AI 中文总结
针对三维可压缩Navier-Stokes-Korteweg方程，证明当α<1/2时，存在光滑初始数据使解形成有限时间内爆奇点，补充了相关整体存在性理论，揭示了独特的小α爆破机制。
AI 中文摘要
Gu-Huang-Meng-Zhou及Huang-Lei-Zhou的前期工作已证明，当参数α处于合适范围时，对任意大的初始数据，可得到远离真空的整体强解。与之相对，本文证明：对于一类小正指数α（α<1/2），存在密度均匀远离真空的光滑初始数据，其对应的解会形成有限时间内爆奇点。本文的构造基于可压缩Euler方程的光滑自相似内爆剖面；在自相似坐标下重新表述系统后，黏性与毛细效应表现为指数衰减的扰动项。通过加权高阶能量估计、线性化算子的稳定-不稳定分解以及不稳定分量的有限维选择，对所得非自治系统进行控制。构造的解在奇异时间前保持光滑，经尺度缩放后收敛到预设的内爆剖面。特别地，在爆破时间T，原点处密度变为无穷大，而有效速度u + dαρ^(α-2)∇ρ在原点的任意邻域内无界。这些结果补充了前述整体存在性理论，并揭示了小α regime中独特的有限时间爆破机制，其中有效体积黏度结构可能不再为正。
英文摘要
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 $α$ lies in a suitable range. In contrast, we show that, for a class of small positive exponents $α$ $(α<\frac{1}{2}$), 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 $u + d αρ^{α-2} \nabla ρ$ 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-$α$ regime, where the effective bulk-viscosity structure may no longer be positive.
Comments90 pages
