# Caltech/UCLA/USC Joint Analysis Seminar

In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. The final goal of this talk is to discuss the construction of two distinct Leray solutions with zero initial velocity and identical body force.

The starting point of our construction is another long-standing problem in fluid dynamics, namely whether the Yudovich uniqueness result for the 2D Euler system can be extended to the class of L^p-integrable vorticity. Several approaches of different nature have been proposed to attack the latter problem and will be discussed in the first part of the talk.

The second part builds on Vishik's answer to the Yudovich uniqueness question in L^p to construct a suitably unstable 3D Navier-Stokes solution. Finally, the unstable manifold associated to this solution is built to obtain the nonuniqueness of Leray solutions, in accordance with the predictions of Jia and Šverák.