On local uniqueness of weak solutions of the Navier--Stokes system with bounded initial data. (Q1413196)

The author considers the Cauchy problem for the Navier-Stokes equations in \(\mathbb{R}^n\), i.e. \[ \begin{alignedat}{2} u_t-\Delta u+ (u\cdot\nabla)u+ \nabla\pi &= 0\quad &&\text{in }\mathbb{R}^n\times (0,T),\\ \text{div\,}u &= 0\quad &&\text{in }\mathbb{R}^n\times (0,T),\\ u(\cdot,0) &= u_0\quad &&\text{in }\mathbb{R}^n,\end{alignedat} \] where the initial function \(u_0\) belongs to \(L^\infty(\mathbb{R}^n, \mathbb{R}^n)\). It is known that this problem has a unique local mild solution, while weak solutions are not unique. The main result of the paper states that \(\widetilde u\) is a weak solution on \([0,T)\) if and only if there exists a function \(\phi\in L^\infty_{\text{loc}}([0, T),\mathbb{R}^n)\) with \(\lim_{t\to 0^+} \phi(t)= 0\) such that \[ \widetilde u(x,t)= u(x- \Phi(t),t)+ \phi(t)\quad\text{a.e. in }\mathbb{R}^n\times (0,T), \] where \(\Phi(t)= \int^t_0 \phi(s)\,ds\).