79 problems

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…

Let $u=u(t,x)$ be defined for $(t,x)in\mathbb{R}_+\times\mathbb{R}$, let $\beta\in(0,1)$ and $\alpha\in(0,2)$, and let $\dot{W}$ denote space-time white noise. Consider the space-t…

Let the initial data for the flow be large and satisfy a suitable non-trapping condition. Large-data non-trapping global well-posedness conjecture. The result of Theorem holds for…

Scaling-critical regularity conjecture. The equation is well-posed in $B^s$ for $s>s_\text{crit}$ and ill-posed in $B^s$ for $s<s_\text{crit}$.

Let $s_c$ be the scaling-critical Sobolev index of a partial differential equation, meaning that the homogeneous $H^{sc}$-norm is invariant under the equation's scaling symmetry.…

Consider the defocusing cubic nonlinear Schrödinger initial value problem … with initial data $\phi(x,0)=\phi_0(x)\in H^s(\mathbb{R}^3)$, where $H^s(\mathbb{R}^3)$ is the usual inh…

The intensity-dependent dispersion nonlinear Schrödinger model is given by equation, with $\Sigma$ the energy space in and $\mathcal{H}$ the exponentially weighted…

Global well-posedness and scattering conjecture. Global well-posedness and scattering hold for $(1.1)$ when $0<s_c<1$. The mass-critical case $s_c=0$ and energy-critical case…

Consider the generalized Benjamin–Ono equation with nonlinearity exponent $k\geq 4$, and let $u$ be a solution. For $\lambda>0$, define the scaling … Then $u_\lambda$ is again a so…

Consider the dissipative modified Korteweg–de Vries equation on the real line … The scaling-critical index of the mKdV equation is $-\frac{1}{2}$, while that of the equation with o…

Quintic NLS well-posedness conjecture. Local and global well-posedness should hold in $H^s(\mathbb T)$ for all $s\geq 0$.

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 $(k,s)\in\mathbb{R}^2$ and let $\mathscr{A}_a$ denote the parameter region associated with the local well-posedness estimates for the Hirota–Satsuma system. For $a=-1/8$, the s…

Let $\mathcal{M}^{p,q}_{s}$ denote a weighted modulation space, with $0<q\leq 1$ allowed, and consider the nonlinear heat equations associated with fractional powers of the general…

Let $\Omega$ be the spatial domain, let $T>0$, and consider the two-phase model with weak solutions defined by quadruples … where $L_0^2(\Omega)$ is the space of square-integrable…

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 $s\geq s_0$.

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…

The Raman scattering process in modulated wavetrains can be modeled by an integro-differential equation of nonlinear Schrödinger type with an oscillatory memory function; applying…