On the global existence of columnar solutions of the Navier-Stokes equations

We consider columnar solutions of the three dimensional incompressible Navier-Stokes equations in a slab type domain. We call the solution u a columnar flow when u = (u(1), u(2), u(3)) = (u(1)(x(1), x(2), t), u(2)(x(1), x(2), t), x(3) gamma(x(1), x(2), t)+ phi(x(1), x(2), t)) for some scalar function gamma and phi In this paper, we obtain the global existence of columnar flows in Sobolev spaces if partial derivative(3)u(3)(x(1), x(2), 0) is nonnegative. We also show that a solution blows up in finite time if the initial data partial derivative(3)u(3)(x(1), x(2), 0) is radial and the negative part of partial derivative(3)u(3)(x(1), x(2), 0) is dominant. Furthermore, we present nontrivial exact blow-up solutions. These exact solutions blow up at every point of the domain and the blow-up rate of these solutions in the W-1,W-p norm for 1 < p < infinity is (T -t)(-1) near the blow-up time t = T > 0.