| In this thesis we obtain three regularity results related to the Navier-Stokes equations. The introduction to those result are given in chapter 1.; In chapter 2, we study radial solutions to a model equation for the Navier-Stokes equations. We show that the model equation has self-similar singular solution if 5 ≤ d ≤ 9. We also show that the solution will blow up if the initial data is radial, large enough and d ≥ 5.; In chapter 3, we consider the Cauchy problem for the Navier-Stokes equations ut + u∇u - Delta u + ∇p = 0, div u = 0 in Rd x R+ with initial data a ∈ Ld( Rd ), and study in some detail the smoothing effect of the equation. We prove that for T < infinity and for any positive integers n and m we have tm +n/2 DmtDnx u ∈ Ld +2( Rd x (0,T)), as long as the uL d+2x,t&parl0;Rdx&parl0;0,T&parr0; &parr0; stays finite.; In chapter 4, we study the interior partial regularity to the Navier-Stokes equations in four spatial dimensions. We get two partial regularity criterions, and prove that the two-dimensional Hausdorff measure of the set of singular points at first the first blow-up time is equal to zero. |
