13 problems

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 $L^1$.

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

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

Let $M\subset\mathbb{R}^2$ be a bounded simply connected open domain, and let $f,g\in\operatorname{SDiff}(M)$ be isotopic states, meaning that there exists a smooth path…

Collisions of the solid produce non-Lipschitz boundaries for the fluid. Collision-avoidance conjecture. An incompressible fluid should prevent collisions of the solid from occurrin…

For non-pressure-robust space discretisations, the corresponding discrete Helmholtz--Hodge projector can introduce a hidden consistency error. High-order-methods conjecture. Such h…

Investigating the material derivative of incompressible Euler flows, the strong-pressure-gradient conjecture. Strong pressure gradients are typical for non-trivial high-Reynolds-nu…

Let the incremental pressure projection methods under consideration include standard, rotational, and vectorial forms. These methods are described as diminishing errors introduced…

Consider the pressure fields produced by the IFE-CGP method, where a noticeable discrepancy near the boundary is attributed to the homogeneous artificial Dirichlet pressure conditi…

Euler sprays conjecture. For sufficiently small $\varepsilon>0$, Euler sprays can be constructed that hit $\Omega_1$ up to a set of measure zero.

The pressure $p$ in incompressible Euler and Navier–Stokes flows satisfies the Poisson equation … where $\zeta$ is the vorticity and $S$ is the rate-of-strain matrix. Gibbon's geom…

Let $X$ be a compact metric space and let $\Omega=X^{\mathbb{R}}$ be the path space. A probability measure $q$ on $\Omega$ is a generalized incompressible flow (GIF) if, for every…

Let $H$ be the left half of the torus, and let $\Psi$ be the flow generated by an incompressible vector field $u$. The image of $H$ under the flow $\Psi$ after time $T$ is called…