A regularity criterion for the Navier–Stokes equations with mass diffusion

In this work, a regularity criterion is proved for local strong solutions of the Navier–Stokes equations in the presence of mass diffusion.     

A new regularity criterion for weak solutions to the Navier–Stokes equations

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier–Stokes equations. We show that if any one component of the velocity field belongs to $L^{\alpha }([0,T);L^{\gamma }({\mathbb{R}}^3))$ with $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.⩽\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $6<\gamma ⩽\infty$, then the weak solution actually is regular and unique.     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {Omega_0} subset bigcuplimits_{k = 1}^N {{Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {left| {{v_r}} right|_{{L_infty }left( {0,T;{L_3}left( {{Omega_k}} right)} right)}} leq {alpha_1},or;{left| {frac{{{v_r}}}{r}} right|_{{L_infty }left( {0,T;{L_{3/2}}left( {{Omega_k}} right)} right)}} leq {alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) left( {v in W_2^{2,1}left( {Omega times left( {0,T} right)} right),nabla p in {L_2}left( {Omega times left( {0,T} right)} right)} right) . Bibliography: 28 titles.     

Boundary partial regularity for the high dimensional Navier–Stokes equations

We consider suitable weak solutions of the incompressible Navier–Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.     

A regularity criterion for 3D Navier–Stokes equations via one component of velocity and vorticity

In this paper we show that a Leray–Hopf weak solution u to 3D Navier–Stokes initial value problem is smooth if there is some \(\alpha \in {{{\mathbb {R}}}}, \alpha \ne 0,\) such that \(\alpha u_3+(-\Delta )^{-1/2}\omega _3\) is suitably smooth, where \(\omega =\text {curl}\,u\).     

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.     

Time regularity of the densities for the Navier–Stokes equations with noise

We prove that the density of the law of any finite-dimensional projection of solutions of the Navier–Stokes equations with noise in dimension three is Hölder continuous in time with values in the natural space L ¹. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected, at least with the known regularity, diffusive scaling.     

The regularity criterion for 3D Navier–Stokes equations involving one velocity gradient component

In this article, we establish sufficient conditions for the regularity of solutions of Navier–Stokes equations based on one of the nine entries of the gradient tensor. We improve the recent results of C.S. Cao, E.S. Titi [C.S. Cao, E.S. Titi, Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal. 202 (2011) 919–932] and Y. Zhou, M. Pokorný [Y. Zhou, M. Pokorný, On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity 23 (2010) 1097–1107].     

A Beale–Kato–Majda blow-up criterion for the 3-D compressible Navier–Stokes equations

We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier–Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is a priori estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.     

Components reduction regularity results for the Navier–Stokes equations in general dimensions

We consider the Cauchy problem of the Navier–Stokes equations in arbitrary dimensions, and establish several new components reduction regularity criteria.