88 problems

Let $f(a_1,p_1,\ldots,a_\kappa,p_\kappa)$ be a density of support $\kappa$, and define its translates by $f_j(a,p)=f(a_{j+1},p_{j+1},\ldots,a_{j+\kappa},p_{j+\kappa})$. For a perio…

Consider the Cauchy problem for the multi-dimensional compressible Euler equations, and call a weak solution entropy/admissible if it satisfies the relevant entropy inequalities. N…

Finite entropy solutions arise in applications including large deviations, the Aviles–Giga functional, and diffusive-dispersive approximations. The isentropic Euler system with…

Kupershmidt's integrability conjecture. The conservation laws $H_1,H_2,\dots$ of the Kupershmidt deformation should commute in an appropriate sense, so that the deformation is inte…

Uniqueness conjecture. The system has only one conservation law defining $w$ by

Generalized Toda integral formula conjecture. Formulae -- are valid not only for $m=2,3$, for which they were explicitly calculated, but also for an arbitrary value of $m$.

Nonsingular characteristic-matrix conjecture. An integrable system $$ must possess higher-order conservation laws for which the matrix ${\mathbf{S}}_\chi$ is nonsingular.

Consider a third-order scalar evolutionary differential equation of the form … A conservation law has a weight as defined by the differential-system and characteristic-cohomology f…

Consider the one-dimensional nonlinear conservation law … with bounded initial data and suppose that the non-degeneracy condition of order $\alpha$ holds. Lions–Perthame–Tadmor con…

Let $\beta\in(0,\frac12)$ parameterize an ${\cal O}(n^{-\beta})$ perturbation of a constant equilibrium profile in the hydrodynamic limit, with time rescaled by $n^{1+\beta}$ and s…

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

The heteroclinic waves arise as component waves in the multi-bump limit of a family of homoclinic orbits. Their associated linearized operators have point spectra, considered in th…

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…

Characteristic conjecture. Any local or potential conservation law of a $(1+1)$-dimensional linear parabolic equation is equivalent to one whose characteristic depends only on the…

Let $\Delta x$ and $\Delta t$ denote the spatial and temporal mesh sizes, respectively, and consider approximate solutions obtained by replacing space and time derivatives in the e…

Rarefaction stability conjecture. The long-time behavior of solutions is governed by the smoothed Riemann solution of the corresponding inviscid system.

Local asymptotic pattern conjecture. Depending on the type of singularity, one of the rescaled families

Let … admits no other independent local conservation laws. The paper presents families of local conservation laws and a Hamiltonian structure, and indicates that a proof that the l…

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 an integrable system of conservation laws that is a deformation of the Riemann--Hopf hierarchy, and transform it by the unique Miura-type transformation to its normal form…

Grabowski–Mathieu conjecture. The Hamiltonian $\hat{H}$ is integrable.

Let $\boldsymbol{u}:\mathbb R^+\times\mathbb R\to\mathbb R^2$ be an $\mathbf L^\infty$ entropy solution of a $2\times 2$ system of conservation laws … with flux…

Let $b$ be a vector field satisfying Assumptions,, and. For each $N$ and configuration $\bm{x}$, let $\sigma_N(t,\bm{x})$ be the corresponding entropy solution, and let…

Consider a deformation of the Riemann hierarchy in conservation-law form, with first Hamiltonian density … Assume $c_2(u)\ne0$, and suppose the existence conditions for a deformati…

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…