10 problems

Ill-posedness conjecture. For any $\gamma\in \mathbb{R}$, there exists $\eta \in L_x^1+L_x^{\infty}(\mathbb{R}^2)$ such that (eq:NLS) is ill-posed in $H_x^\gamma(\mathbb{R}^2)$.

Strong ill-posedness conjecture. The Euler equations with Riesz forcing are strongly ill-posed in $L^\infty$. The corresponding equations are a model for fluid-dynamical systems co…

Eigenfunction-induced turning-point conjecture. The turning point at $\alpha=11/3$ is induced by properties of eigenfunctions on the spheres, rather than by technical reasons. This…

Sharp regularity threshold conjecture. There exists $s_0\in\mathbb R$ such that the equation is well-posed in $H^s$ if and only if $s\geq s_0$.

Let the nonlinear Schrödinger equation on the Sierpinski gasket be the equation considered in the paper, and let its flow map assign to initial data the corresponding solution. Non…

Let $T>0$ and let $b1'$ be such that $1-2Tb1'>0$. An exponential weight is a factor of the form $e^{b2|v|^2}$ in the velocity variable $v$. Critical exponential-weight conjecture.…

Consider the system … … with vorticity formulation … … and $-{u_1}_y=R_2^2\omega$, where $R_2$ is the Riesz transform with symbol $\frac{-i\xi_2}{|\xi|}$. Strong ill-posedness conj…

For weak solutions of the Euler equations, the relevant Hölder regularity is measured in the space $C^{\alpha}_{t,x}$. Euler nonuniqueness conjecture. Nonuniqueness should hold for…

Consider the three-dimensional incompressible Navier–Stokes equations on $\mathbb{R}^3\times(0,\infty)$ with divergence-free, scale-invariant initial data…

Reciprocal structured condition-number conjecture. For traditional arithmetic, (1) the reciprocal of the structured condition number is an approximation of the relative distance fr…