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

Abstract

Previous works of Gu-Huang-Meng-Zhou~\cite{Gu-Huang-Meng-Zhou} and Huang-Lei-Zhou~\cite{Huang-Lei-Zhou} established global strong solutions away from vacuum for arbitrarily large initial data when $\alpha$ lies in a suitable range. In contrast, we show that, for a class of small positive exponents $\alpha$ $(\alpha<\frac{1}{2}$), there exist smooth initial data with density uniformly separated from vacuum whose corresponding solutions develop finite-time implosion singularities. Our construction is based on smooth self-similar imploding profiles of the compressible Euler equations. After reformulating the system in self-similar coordinates, the viscous and capillary effects appear as exponentially decaying perturbations. We control the resulting non-autonomous system through weighted high-order energy estimates, a stable-unstable decomposition of the linearized operator, and a finite-dimensional selection of the unstable components. The constructed solutions remain smooth before the singular time and converge, after rescaling, to the prescribed imploding profile. In particular, at the blowup time $T$, the density becomes infinite at the origin, while the effective velocity $u + d \alpha \rho^{\alpha-2} \nabla \rho$ is unbounded in every neighborhood of the origin. These results complement the aforementioned global existence theory and exhibit a distinct finite-time blowup mechanism for the small-$\alpha$ regime, where the effective bulk-viscosity structure may no longer be positive.