7 problems

Let $u$ solve the incompressible Navier–Stokes equations on $[0,1]\times\mathbb{R}^3$, … For any $\alpha\in[0,1/2)$, consider initial data $u_0\in H^\alpha\cap L^{3,\infty}$ and so…

Let a fluid dynamics model with zero initial datum possess infinitely many weak solutions. Couple it with a transport equation and subject the velocity field to multiplicative and…

Let $2<p<\infty$. Consider the two-dimensional Euler equation with initial velocity $v^\circ\in L^2$ and initial vorticity $\omega^\circ\in L^1\cap L^p$. A weak solution has veloci…

Supercritical non-uniqueness conjecture. Non-uniqueness should hold throughout the whole supercritical regime determined by the Ladyzhenskaya–Prodi–Serrin condition.

Cheskidov–Luo's conjecture. Non-uniqueness of weak solutions should hold throughout the full supercritical regime

Forward self-similar solutions are solutions of the Navier–Stokes equations invariant under the forward self-similar scaling and defined for $t>0$; a Leray weak solution is a weak…

Non-uniqueness conjecture. For sufficiently large scale-invariant initial data, the corresponding scale-invariant Navier–Stokes solution is not unique.