72 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…

Taylor's conjecture. Magnetic helicity is approximately conserved for small resistivities. This conjecture concerns the robustness of magnetic helicity in turbulent regimes, and th…

Turbulence is considered in the setting of the Boussinesq equations and their Lagrangian process, where deterministic and stochastic descriptions of the fluid dynamics are compared…

Let $D_\lambda$ denote the diffusion coefficient measured at scale length $\lambda$ in turbulent air. Richardson's conjecture. The diffusion coefficient $D_\lambda$ depends on the…

Let $u$ be a weak solution of the Euler equations whose spatial regularity is below $1/3$ of a derivative, or whose time integrability is below $q=\infty$, and let $e(t)$ denote it…

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…

Barenblatt's conjecture. The structure-function exponents may depend on the Reynolds number.

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 $boldsymbol{u}^ u,boldsymbol{ abla}boldsymbol{times}boldsymbol{u}^ u=boldsymbol{ abla}timesboldsymbol{u}^ u$ 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 $r$ denote a spatial scale, let $u$ be the kinematic viscosity, let $S{ij}$ be the strain-rate tensor, and let the energy dissipation field be $2nu S{ij}S{ij}$. Let $alpha$ be…

In turbulent flow, intense vorticity can generate local swirling regions, and transverse structure functions are especially sensitive to anti-aligned velocity vectors across vortex…

Quantum-state preparation conjecture. The observable $u_{0,\xi^*}$ can be efficiently prepared as a quantum state. The claimed efficient preparation is relevant to the quantum algo…

The models under consideration are stochastic differential equations with dissipative dynamics; stronger dissipation may be induced by an operator such as the standard Stokes opera…

Farey-sector mode-locking conjecture. The exact autonomous flow dynamically selects the rational Farey sector by a mode-locking mechanism analogous to Arnol'd tongues.

Mode-locking conjecture. The autonomous flow mode-locks the planar fixed point onto commensurate angular data, with the strongest preference for the equal-step rational sectors.

Bizon–Rostworowski's resonant-instability conjecture. The instability is driven by a mechanism of resonant interactions of spherically symmetric modes, leading to weakly turbulent…

Viscosity-assumption conjecture. The assumption

Let $\psi_1(x,y,t)$ and $\psi_2(x,y,t)$ be solutions of the two-dimensional Navier–Stokes equation for turbulent Kolmogorov flow with an even forcing wavenumber $n_K$, subject to t…

Let the Euler ensemble be the fixed-point, decaying solution of the loop diffusion equation in the theory. Its distinct states are the states that can contribute to long-time avera…

The authors' conjecture. The observed power law is influenced by the limited small-scale separation accessible in the computations and would not occur under the assumptions conside…

Let $\lambda_1>0$ be the positive (maximum) Lyapunov exponent of three-dimensional turbulence, let $\nu$ be the kinematic viscosity, and let $\epsilon$ be the kinetic energy dissip…

Geometric-scale separation problem. Prove anomalous diffusion for a variant of the construction with geometrically separated active scales.