6 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 $L^1$.

Let $M\subset\mathbb{R}^2$ be a bounded planar domain, let $\omega_0\in L^{\infty}(M)$ be initial vorticity, and let $\{\omega(t)\}_{t\in\mathbb{R}}$ be its inviscid incompressible…

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…

Let $\theta$ be a passive scalar transported on $\mathbf{R}^d$ by a time-dependent divergence-free velocity field $u$, and let $\theta(t,\cdot)$ denote the resulting scalar. Write…

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…

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…