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We consider the long-time behavior and optimal decay rates of global strong solutions to the isentropic compressible Navier-Stokes-Korteweg system in R 3 . When the regular initial data belong to the Sobolev space) with l ≥ 3 and s ∈ [0,1], we show that the density and momentum of the system converges to its equilibrium state at the ratesin the L ∞ -norm, respectively, which are proved to be optimal for the compressible Navier-Stokes-Korteweg system.

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We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By Aharmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.for all x, x ∈ Ω, u, ũ ∈ R N , and p ∈ R nN ; without loss of generality we take K ≥ 1.

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