Global existence of solutions to the Proudman-Johnson equation

Abstract

We show that there is no blow-up solutions, for positive viscosity constant $\nu$, to the equation $f_{xxt} - \nu f_{xxxx} + f f_{xxx} - f_xf_{xx} =0$, $x \in (0,1)$, $t > 0$ with (i) periodic boundary condition, or (ii) Dirichlet boundary condition $f = f_x = 0$ or (iii) Neumann boundary condition $f = f_{xx} = 0$ on the boundary $x = 0, 1$. Furthermore we show that every solution decays to the trivial steady state as $t$ goes to infinity.