## On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates.(English)Zbl 0916.35084

Arch. Ration. Mech. Anal. 143, No. 1, 29-51 (1998); erratum ibid. 147, 363 (1999).
The author considers self-similar solutions of the Navier-Stokes equations $u_t- v\Delta u+ u\nabla u+\nabla p= 0,\quad\text{div }u= 0\quad\text{in }\mathbb{R}^3\times (t_1,t_2)$ having the form $$u(x, t)= \lambda(t)\cdot U(\lambda(t)x)$$, $$p(x,t)= \lambda^2(t)\cdot P(\lambda(t)x)$$, where $$\lambda(t)= 1/\sqrt{2a(T- t)}$$ and $$a>0$$, $$T>0$$. The main theorem states that any weak self-similar solution which satisfies local energy estimates in the cylinder $$B_1(0)\times(T- 1,T)$$, is identically zero.
Reviewer: O.Titow (Berlin)

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids

### Keywords:

weak self-similar solution
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