14 problems

For a first-order deformation determined by the inhomogeneous Jacobi field $u=\mathrm{i}v_{\rho}+\mathrm{i}\delta v_0$, let $\psi_j$, $j=0,1,\ldots$, be the sequence of $1$-forms o…

In the terminology of Constantin Carathéodory, two problems are Carathéodory-equivalent when their respective Lagrangians differ by a total derivative; the corresponding extremals…

Consider the scalar conservation law … Let $u$ solve the fully nonlinear fourth-order approximation … where $f$ is strictly convex, $g$ is concave, and $g(0)=0$. Strong convergence…

Consider an integrable hierarchy of scalar conservation laws … which is a perturbation of the Riemann--Hopf hierarchy, and transform it to the Arsie--Lorenzoni--Moro normal form ……

Consider a deformation of the Riemann hierarchy … where the flow with respect to $t_1$ is in normal form: … with $c_\lambda(u)\in\mathbb C[[u]]$. Arsie–Lorenzoni–Moro conjecture. I…

Let $\theta$ be a weak solution of the active scalar equation, let $\delta$ be the parameter appearing in the equation, and set … The Hamiltonian is … with $T_0$ the Fourier multip…

Let $\mu\in\mathscr{N}_+$ and write $v:=\partial_x g:=\partial_x\mathsf{H}[\mu]$, where $v$ is a tractable solution. Non-entropy shock comparison conjecture. The measure $\mu$ rest…

Consider the class of two-dimensional shallow water equations with variable bottom topography, denoted by $\mathcal L$, and its spaces of zeroth-order conservation laws. Two cases…

Even-power and independence conjecture. The quasilinear part of an integrable dispersive conservation law contains only even powers of $\epsilon$, and all coefficients of the quasil…

Viscous central-invariant conjecture. Two integrable viscous conservation laws admitting the same viscous central invariant $a(u)$ are Miura equivalent.

Quasilinear-part conjecture. The quasilinear part of a viscous conservation law is uniquely determined by $a(u)$.

Let $a(u)$ be a non-vanishing functional parameter, and consider integrable viscous conservation laws obtained from the deformation procedure extended to arbitrary order in…

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

Convergence conjecture. As $q\to\infty$, the kinetic function $\psi^{\flat,q}_\alpha$ converges to the exact kinetic function $\varphi^\flat_\alpha$: