6 - Ladder results for the Navier-Stokes equations

Summary

Introduction

In the previous chapter it was shown how solutions of the Navier-Stokes equations could be constructed. Uniqueness of those solutions requires more regularity, however, than that which follows directly from their construction via the Galerkin approximations. In this chapter we shall begin to see how much regularity is needed to ensure smoothness of solutions of the Navier-Stokes equations. The minimum requirements can be reached for the 2d problem, but the problem remains open for the 3d case.

This chapter is devoted to the statement and proof of what will be referred to as the ladder theorem for the Navier-Stokes equations on ω = [0, L]^d with periodic boundary conditions and zero mean, and a discussion of its consequences in both 2d and 3d. This will enable us to relate the evolution of a seminorm of solutions of the Navier-Stokes equations, containing a given number of derivatives, to one containing a lower number of derivatives. In sections 6.3 and 6.4, it is shown how the ladder leads to the identification of length scales in the solutions. Subsequently, section 6.5 contains those estimates that can be gleaned via the ladder from the 2d and 3d Navier-Stokes equations where no assumptions have been made. Finally, to show how forcing fields can be handled differently from the static spatial forcing f(x) of previous chapters, section 6.6 briefly shows how a ladder may be derived for thermal convection.

To derive the ladder theorem, it is necessary to introduce the idea of seminorms which contain derivatives of the velocity field higher than unity.