We study the Navier-Stokes equations for compressible barotropic
fluids in a bounded or unbounded domain $\Omega $ of $
\mathbf{R}^3$. We first prove the local existence of solutions
$(\rho, u)$
 in $C([0,T_* ]; ( \rho^\infty + H^3 (\Omega ) ) \times ( D_0^1 \cap D^3
)(\Omega ) )$ under the assumption that the data satisfies a natural
compatibility condition. Then deriving the smoothing effect of the
velocity $u$ in $t>0$, we conclude that $(\rho , u)$ is a classical
solution in $(0,T_{**}) \times \Omega $ for some $T_{**} \in (0,
T_* ]$. For these results, the initial density needs not be bounded
below away from zero and may vanish in an open subset ({\it vacuum})
of $\Omega$.

/pdf/on-classical-solutions-of-the-compressible-navier-stokes-3mbnz906dt.pdf

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

In this paper, we study the Cauchy problem for incompressible density-dependent magnetohydrodynamics system in $$\mathbb R^N\ (N \ge 2)$$
 . We prove the regularity criterion of solutions in the critical Besov spaces. For the proof of the results we use the commutator estimate and product estimate for nonlinear term.

A Regularity Criterion for the Density-Dependent Magnetohydrodynamics System in Critical Besov Spaces

The global regularity for 3D inhomogeneous incompressible fluids with vacuum

A H\"older type inequality in Besov spaces is established and applied to show that every strong solution $u(t,x)$ on (0,T) of the Navier-Stokes equations can be continued beyond $t>T$ provided that the vorticity $\omega(t,x)\in L^{\frac 2{2-\alpha}}(0,T;\dot{B}^{-\alpha}_{\infty,\infty}(\mr^3))\cap L^{\frac2{1-\alpha}}(0,T;\dot{B}^{-1-\alpha}_{\infty,\infty}(\mr^3))$ for $0<\alpha<1$.

Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices

Regularity criteria and Liouville theorem for 3D inhomogeneous Navier–Stokes flows with vacuum

In this paper, we study the three-dimensional inhomogeneous incompressible Navier-Stokes equations, and establish several regularity criteria in terms of only velocity which allow the initial density to contain vacuum. Therefore, our results can be considered as further improvement to the previous results.

A Note on Blow-up Criterion of Strong Solutions for the 3D Inhomogeneous Incompressible Navier-Stokes Equations with Vacuum

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A Note on Blow-up Criterion of Strong Solutions for the 3D Inhomogeneous Incompressible Navier-Stokes Equations with Vacuum