### Two component regularity for the Navier-Stokes equations.

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We consider the compressible Navier-Stokes-Korteweg (N-S-K) equations. Through a remarkable identity, we reveal a relationship between the quantum hydrodynamic system and capillary fluids. Using some interesting inequalities from quantum fluids theory, we prove the stability of weak solutions for the N-S-K equations in the periodic domain $\Omega {=}^{N}$, when N=2,3.

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