Abstract
In this short paper, we consider the conditional regularity for the 3D inhomogeneous incompressible Navier–Stokes equations in Vishik spaces and give regularity criterion of strong solutions.
Highlights
We consider the regularity issue for solutions ðρ, u, ΠÞ: QT ⟶ R × R3 × R to 3D inhomogeneous incompressible Navier–Stokes equations for QT ≔ R3 × 1⁄20, TÞ:∂tρ + u · ∇ρ = 0, ð1Þ ρut − Δu + ρðu · ∇Þu+∇Π = 0, ð2Þ div u = 0: Here, ρ is the density function of flow velocity, u is the flow velocity, and Π is the pressure
We consider the initial value problem of (1), which requires initial ρðx, 0Þ = ρ0ðxÞ, ð3Þ uðx, 0Þ = u0ðxÞ, x ∈ R3: There is a very rich literature dedicated to the study of the above system
In the case of smooth data with no vacuum, Kazhikov [1] proved that the nonhomogeneous Navier– Stokes equations have at least one global weak solution in the energy space
Summary
In the case of smooth data with no vacuum, Kazhikov [1] proved that the nonhomogeneous Navier– Stokes equations have at least one global weak solution in the energy space. When the initial data may contain vacuum states, Simon [2] proved the global existence of a weak solution to the equations of incompressible, viscous, nonhomogeneous fluid flow in a bounded domain of two or three spaces, under the no-slip boundary condition. Choe and Kim [3] proposed a compatibility condition and investigated the local existence of strong solutions.
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