## 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 $$\mathbb R^3$$.

### MSC:

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

Leray’s solution
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### References:

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