8 problems

In the setting of fluid mechanics, consider solutions of the three-dimensional Navier–Stokes equation arising from initial data that are sufficiently large in an appropriate sense.…

Physical relevance conjecture. The solutions constructed by the numerical scheme are physically relevant.

Let ${\bf u}_1({\bf r},t)$ and ${\bf u}_2({\bf r},t)$ denote solutions of the Navier–Stokes equations with almost identical initial conditions satisfying … for arbitrarily small…

Wild non-uniqueness conjecture. For any such Riemann initial data, there exists a smooth pressure law $p$ with $p'>0$ such that the corresponding system admits infinitely many boun…

The three-dimensional Navier–Stokes equation admits self-similar steady states, and consider solutions of the form $\beta\bar{\omega}$, where $\beta$ is a scalar and $\bar{\omega}$…

Supercritical nonuniqueness and non-Leray-Hopf conjecture. Then there exist two weak solutions $u,v\in X^{p,q}([0,T];\mathbb{T}^d)$ of the Navier–Stokes equations with

Let $(\rho^0,v^0)$ be the piecewise constant initial data … for the isentropic Euler system with $p(\rho)=\rho^2$, and suppose these data allow the nonuniqueness property of Theore…

Let the Cauchy problem for the KdV equation be posed with measure initial data. The Miura transform is given by $\mathcal{M}(u)=\partial_x u+u^2$ and maps suitable $L^2$ solutions…