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A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li — 2021

Applications of Mathematics

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ρ and velocity field u satisfy ρ L ( 0 , T ; W 1 , q ) + u L s ( 0 , T ; L ω r ) < for some q > 3 and any ( r , s ) satisfying 2 / s + 3 / r 1 , 3 < r , then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [ 0 , T ] . Here L ω r denotes the weak L r space.

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