Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity∗
نویسنده
چکیده
In this short note, we give a link between the regularity of the solution u to the 3D Navier-Stokes equation, and the behavior of the direction of the velocity u/|u|. It is shown that the control of div(u/|u|) in a suitable Lpt (L q x) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. But in this case the condition is not on the vorticity, but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.
منابع مشابه
Sufficient Conditions for the Regularity to the 3d Navier–stokes Equations
In this paper we consider the three–dimensional Navier–Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one direction derivative of the velocity field, namely, uz , for the regularity of strong solutions to the three-dimensional Navier–Stokes equations.
متن کاملGlobal Regularity Criterion for the 3d Navier–stokes Equations Involving One Entry of the Velocity Gradient Tensor
In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three–dimensional Navier–Stokes equations in the whole space, as well as for the case of periodic boundary conditions. AMS Subject Classifications: 35Q35, 65M70
متن کاملInvestigation of instable fluid velocity in pipes with internal nanofluid flow based on Navier-Stokes equations
In this article, the instable fluid velocity in the pipes with internal nanofluid is studied. The fluid is mixed by SiO2, AL2O3, CuO and TiO2 nanoparticles in which the equivalent characteristic of nanofluid is calculated by rule of mixture. The force induced by the nanofluid is assumed in radial direction and is obtained by Navier-Stokes equation considering viscosity of nanofluid. The displac...
متن کاملOn the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations
Here u, b describe the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity, η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field with ∇ · u0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. As same as the 3D Navier-Stokes equations, the regularity of ...
متن کاملRegularity of Generalized Naveir-stokes Equations in Terms of Direction of the Velocity
In this article, the author studies the regularity of 3D generalized Navier-Stokes (GNS) equations with fractional dissipative terms (−∆)αu. It is proved that if div(u/|u|) ∈ Lp(0, T ;Lq(R3)) with 2α p + 3 q ≤ 2α− 3 2 , 6 4α− 3 < q ≤ ∞. then any smooth on GNS in [0, T ) remains smooth on [0, T ].
متن کامل