10 problems

BaBoDeMa conjecture. Failure of the relaxed Sard property is equivalent to anomalous dissipation.

Armstrong–Vicol conjecture. Using their approach together with large-scale regularity techniques from quantitative homogenization theory, one can reach $C^0_tC^\frac{1-\beta}{2}_x$…

Let $N^\nu$ be a positive frequency scale and let $(\theta^\nu)_{\nu\in(0,1)}$ be a family of noise coefficients such that … Assume that anomalous dissipation of energy holds for t…

Let $\theta^\nu$ denote the viscosity-dependent transport-noise coefficients in the two-dimensional stochastic velocity equation. The cited theorem uses a particular scaling of…

Let $theta^ u=(thetak^ u){kinmathbb Z0^2}$ be the noise coefficients, and write … The notation $|k|\eqsim\nu^{-\delta}$ means that the active frequencies are comparable to…

Let $\nu$ be the viscosity, $\omega_\alpha$ the vorticity components, and $\mathcal E$ the enstrophy dissipation appearing in … In the turbulent limit, the paper studies dissipatio…

$5/6$ threshold conjecture. For every $s<\frac{5}{6}$, there are weak solutions to the Euler equation in $C_tH^s_x$ with strictly decreasing kinetic energy.

Let $\theta^\kappa$ be a solution of the viscous transport equation on $\mathbb{T}^d$ with velocity field $u\in L^1([0,\infty),W^{1,1}(\mathbb{T}^d))$ and smooth initial data…

Let $\theta_0$ be initial data for an active scalar, and suppose that spontaneous stochasticity occurs for the associated limiting particle trajectories \tilde{\text{\boldmath xi…

Let $u_\epsilon$ be a family of approximating flows converging weakly to a limit $\overline u$, and let $RT$ denote the Reynolds-stress term arising from the weak limit of…