For alpha less than 1/2, the 3D Navier-Stokes-Korteweg system has smooth initial data whose solutions form a finite-time implosion: the density diverges at a point and the effective velocity blows up.
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Formation of Implosion Singularities in 3D Compressible Navier-Stokes-Korteweg Equation
For alpha less than 1/2, the 3D Navier-Stokes-Korteweg system has smooth initial data whose solutions form a finite-time implosion: the density diverges at a point and the effective velocity blows up.