9 problems

Let $\Omega$ be a domain, let $\boldsymbol{\beta}\in [L^\infty(\Omega)]^d$ be divergence-free, and let $L$ be the transport operator with inflow boundary conditions, … For a discre…

Let $p=2$ and let $\mathbf{v}\in L^2(\mathbb{R}^d)$ be a compactly supported, divergence-free vector field. Define … as an operator on $L^2(\mathbb{R}^d)$. Nelson's conjecture. The…

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 $\theta^\kappa$ 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 $\breve{\boldsymbol{U}}_{\tilde{\mathcal{T}}}^{\delta}$ be the discrete trial space on the refined partition…

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 $b$ be a bounded, nearly incompressible vector field in $\operatorname{BV}_{\rm loc}(\mathbb{R} \times \mathbb{R}^{d})$. A bounded, nearly incompressible vector field…

Constant-$C_1$ conjecture. In both cases considered in the cited works, the constant $C_1$ can be taken equal to $1$.

Bressan's compactness conjecture. Under these assumptions, the sequence $\{X_k\}$ is strongly precompact in $L^1_{\operatorname{loc}}([0,+\infty[\times\mathbb{R}^d)$.