12 problems

Uniqueness conjecture. Entropy solutions of the equation in this more general setting should be unique.

Scaling-law persistence conjecture. The scaling laws proven for smooth flux functions remain true for locally Lipschitz continuous flux functions such as the modular Burgers' equat…

Realizability conjecture. For any probability measure $\mu$ on $[\alpha,\beta]$, there exist initial data $u_0\in L^\infty(\mathbb{R})$ satisfying such that $\omega(u_0)$ contains…

Let $TV_a$, $TV_{is}$, and the dual discrete definition of total variation be applied to numerical solutions of two-dimensional scalar conservation laws computed by a second-order…

Let $\rho(x,t)$ be the entropy solution of the scalar conservation law with initial data $\xi$. The initial data is either a white noise derived from a spectrally positive Lévy pro…

Let $f$ be a flux with polynomial degeneracy $p\in\mathbb{N}$, meaning that its zero set of $f”$ is finite and, at each zero $w$, the least integer $p_w\geq 2$ such that…

Let $u_0\in L^\infty(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$, with $\\|u_0\\|_{L^\infty}<R_0$, and let $f=(f_1,\ldots,f_d)$ be a flux whose components are $C^1$ and satisfy the nondeg…

The generator $\mathcal{A}$ of a stationary, spectrally negative Feller process acts on test functions $J\in C^\infty_c(\mathbb{R})$ by … Allowing $b$ and $f$ to depend on time giv…

Let $\Omega$ be the domain in the variational problem, let $\partial\Omega$ denote its boundary, and let $\nabla^{\perp}dist(\cdot,\partial\Omega)$ be the rotated gradient of the d…

Let $H$ and $H^{JV}$ be the functionals associated with the scalar conservation law and its companion formulation, and let $f(u)=u(1-u)$. The totally asymmetric simple exclusion pr…

Whole-sequence convergence conjecture. The whole sequence $u^\epsilon$ should converge in Case (1), rather than only a subsequence.

Self-preparation conjecture. The solution $u^{\varepsilon}$ re-prepares itself in order to match the microscopic profile dictated by the equation.