32 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…

Bressan's compactness conjecture. The sequence $(X_n)$ is strongly precompact in $L^1_{\mathrm{loc}}(I\times\mathbb R^d;\mathbb R^d)$. This conjecture concerns compactness of flows…

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…

Let $1\leqslant p<\infty$, let $X_p$ be the underlying $L^p$ phase-space space, and let $L^p_+$ and $L^p_-$ denote the outgoing and incoming boundary spaces. Let…

Let $N>0$, and let $m_i:=\int_0^{\delta}v^i g(v)\,\mathrm{d}v$ for $i\in\{1,2,3\}$. The functions $b$, $u^{(N)}(t,x)$, $u^{(N)}_{\partial}(t)$, and the initial datum $n_0(x)$ be as…

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…

Consider the conservative cascade semi-Lagrangian method applied to the two-dimensional linear transport equation … with divergence-free velocity field $\mathbf{a}=(a_1,a_2)$. In t…

Homogenization conjecture. There exists $\overline{V}\in \operatorname{Lip}_{\mathrm{loc}}(\mathbb{R}^n\times[0,\infty))$ such that $V^\varepsilon$ converges to $\overline{V}$ in t…

Let $G$ be the spatial domain, $S$ the direction space, $I$ the energy interval, $E_m$ the upper energy endpoint, and $\Gamma_-$ the inflow boundary. For…

The CSLDG method is a semi-Lagrangian discontinuous Galerkin scheme for transport equations; its numerical experiments exhibit both superconvergence and fluctuations in convergence…

Transport-limit approximation conjecture. The rescaled model is approximated, as $N\to+\infty$, by

Let $\mathbf{v}\in L^2(\mathbb{R}^2)$ be a divergence-free vector field. The chain rule property $\operatorname{CRP}_\infty$ is the stationary analogue of the renormalization prope…

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 $\rho$ be a passive scalar transported by a Sobolev vector field $u$, with initial datum $\rho_0$, and let $p$, $q$, and $r$ be the integrability exponents appearing in the pre…

Regularity-gap conjecture. Not every initial condition in $H^1({\mathbb T})\times L^2({\mathbb T})$ can be steered to $0$ in time $T$ with $L^2$ controls.

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…

Let $E_0$ be the threshold from the assumptions, let $G_{,E}$ be the previously defined constant, and let $\Phi'$ and $\mathcal{T}$ be the functions related by…

Consider the nonlocal transport equation with parameters $\alpha$ and $\gamma$, where the equation and its solution class are as specified in the source. Li–Rodrigo's singularity c…

Let $G=\{x\in U:r(x)>0\}$ be a domain with $r\in C^1(U)$, smooth noncharacteristic boundary away from the grazing set, and suppose the transport problem … has homogeneous inflow da…

Let $G=B(0,R)\subset\mathbb{R}^3$, let $S$ be the sphere of transport directions, and let $I^\circ$ denote the interior of the energy interval. Consider the solution…

Arbitrary-solution well-posedness conjecture. The same well-posedness statement holds for any solution, not only for positive solutions. This is described as an even more daring co…

Well-posedness conjecture. For positive solutions of the continuity equations, well-posedness holds whenever

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…

The PD-WG method is tested for a transport problem on the unit square with exact solution $u=\sin(x)\cos(y)$, convection vector ${\boldsymbol{\beta}}=[y-0.5,-x+0.5]$, reaction coef…

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