24 problems

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…

Carbuncle conjecture. Carbuncles are a special class of non-physical entropy solutions.

Vanishing viscosity conjecture. The entropic solutions of the hyperbolic system (1.1) actually coincide with the limits of solutions $u^\varepsilon$ to the parabolic system as the…

Vanishing viscosity conjecture. The entropic solutions of the hyperbolic system (1.1) actually coincide with the limits of solutions to the parabolic system

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…

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

Let the minimizing movements scheme for the relaxed macroscopic IPM problem be defined by time steps of size $h>0$, with iterates obtained by minimizing the sum of the Wasserstein-…

Orsina–De Lellis conjecture. $\mathcal H^1$-almost every point $x$ satisfying $\nu(B_r(x))/r\to0$ as $r\to0^+$ is a Lebesgue point of $m$. This means that there exists $m(x)$ such…

The discussion concerns entropy dissipation in numerical approximations of conservation laws, where Godunov's method is compared with more dissipative numerical fluxes such as the…

Consider the nonlocal transport equation … allows for non-uniqueness. The paper leaves the question for future investigation; multiple weak solutions are known when the support of…

Let $u$ be a solution of the grid-function formulation, let $u^{!astphi(u)}{(t,x)}$ denote the associated measure, let $Omega{mathbb{X}}$ be the hyperfinite spatial domain, and le…

For an isentropic flow on a network, let entropy flux inequalities at a junction be conditions imposed on the entropy flux traces of the solution. Uniqueness conjecture. A sufficie…

For a first-order conservation law, let $u$ denote its entropy solution and consider the temporal $L^1$ error between $u(t+\Delta t)$ and $u(t)$ as $\Delta t\to 0$. Chen–Ding–Karls…

Let $\bar u=(\bar r,\bar p)$ be a limit solution of the hyperbolic system with boundary conditions, satisfying the Clausius inequality … for all $t\in[0,T]$, where $W(t)$ is the wo…

Let $r^{\bar\delta}(t,x)$ and $p^{\bar\delta}(t,x)$ solve the viscous system with parameters $\bar\delta=(\delta_1,\delta_2)$, where $\delta_j>0$. Consider any limit point…

Let $\rho^{n,m}$ be the discrete densities generated by the particle scheme, and suppose that, under the stated assumptions, a subsequence converges almost everywhere and in…

The scheme approximates a density $\rho$ for a vehicular-flow initial-boundary value problem, with entropy solutions understood through the boundary trace condition associated with…

Compressible potential flow is described by a conservation-law system for the density $\varrho$ and velocity $\mathbf v$, with entropy weak solutions understood in the usual distri…

Let $u$ be a space-time velocity field on ${\mathbb R}\times[0,t_f]$, let $u_0$ be an initial condition on ${\mathbb R}$, and let $\tilde{x}(\tau)$ be a stochastic process satisfyi…

Slemrod's conjecture. The energy dissipation obtained when the compressible Euler equations are viewed from the perspective of kinetic theory has additional terms relative to the e…

Strong zero-dissipation limit conjecture. A general weak entropy solution to the inviscid flow should be the strong limit of the solution to the corresponding viscous flows with th…

Carbuncles are numerical structures arising near shock waves in discretizations of fluid equations. Elling's conjecture. Carbuncles can be related to a special class of non-physica…

Extension conjecture. The solution can be defined for $x\geq X_\infty$ as an entropy locally piecewise smooth solution outside the boundary, with infinite velocity on

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