Abstract:
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In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. The question of whether Leray solutions are unique is fundamental in fluid mechanics. This series of lectures aims to present recent developments in our understanding of this question.
In the second and third lectures, we outline the strategy of proof of the latter result. After a brief discussion on a recent work of Vishik (Vishik arXiv:1805.09440), we construct a new linear unstable self-similar solution to the 3D Navier-Stokes with force. We then employ linear instability to build a trajectory on the unstable manifold in similarity variables. This produces non-uniqueness, in complete agreement with the predictions of Jia and ?verák.
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