23 problems

Let a nonlinear dispersive equation have dispersion of order $l$ and a nonlinearity of order $k\ge 3$ with a loss of $m$ derivatives. Let $s_0$ be such that, for $s>s_0$, the $H^s$…

Small-data critical well-posedness conjecture. The Cauchy problem is well-posed for small data in $\dot H^{\frac{n}{2}}$.

Consider the cubic nonlinear Schrödinger equation referred to as, with initial data in the Sobolev space $H^s$. Local well-posedness conjecture. The cubic NLS equation is locally w…

Null-condition well-posedness conjecture. If the null condition holds, then (GNLW) is well-posed in $H^s\times H^{s-1}$ for some $s<\frac n2+\frac34$ when $n=2$, respectively for s…

Let $\mathcal G$ be a compact Lie group with Lie algebra $\mathbf g$, let $s_0=\frac{n-2}{2}$ be the scaling index, and consider the Yang–Mills equation … for a connection with cur…

Let $(M,g)$ be a complete Riemannian manifold, and consider wave maps from $\mathbb R^2\times\mathbb R$ into $(M,g)$. The critical initial-data space is $H^1\times L^2$. Wave maps…

Let $n\geq 4$, let $0\leq s\leq \frac12$, and consider the power nonlinear wave equation … The initial data lie in $H^s\times H^{s-1}$. Local well-posedness conjecture. The equatio…

For a semilinear wave equation, let $H^s \times H^{s-1}$ denote the Sobolev data space, and let $s_0$ be the scaling index determined by the equation's scaling. Below-scaling ill-p…

The systems under consideration are basic field theories with critical well-posedness exponent $s_c$, meaning that the $dot H^{s_c} \times dot H^{s_c-1}$ norm of the initial data i…

Let $k\geq 3$ and consider the $k$-generalized Korteweg-de Vries equation on the periodic domain, with initial data in the Sobolev space $H^s(\mathbb T)$. Unconditional well-posedn…

Let $L^2_{+}(\mathbb{T})$ and $L^2_{+}(\mathbb{R})$ denote the positive-frequency Hardy subspaces of $L^2(\mathbb{T})$ and $L^2(\mathbb{R})$, respectively. Consider the defocusing…

Higher-spatial-regularity conjecture. The Augmented formula is well-posed in $\mathscr{C}^k\left([0,T),\mathbb{H}^n\times\mathbb{R}^4\right)$, with the corresponding weighted norm…

Let $k\in\mathbb{N}$, and let $T_{\varepsilon,\kappa,R}$ denote the time scale introduced for the Augmented formula. The solution takes values in the product of the Sobolev space…

Let the initial data be non-trapping. Global well-posedness conjecture for non-trapping initial data. The result in Theorem holds for all non-trapping initial data. The source give…

Let $d$ be the spatial dimension, let $\frac{d}{2}+1<\sigma<\infty$, and let $(v,B,\Gamma)\in C([0,T);\mathbf{H}^{\sigma})$ be a solution to the free boundary MHD equations. Write…

Global existence and scattering conjecture. All maximal-lifespan solutions are globally well-posed and scatter.

Let $u=u(t,x)$ solve the derivative nonlinear Schrödinger equation on $\mathbb{R}$, … For $s\in\mathbb{R}$, let $H^s(\mathbb{R})$ denote the Sobolev space of regularity $s$. DNLS w…

Equicontinuity conjecture. For any $Q\subseteq\mathcal{S}$ that is $L^2$-bounded and equicontinuous, the set $Q_*$ is also $L^2$-equicontinuous.

Let $r$ and $d$ be positive integers, and let $s_p=-\frac{1}{2r}$ be the critical threshold for probabilistic scaling. Consider the nonlinear Schrödinger equation referred to as…

Regularity propagation conjecture. The local solution $f(t)$ carries the $\frac{1}{2}+$ regularity of the initial data up to time $T_0$; in particular, for every $T\in(0,T_0)$,

Consider the defocusing Schrödinger initial value problem for $u:I\times\mathbb{R}^{d}\to\mathbb{C}$, … where $s_c\geq 0$ and $p=\frac{4}{d-2s_c}$. The critical norm conjecture ass…

Let $v$ solve the mass-critical nonlinear Schrödinger equation on ${\mathbb R}\times{\mathbb R}$, let $v_0\in L^2_x({\mathbb R})$ be its initial data, let $M(v_0)$ denote its mass,…

Let $u$ solve the mass-critical generalized KdV equation on ${\mathbb R}\times{\mathbb R}$, let $u_0\in L^2_x({\mathbb R})$ be its initial data, let $M(u_0)$ denote its mass, and l…