derived ranges, setting the stage for future research.

In summary, this paper represents a major breakthrough in the mathematical theory of compressible fluids with capillarity, solving a problem that has remained open for decades by introducing innovative analytical tools to handle large data and strong nonlinearities.