Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor
- First Online:
- 441 Downloads
In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, that is, 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.
Unable to display preview. Download preview PDF.
- Ladyzhenskaya, O.A.: Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, English translation, 2nd ed., 1969Google Scholar
- Ladyzhenskaya, O.A.: The sixth millennium problem: Navier–Stokes equations, existence and smoothness. (Russian) Uspekhi Mat. Nauk. 58(2), 45–78 (2003); translation in Russian Math. Surveys 58(2), 251–286 (2003)Google Scholar
- Lions P.L.: Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models. Oxford University Press, Oxford (1996)Google Scholar
- Pokorný M.: On the result of He concerning the smoothness of solutions to the Navier–Stokes equations. Electron. J. Diff. Equ. 11, 1–8 (2003)Google Scholar
- Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North–Holland, 1984Google Scholar
- Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. 2nd edn, SIAM, 1995Google Scholar
- Temam, R.: Some developments on Navier–Stokes equations in the second half of the 20th century. Development of Mathematics 1950–2000. Birkhauser, Basel, 1049–1106, 2000Google Scholar