Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier--Stokes Equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$

Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Solutions to Three-dimensional Navier-stokes Equations for Incompressible Fluids

This article gives explicit solutions to the space-periodic NavierStokes problem with non-periodic pressure. These type of solutions are not unique and by using such solutions one can construct a periodic, smooth, divergence-free initial vector field allowing a space-periodic and time-bounded external force such that there exists a smooth solution to the 3-dimensional Navier-Stokes equations fo...

Three-dimensional Si/Ge quantum dot crystals.

Modern nanotechnology offers routes to create new artificial materials, widening the functionality of devices in physics, chemistry, and biology. Templated self-organization has been recognized as a possible route to achieve exact positioning of quantum dots to create quantum dot arrays, molecules, and crystals. Here we employ extreme ultraviolet interference lithography (EUV-IL) at a wavelengt...

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier-Stokes Equations II

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier-Stokes Equations II Chiun-Chuan Chen a; Robert M. Strain b; Tai-Peng Tsai c; Horng-Tzer Yau b a Department of Mathematics and Taida Institute for Mathematical Sciences, National Taiwan University and National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan b Department of Mathematics, Harvard University, Cambridge, Massa...

Equations and Dot - Depth One

This paper studies the fine structure of the Straubing hierarchy of star-free languages. Sequences of equations are defined and are shown to be sufficiently strong to characterize completely the monoid varieties of a natural subhierarchy of level one. In a few cases, it is also shown that those sequences of equations are equivalent to finite ones. Extensions to a natural sublevel of level two a...