Transport, Fluids and Mixing

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations 1 and 2

speaker: Maria Colombo (École polytechnique fédérale de Lausanne )

abstract: In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In these talks we exhibit two distinct Leray solutions with zero initial velocity and identical body force. The starting point of our construction is Vishik's answer to 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. Building on Vishik's work, we construct a `background' solution which is unstable for the 3D Navier-Stokes dynamics; the second solution from the same initial datum is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák.

timetable:
Tue 25 Jan, 15:00 - 15:50,
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