Formation of Implosion Singularities in 3D Compressible Navier-Stokes-Korteweg Equation

Technical Summary: Formation of Implosion Singularities in 3D Compressible Navier–Stokes–Korteweg Equations

Problem Statement
This paper investigates the finite-time singularity formation for the three-dimensional compressible Navier–Stokes–Korteweg (NSK) system with density-dependent viscosity and capillarity coefficients. The system is defined by:
$$\begin{cases} \rho_t + \text{div}(\rho u) = 0, \\ (\rho u)_t + \text{div}(\rho u \otimes u) + \nabla P = \text{div}(2\mu(\rho)D(u)) + \nabla(\lambda(\rho)\text{div} u) + \text{div} K, \end{cases}$$
where the coefficients satisfy the specific power-law relations:
$$\mu(\rho) = \nu\rho^\alpha, \quad \lambda(\rho) = 2\nu(\alpha - 1)\rho^\alpha, \quad \kappa(\rho) = \varepsilon^2\alpha^2\rho^{2\alpha-3}.$$
Previous works by Gu–Huang–Meng–Zhou [19] and Huang–Lei–Zhou [21] established global strong solutions for arbitrarily large initial data (away from vacuum) when the exponent $\alpha$ lies in a specific suitable range (e.g., $\alpha \in (\frac{9\sqrt{3}-4\sqrt{2}}{9\sqrt{3}-2\sqrt{2}}, 1)$ for 3D). The central question addressed here is whether solutions exist globally or blow up in finite time when $\alpha$ falls outside this range, specifically for small positive exponents ($\alpha < 1/2$). In this regime, the effective bulk viscosity becomes negative ($\lambda(\rho) + \frac{2}{3}\mu(\rho) < 0$), a feature associated with non-equilibrium acoustic media and active matter, which challenges standard global existence theories.

Methodology
The authors employ a perturbative approach based on the construction of smooth, self-similar imploding profiles derived from the compressible Euler equations. The methodology proceeds through several key steps:

1. Effective Velocity Reformulation: To handle the third-order Korteweg tensor, the system is reformulated using an effective velocity $v = u + d\rho^{\alpha-2}\nabla\rho$ (where $d = \nu - \sqrt{\nu^2-\varepsilon^2}$). This transformation recasts the system into a parabolic-parabolic structure for $(\rho, v)$, revealing a hidden degenerate diffusion term $d\Delta(\rho^\alpha)$ in the mass equation.
2. Self-Similar Scaling: The problem is transformed into self-similar coordinates $(\tau, y)$ where $\tau = -\log(T-t)/\Lambda$ and $y = x/(T-t)^{1/\Lambda}$. In these coordinates, the viscous and capillary effects appear as exponentially decaying perturbations ($e^{-f_1\tau}$) provided the scaling parameter $\Lambda$ satisfies a specific condition ($f_1 > 0$).
3. Stability Analysis of Perturbations: The core of the proof involves analyzing the nonlinear stability of the system around a smooth, spherically symmetric self-similar Euler profile $(\bar{S}, \bar{V})$. The perturbation equations are derived, and a bootstrap argument is constructed to control the solution.
4. Energy Estimates and Spectral Decomposition:
   • Weighted Energy Estimates: High-order weighted energy estimates are established to control the decay of perturbations. The authors utilize a specific weight function $\phi$ and a hierarchy of parameters to manage the interplay between transport and dissipation.
   • Bootstrap Argument: A rigorous bootstrap scheme is designed to close the estimates for lower-order derivatives (specifically the density lower bound) and higher-order derivatives. A critical innovation is the use of both transport and dissipation structures to establish a lower bound for the density, overcoming the difficulty that the density equation possesses a dissipation structure in addition to transport.
   • Unstable Mode Control: The linearized operator is decomposed into stable and unstable subspaces. The unstable components, which correspond to a finite-dimensional space, are controlled by selecting specific initial data coefficients. This selection ensures that the solution remains on a finite-codimensional manifold that leads to blow-up.

Key Contributions and Results
The paper proves the existence of smooth initial data with density uniformly separated from vacuum that evolve into finite-time implosion singularities for the 3D NSK system when $\alpha$ is sufficiently small ($\alpha < 1/2$).

• Main Theorem (Theorem 1.1): For parameters satisfying either:
  • (P1) $1 < \gamma < 1 + 2/\sqrt{3}$, $0 < \alpha < 1/2$, and $f_1 > 0$; or
  • (P2) $\gamma \in (1, 1 + 2/\sqrt{3}) \setminus \mathcal{J}$ (where $\mathcal{J}$ is a countable exceptional set) and $0 < \alpha < \alpha^*(\gamma)$,
    there exist $C^\infty$ initial data $(\rho_0, u_0)$ such that the corresponding solution $(\rho, u)$ exists on $[0, T)$ and develops an implosion singularity at $t=T$.
• Singularity Characteristics: At the blow-up time $T$:
  • The density becomes infinite at the origin: $\lim_{t \to T^-} \rho(t, 0) = +\infty$.
  • The effective velocity $v$ is unbounded in every neighborhood of the origin.
  • The solution converges, after rescaling, to the prescribed self-similar imploding profile.
• Corollary 1.1: The original system develops a finite-time singularity, specifically characterized by the density blow-up ($\lim_{t \to T^-} \rho(t, 0) = +\infty$). However, the blow-up of the physical velocity $u$ cannot be directly concluded from the effective velocity blow-up, as the gradient term $\nabla\rho$ may also become singular near the origin, and a possible cancellation in the relation $v = u + d\rho^{\alpha-2}\nabla\rho$ cannot be ruled out.

Significance
The paper claims to complement the existing global existence theory for the NSK system. While previous results demonstrated global well-posedness for a specific range of $\alpha$, this work identifies a distinct finite-time blow-up mechanism for the "small-$\alpha$" regime. This regime is characterized by a negative effective bulk-viscosity structure, a phenomenon relevant to non-equilibrium physics. The results demonstrate that even with smooth initial data far from vacuum, the interplay of density-dependent viscosity, capillarity, and negative bulk viscosity can lead to implosion singularities where the density and effective velocity fields themselves blow up, rather than just their derivatives (shock formation). The construction relies on the stability of self-similar Euler profiles, extending the implosion mechanism known for the Euler equations to the viscous and capillary NSK setting.