16 problems

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…

Eyink's conjecture. Both enstrophy defects are well-defined, they give rise to the same distribution in the limit, and they do not always vanish.

Let a viscosity solution be a 2D Euler solution obtained as a zero-viscosity limit of 2D Navier–Stokes solutions, and call it dissipative when $Z_h(\omega)\geq 0$ for convex $h$. U…

Let $boldsymbol{u}^ u,boldsymbol{ abla}boldsymbol{times}boldsymbol{u}^ u=boldsymbol{ abla}timesboldsymbol{u}^ u$ be a sequence of 2D Navier–Stokes solutions satisfying the paper's…

Let a viscosity solution be a 2D Euler solution obtained as a zero-viscosity limit of Leray solutions of 2D Navier–Stokes whose energy spectra satisfy a uniform upper bound by the…

Let two-dimensional turbulence be considered in the infinite-Reynolds-number limit, and let the turbulent cascade be the transfer of enstrophy to small length-scales. Onsager's con…

Let the weak solutions be the distributional 2D incompressible Euler solutions considered in the paper, obtained as zero-viscosity limits of 2D Navier–Stokes solutions with the sta…

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 $W$ be a noise satisfying Assumption … {rm d} omega + operatorname{curl}^{-1} omegacdot nabla omega, {rm d} t + circ {rm d} Wcdot nabla omega = 0 … such that these stochastic 2…

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…

Let $M$ be a bounded planar fluid domain, let $\omega_*$ be a steady Euler state in $C^{\alpha}$, and let $\omega(t)$ denote the vorticity evolving from initial data $\omega_0$. Yu…

Let $M$ be a fluid domain and let a steady Euler state have vorticity in $C^{\alpha}$. Instability conjecture. All steady states are unstable in $C^{\alpha}$. This is motivated by…

High-Reynolds-number damping conjecture. For high-Reynolds-number flows, an appropriate analogue of Theorem should hold for initial data

Generic inviscid damping conjecture. In regularities lower than Gevrey-$2$—with $H^s$ for $s>3/2$ suggested as one natural choice—“most” sufficiently small initial data should damp…

Intermediate-behavior conjecture. Neither of these scenarios is quite true. Instead, non-trivial $L^2$-precompact trajectories will exist, but initial data leading to them will not…

Continuity conjecture. Kuksin measures are in fact supported on continuous vorticities.