Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\subset\mathbb{R}^3$

http://dx.doi.org/10.4153/CMB-2004-005-3
Canad. Math. Bull. 47(2004), 30-37

  Published:2004-03-01
 Printed: Mar 2004

Leray's self-similar solution of the Navier-Stokes equations is defined by $$ u(x,t) = U(y)/\sqrt{2\sigma (t^*-t)}, $$ where $y = x/\sqrt{2\sigma (t^*-t)}$, $\sigma>0$. Consider the equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of $\mathbb{R}^3$ with non-zero boundary condition: \begin{gather*} -\nu \bigtriangleup U + \sigma U +\sigma y \cdot \nabla U + U\cdot \nabla U + \nabla P = 0,\quad y \in \mathcal{D}, \\ \nabla \cdot U = 0, \quad y \in \mathcal{D}, \\ U = \mathcal{G}(y), \quad y \in \partial \mathcal{D}. \end{gather*} We prove an existence theorem for the Dirichlet problem in Sobolev space $W^{1,2} (\mathcal{D})$. This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at $t=t^*$ with $t^* < +\infty$, provided the function $\mathcal{G}(y)$ is permissible.