33 problems

Let $b\in L^{\infty}([0,\infty),W^{1,1}(\mathbb{T}^d,\mathbb{R}^d))$ be divergence free, and let $\rho$ denote the mixing rate associated with an initial datum $u_0$. Bressan's mix…

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

Generic flow-map non-precompactness conjecture. For a residual subset of $\mathcal{U}_p$, the associated flow maps $\{\Phi_t\}_{t\geq 0}$ are not precompact in $L^1$.

Let $(\boldsymbol{\mathrm{P}}\!_n)_{n\geq 1}$ be a sequence of measures on finite product spaces, and let $\Gamma_n(\boldsymbol{\mathrm{P}}\!_n)$ and…

Shape-detection conjecture. If all mixing shapes of cardinality $r$ for $\alpha$ are mixing, then $\alpha$ is mixing of order $r$: whenever…

Let $X_0,X_1,\ldots,X_r$ be vector fields on a manifold, and consider an operator … Let the minimal time for mixing mean the least time at which the associated kernel $P_t$ has uni…

Bressan's rearrangement cost conjecture. The optimal lower bound for the decay of this norm should be

A suspension over a staircase transformation is the flow obtained by suspending a staircase transformation with an appropriate roof function; a translation flow is the flow induced…

Let $(S_n)_{n\in\mathbb{N}}$ be the sequence of service times and let $Z_1$ be the relevant interarrival-time random variable, under Assumptions on $(S_n)_{n\in\mathbb{N}}$ and…

The residence-time distribution (RTD) method measures tracer concentration over time at the reactor outlet to characterize the reactor's mixing performance. RTD-fidelity conjecture…

Let $b$ be a divergence-free vector field on the torus, let $u_0$ be mean-zero initial data, and let $u$ solve the transport equation … The geometric mixing scale measures the size…

Ferenczi's conjecture. The example described by Ferenczi, whose word complexity satisfies

Mix-norm conjecture. The mix-norm may be used to identify organized initial perturbations that lead to a flow evolution that is optimal for mixing, for a given initial energy cost;…

Time-averaged energy-growth conjecture. An appropriate way to enhance mixing is through maximization of the time-averaged energy growth using a cost functional approach.

Let $G_N\subseteq \mathbb{T}^2$ be the discrete grid used in the paper, let $\rho_0\in L^\infty(G_N)$ be the initial datum, and let the alternating horizontal and vertical wedge fl…

Let a piecewise isometry (PWI) be an area-preserving map with finite-perimeter atoms, and let its return plot encode the area and connectivity of the potentially ergodic region. Le…

Let $1<t\leq\infty$, let $M=L(\Bbb F_t)$, and let $B\subset M$ be a maximal amenable von Neumann subalgebra. Multiple mixing conjecture. The subalgebra $B$ is multiple mixing in…

For a higher-dimensional periodic billiard, let the growth of complexity denote the complexity growth associated with its dynamics. Bounded-growth conjecture. Growth of complexity…

Faster-decay conjecture. The minimal distance of two trajectories decays faster when both trajectories start in the same invariant set than when they start in different invariant s…

Let HCCSs denote hyperbolic covariant coherent structures, and consider an instantaneous HCCS associated with a two-dimensional flow. In particular cases, such as the merging of tw…

Let $S$ be a surface of genus $g=1$, and consider a typical locally Hamiltonian flow preserving a probability measure. Suppose that the flow has at least one periodic component, eq…

A random walk $(X_t)$ on ${\mathbb{Z}}^d$ has transition probabilities depending on a dynamic random environment, and uniform mixing environments are known to exhibit diffusive beh…

Let $\eta$ be a stationary max-stable random field, and let $C(0)$ denote its cell containing the origin. The field is strongly $\beta$-mixing if for every compact set…

Consider a fast-slow partially hyperbolic system with small parameter $\varepsilon$, in the case where the averaged slow motion is a rotation because the averaged vector field has…

Let $r\geq 2$ and consider the set of $\mathcal{C}^{r}$ Axiom A flows, with exponential mixing understood for the relevant equilibrium or SRB measures and Hölder observables. Dolgo…