On Leray’s self-similar solutions of the Navier-Stokes equations. (English) Zbl 0884.35115

Leray’s problem of the existence of self-similar solutions of the Navier-Stokes equations \[ \frac{\partial u}{\partial t}+u\cdot\nabla u -\nu\Delta u +\nabla p= 0, \quad \nabla\cdot u=0\;\text{in}\;\mathbb R^3\times(t_1,t_2) \] is considered. These are solutions of the form \[ u(x,t)=\frac 1{\sqrt{2a(T-t)}}U\left(\frac x{\sqrt{2a(T-t)}}\right) \] where \(T\in \mathbb R,\;a>0,\;U:\mathbb R^3\to \mathbb R^3\). \(U\) satisfies the system \[ -\nu\Delta U +aU+a(y\cdot\nabla)U+(U\cdot\nabla)U+ \nabla P=0,\qquad \nabla U=0. \tag{1} \] The main result of the paper is the following
Theorem. Let \(U\) be a weak solution of (1) belonging to \(L^3(\mathbb R^3)\). Then \(U\equiv 0\) in \(\mathbb R^3\).


35Q30 Navier-Stokes equations
35C06 Self-similar solutions to PDEs
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