Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces (Q6103929)

In this paper the authors consider the Cauchy problem for the Navier-Stokes equatios in \((0,\infty)\times \mathbb{R}^3\), \[ \partial_t+u\cdot \nabla u=\Delta u-\nabla p, \quad \nabla \cdot u=0, \quad u(0, \cdot )=u_0, \] and discus the problem of uniqueness of the solutions. \textit{H. Jia} and \textit{V. Šverák} [Invent. Math. 196, No. 1, 233--265 (2014; Zbl 1301.35089)] proved that if \(u_0 \in C^{\infty}(\mathbb{R}^3\setminus {0})\) is scale-invariant, then there exists at least one scale-invariant solution \(u\in C^{\infty}\big( (0,\infty)\times \mathbb{R}^3\big)\) of \((1)\). The scale-invariant solution \(u\) is unique for any small scale-invariant initial datum. On the other hand, Jia and Šverák conjectured that this is not the case for large scale-invariant data. In this paper a numerical evidence of this conjecture is presented.