285 problems

Let $u$ be a weak solution of the incompressible fluid equations with Hölder regularity $C^\alpha$. Compared with traditional statistical theory, Onsager proposed that energy dissi…

Let $(\omega,u)$ be a solution of the two-dimensional incompressible Euler equation on $\mathbb{T}^2$, and consider its weak limit set as $t\to\pm\infty$. A solution lies on a comp…

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…

Flat Darboux coordinate conjecture. These quantities are flat Darboux coordinates for the reduced Poisson structure. The source states that the associated result was explicitly pro…

Low-frequency spectral asymptotics.

Consider smooth solutions of the axisymmetric Euler equations with swirl on a periodic cylinder. Hou–Luo conjecture. Such solutions could develop finite-time singularities at the b…

Finn's conjecture. If $a$ is sufficiently small, the starting problem admits a solution that tends to a stationary solution as $t\to\infty$.

Strong ill-posedness conjecture. The Euler equations with Riesz forcing are strongly ill-posed in $L^\infty$. The corresponding equations are a model for fluid-dynamical systems co…

Consider the De Gregorio model on $S^1$, given by … where $H$ is the Hilbert transform and smooth initial data are prescribed. Global regularity conjecture for the De Gregorio mode…

Let $\omega$ and $u$ solve the De Gregorio model … where $H$ is the Hilbert transform and $\Lambda^{-1}$ is defined by the displayed relation. De Gregorio's global regularity conje…

Scale-invariant solutions are solutions of fluid equations whose spatial behavior is unchanged under the relevant scaling; the surrounding discussion concerns solutions with scale-…

Let the one-dimensional flow model have particle trajectories governed by $dx/dt=v=-H(\theta)$, where $H(\theta)$ is antisymmetric, so that a particle initially at the origin remai…

Let $R$ be the Reynolds number for a periodically forced Navier–Stokes fluid, let $\mu_R$ be the probability distribution describing the time-average statistics of the irreversible…

Let $p$ denote the order of a moment of the velocity field, and let the viscosity tend to zero. Higher-moment divergence conjecture. The higher-order moments with $p\ge 4$ diverge…

Let $Omega$ be a compact region in $\mathbb{R}^n$ with smooth boundary. Consider solutions of the Navier–Stokes equations in $Omega$ and solutions of the Euler equations with the s…

Similarity-form solution conjecture. The correction term $\psi_0$ may be similarity and connected to $x^\alpha f(yx^{-\beta})$ and $\mathcal{U}(x)$. This conjecture concerns extend…

The 2D quasi-geostrophic equation is considered with an initial scalar field whose level sets are of hyperbolic saddle type. Constantin–Majda–Tabak conjecture. Such a solution is l…

A vortex patch is a solution of the three-dimensional incompressible Euler equations whose initial vorticity has patch-like, conormally smooth structure; its boundary is initially…

Equal-energy controllability conjecture. For any two flows with equal energies, the conclusion of Theorem 4 is true.

Electric-field insufficiency conjecture. When $cmu/(aQ)>cpi^2/2$, the electric field is too weak to prevent breakthrough caused by the hydrodynamic dipole. The claim concerns the a…

In an inviscid heterogeneous parallel shear flow, let an unstable wave have growth rate determined by the linear instability problem, and let its wavelength decrease to zero. Here…

Perelman's conjecture. Every smooth rotated backward self-similar profile with $\alpha\neq 0$ and the stated decay is trivial.

One-component regularity conjecture. If

Critical recurrence diagnostic conjecture. There exist relative filter scales $\ell_n\in[\ell_-,\ell_+]\subset(0,1)$, a subsequence, and a constant $c>0$ such that

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…