22 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 $H_x^\gamma(\mathbb{R}^2)$.

The Strichartz conjecture on $\mathbb{R}\times\mathbb{S}^3$. One has

Let $N\geq 1$, let $\varphi$ be the time cutoff appearing in the estimate, and let $\phi\in L^2(\mathbb{R}\times\mathbb{T})$. For $\xi_0\in\mathbb{R}\times\mathbb{Z}$, write…

Let $n\geq2$, let $(1/p,1/q)$ lie on the line segment $(O,A]$, and set $s$ by … For an orthonormal family $\{f_j\}_{j\in J}\subset\dot{H}^{s}(\mathbb{R}^n)$ and coefficients…

Let $n\geq1$, and let $A=(1/p,1/q)=((n-1)/(n+1),n/(n+1))$. For an orthonormal family $\{f_j\}_{j\in J}\subset L^2(\mathbb{R}^n)$ and a sequence…

Let $I$ be a compact interval. For $p,q\in[1,\infty]$, write $L^q_xL^\infty_t(\mathbb{R}\times I)$ for the corresponding mixed-norm space. For $\alpha\in\mathbb{R}$, let…

Generic Schrödinger control conjecture. Generically with respect to the lengths $(\ell_i)_{i=1,\dots,N}$, (GGCC) is a necessary and sufficient condition for the internal control of…

Bilinear space-time estimate conjecture. For $2\leq p,r<\infty$, the estimate

Lee–Rogers–Seeger local space-time conjecture. The estimate

Planchon's space-time conjecture. One has

Let … where $a_m$ is supported in … For $\lambda>0$, define the superlevel set $U_\lambda(\nu)=\{(x,t):|\nu(x,t)|\geq\lambda\}$. Periodic restriction conjecture. If $|a_m|\leq 1$ f…

Let $d\geq 2$, $N\geq 1$, and define … … Let $\mathcal{M}$ be a real analytic hypersurface in $\mathbb{T}^d$ with nonnegative curvature and surface measure $\sigma$. Paraboloid res…

Controllability-diameter conjecture. For every such Hamiltonian, the implication

Uniqueness conjecture for the Schrödinger potential. The knowledge of $\Lambda_{g,q}$ uniquely determines $q$. This is the potential-recovery version of the Calderón problem in a k…

Restricted-type endpoint conjecture. The estimate

Improved symmetric-space Strichartz conjecture. Under these assumptions, this estimate holds for all $p>2+\frac{4}{r}$. This strengthens the preceding conjectured range…

Let $G/K$ be a Riemannian globally symmetric space of compact type, equipped with the negative of the Cartan--Killing form as the canonical Riemannian metric $g$. Let $d$ and $r$ b…

Let $G$ be a compact Lie group equipped with a rational metric $g$. Let $d$ be the dimension of $G$ and $r$ the rank of $G$. Let $I\subset\mathbb{R}$ be a finite time interval. Com…

Gaussian extremizer conjecture. A function $f(\boldsymbol{x})$ maximizes the Strichartz estimate if and only if it has the form

Generic Strichartz conjecture. For arbitrarily small $\varepsilon>0$, one has

Let $\psi_0$ be the initial wave function of an $N$-particle system, with initial density $n_0$ on a domain $\Omega$. Assume that the external potential is time-analytic, that…

Let $A(1)=\{\xi:|\xi|\sim 1\}$. For $R\geq 1$, let $B(0,R)$ denote the ball of radius $R$ centered at the origin, and let $e^{it\Delta}f$ be the free Schrödinger evolution of a fun…