| Abstract: |
| I will overview some resent results on non-uniqueness of solutions to the Navier-Stokes and related equations. I will describe a recent instantaneous blow-up construction, where for any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type I blow-up of the L^\infty norm, while remaining smooth. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier-Stokes flow, which transfers energy from infinitely high to low frequencies. This is a joint work with Mimi Dai and Stan Palasek. |
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