Non-uniqueness of Leray-Hopf solutions for a dyadic mode

Abstract

The dyadic model ${\dot{u}}_n+{\lambda }^{2n}{u}_n-{\lambda }^{\beta n}{u}_{n-1}^2+{\lambda }^{\beta (n+1)}{u}_n{u}_{n+1}=f_n$, $u_n(0)=0$, is considered. It is shown that in the case of non-trivial right hand side the system can have two different Leray-Hopf solutions.