On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates
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.
scientific article; zbMATH DE number 1218195