In this paper we consider the Navier–Stokes–Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary to previous results regarding this system, vacuum regions are considered in the definition of weak solutions and no additional damping terms are considered. The convergence of the approximating solutions is obtained by introducing suitable truncations of the velocity field and the mass density at different scales in the momentum equations and use only the a priori bounds obtained by the energy and the Bresch–Desjardins entropy. Moreover, the approximating solutions enjoy only a limited amount of regularity, and the derivation of the truncations of the velocity and the density is performed by a suitable regularization procedure.
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Paolo Antonelli, Stefano Spirito, Global existence of weak solutions to the Navier–Stokes–Korteweg equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 1, pp. 171–200DOI 10.4171/AIHPC/5