Time: 14:00-15:30, May 28 2025

Venue: E4-221, Yungu Campus

Speaker: Jeaheang Bang, Westlake University

Title: Non-Uniqueness of Smooth Solutions to 3D Navier-Stokes Equations

Abstract: We will mainly read the recent paper by Coiculescu-Palasek on Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data. They recently constructed a BMO^{-1} initial data which admits two distinct smooth solutions.

This time, we will review their main ideas: their initial data consists of an infinite summation of initial-time flows v_k^0 which are localized in the Fourier space. Each initial-time flow's frequencies are localized far awar from each other. For a fixed v_k^0, one can first utilize a heat-dominated flow v_k. Then one can use the (k-1)-th initial data, v_{k-1}^0, to annihilate the self-interaction of v_k, which signifies inverse cascade-dominated flows. Let us denote this flow by u_{k-1}. Carefully choosing the initial data by using Mikado flows will enable this annihilation process . Then by the nature of this construction, one can choose two distinct solutions v_0+u_1+v_2+u_3...; u_0+v_1+u_2+v_3, which is equivalent to choosing only even indicies or odd indicies for v_k and then choosing u_k accordingly. Lastly, one can handle all the other nonlinear interactions by using a perturbation argument.

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