17 problems

Uniform shifted-sum conjecture. The shifted sum satisfies estimate uniformly in $z_0$. Accordingly, for $d\geq 3$ and $a,b\in\mathbb{R}^d$, the sum

Uniqueness conjecture. The sequence $bm{nu}^{\infty}={nu_{k}}_{k}$ is the unique solution in $L^{\infty}([0,\infty),\mathcal{X})$ to the annihilated Boltzmann hierarchy in the sense o…

Let $M$ be the state manifold, let $R>0$, let $v$ denote the velocity variable, and let ${\mathcal K}$ be the interaction kernel. Let $\mu_t^N$ be the empirical measure of the norm…

Consider a conservative long-range interacting particle system at thermal equilibrium with a tagged particle whose mean-field evolution is trivial. Let the linearized mean-field op…

Let $n\geq 2$, let $\rho^n$ denote the function associated with equilibria whose largest eigenspace has dimension $d=1$, and let $\rho^*>0$ be the threshold density. For…

Let $\mathsf{T}_{\mathsf{H}}$ be the generator of the $C_{0}$-semigroup $\left(U_{\mathsf{H}}(t)\right)_{t\geqslant0}$, let $\mathbf{P}_{0}$ be the spectral projection associated w…

Let $\Phi$ be a scale-invariant potential of the form … and consider the scaling … with $\gamma>0$. Scale-invariant potentials conjecture. If $s\in(\frac12,1)$, then the one-partic…

Let $\Phi$ be a smooth decaying potential and consider the scaling … with $\varepsilon\to0$, where $N$ is the average number of particles per macroscopic unit volume. Smooth decayi…

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 $T\in(0,T_0)$,

Let $B$ be a collision cross-section satisfying condition with homogeneity $\gamma<\gamma_{\mathrm{crit}}=-\frac32$, and let $g(t,w)$ be a weak solution of the cylindrical-dilatati…

Let $B$ be a collision cross-section satisfying condition, and let $\gamma$ denote its homogeneity. Let $Q(t)$ and $L_0$ be the matrix-valued quantities appearing in the homoenerge…

Let $B$ be a Boltzmann collision kernel with parameters $\gamma\in(-d,\infty)$ and $\alpha\in[0,2)$. A spectral gap means that the associated linearized Boltzmann collision operato…

Let $0<\nu<1$ be the order of the singular cross section kernel, and let $f$ be a weak solution of the non-cutoff homogeneous Boltzmann equation in $\mathbb{R}^d$ with initial datu…

Let the particles of Saturn's Ring be modeled by a kinetic Boltzmann equation associated with the many-body system referred to as system (15). A fractal solution means a solution e…

Kinetic equation of state hyperbolicity and causality conjecture. Under this equation of state, there exists a smooth function $f_{\rm kinetic}$ such that

Kinetic equation of state auto-diffeomorphism conjecture. The map $(n,\beta)\leftrightarrow(\eta,\rho)$ given implicitly by these equations is an auto-diffeomorphism of…

The limiting-process conjecture. The limiting process $\mathbf{Q}(\tau)$ is (a) straight motion if $\beta=\alpha+1/2-\epsilon$; (b) the integral of an Ornstein–Uhlenbeck process,