15 problems

Šverák's conjecture. For a generic vorticity $\omega_0\in L^\infty(\mathbb{T}^2)$, the vorticity orbit

Let $u$ be an inviscid incompressible flow on $\mathbb{R}^2$ with scalar vorticity $\omega=\partial_1u^2-\partial_2u^1$, recovered from $\omega$ by the Biot–Savart law…

Let a generic smooth solution of the two-dimensional Euler equation be given, with vorticity field $\omega$. Linear growth conjecture. The gradient of vorticity exhibits linear gro…

Let $\delta\in(0,1)$ and $p>1$. On the two-dimensional torus $\mathbb{T}^2$, let $u_0$ be a divergence-free vector field in $L^2(\mathbb{T}^2)$ with vorticity…

The universal amplitude equation derived for weakly nonlinear disturbances of vorticity interfaces describes the wave evolution for a straight interface, a circular patch in the pl…

Ionescu–Iyer–Jia sharp extremal-layer bounds conjecture. The same inequalities should hold with $2^-$ and $1^-$ in place of $7/4$ and $7/8$, respectively. These bounds quantify the…

Bouchet–Morita depletion-property conjecture. The limiting vorticity profile satisfies

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…

Consider a two-layer internal wave with densities $\rho_1$ and $\rho_2$, whose constant vorticities $\boldsymbol{\omega}_1$ and $\boldsymbol{\omega}_2$ are orthogonal, with…

Let $M$ be the fluid domain, let $YU(M)$ denote the space of flows under consideration, and let $\mathscr{GM}$ be the set of generalized minimal flows: those $u\in YU(M)$ for which…

Let $r_\star$ be the threshold such that, for sufficiently large Bernoulli constant $r$, the corresponding bifurcation curve of Stokes waves reaches an extreme Stokes wave. A highe…

Taylor's conjecture. Stretching of the vortex filaments was the principal physical mechanism behind the phenomenon of turbulent dissipation.

Let $u$ be a solution of the three-dimensional Navier–Stokes equations, let $T$ be a possible singular time, and let $\lambda_p$ denote the frequency scale associated with the $p$-…

The pressure $p$ in incompressible Euler and Navier–Stokes flows satisfies the Poisson equation … where $\zeta$ is the vorticity and $S$ is the rate-of-strain matrix. Gibbon's geom…

Let $\omega$ be a vorticity distribution satisfying conditions (ii), and let $r$ and $r_0$ be the parameters defined in the paper. A non-existence conjecture for stream solutions a…