Having considered the linear Stokes equations, we will now bring back the nonlinear term and consider the nonlinear version of the Stokes system. The solution of these equations can be viewed as the limit to which the solution of the full N-S equations tends as t tends to infinity. Of course no one knows at this point if the solutions of the N-S equations do tend to some limit as t tends to inf...

In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations with periodic boundary conditions using the same strategy. In this paper, we mo...

A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical meth...

We study how the number of numerically determining modes in the Navier–Stokes equations depends on the Grashof number. Consider the two-dimensional incompressible Navier–Stokes equations in a periodic domain with a fixed time-independent forcing function. We increase the Grashof number by rescaling the forcing and observe through numerical computation that the number of numerically determining ...

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when...

We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential was defined by Katz includes classical equation. In addition, it which not rigid. Our approach is based on topological computation enhanced Fourier-Sato transform a perverse sheaf due D'Agnolo, Hien, Morando Sabbah.

It is shown that expansion waves for the compressible Navier-Stokes equations are nonlinearly stable. The expansion waves are constructed for the compressible Euler equations based on the inviscid Burgers equation. Our result shows that Navier-Stokes equations and Euler equations are timeasymptotically equivalent on the level of expansion waves. The result is proved using the energy method, mak...