Abstract view
Existence of Leray's SelfSimilar Solutions of the NavierStokes Equations In $\mathcal{D}\subset\mathbb{R}^3$


Published:20040301
Printed: Mar 2004
Abstract
Leray's selfsimilar solution of the NavierStokes 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 nonzero 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 selfsimilar solution of the NavierStokes equations which blows
up at $t=t^*$ with $t^* < +\infty$, provided the function
$\mathcal{G}(y)$ is permissible.