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

Tsai, Tai-Peng

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