135 problems

Let $\mathcal{L}_N$ be Kac's master-equation operator on $L^2\left(\mathbb{S}^{N-1}\left(\sqrt{N}\right),d\sigma_N\right)$, and let … be its spectral gap. Kac's uniform spectral-ga…

Self-similar long-time behavior conjecture. In this regime, the long-time behavior of the solution should be given by a self-similar solution of the displayed approximate equation.…

Cercignani's conjecture. The entropy production term of the Boltzmann equation should be controlled by the relative entropy between $f$ and $\mathcal{M}_0(f)$ through an entropy-en…

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 $F_N$ range over probability densities on Kac's sphere, and define the entropy-entropy production ratio by … where $H_N\left(F_N\right)$ is the relative entropy and…

Desvillettes–Villani conjecture. Time oscillations should occur in the evolution of the relative local entropy. This conjecture concerns the long-time behavior of solutions to the…

A neural network consists of neurons receiving excitatory and inhibitory inputs. A network is dynamically balanced when these inputs are sufficiently numerous that their network-wi…

Let $f(t,v)$ be the velocity profile of a granular gas evolving according to the inelastic Boltzmann equation. A self-similar solution has the form … where…

The relativistic Boltzmann equation in the relaxation-time model describes non-equilibrium dynamics, and its hydrodynamic gradient expansion is generally asymptotic rather than con…

Let $Q(g,f)$ denote the Boltzmann collision operator without angular cutoff, and let $s$ be the exponent determined by the angular singularity. For a fixed function $g$, write…

Let $f_j^N$ be the $j$-particle marginal, with initial data $f_j^N(0)=f_0^{\otimes j}$, where $f_0$ is a given one-particle density. The difficulty in deriving the Landau equation…

Let $\mu_d(v)=(2\pi)^{-d/2}e^{-|v|^2/2}$ be the Maxwellian equilibrium distribution on $\mathbb{R}^d$, let $Q$ denote the Boltzmann collision operator, and let $s$ with $0<s<1$ be…

Consider the non-cutoff Boltzmann collision operator … where $B(\lvert v-v_*\rvert,\sigma)=\Phi(\lvert v-v_*\rvert)b(\cos\theta)$,…

Convergence conjecture. The series converges when

Let $\mathcal{L}$ be the linear Boltzmann operator, let $\mathfrak{S}(\mathcal{L})$ denote its spectrum, and let … where $\sigma$ is the collision frequency. The spectral descripti…

Discrete-spectrum conjecture. If $\gamma+\alpha>0$ and $\alpha>0$, the spectrum is purely discrete; moreover, this is also conjectured to hold in the non-cutoff borderline case…

Cross-section regularity conjecture. Not all derivatives of the kinetic part of the cross section are needed, probably only one derivative is enough, and the angular part need not…

Counterexample conjecture. The sole assumption $f_{in}\in\bigcap_{s>0}L^1_s$ may not be sufficient to propagate uniformly the $L^1_s$ norm for $s>2$; it may be possible to construc…

Algebraic-tail conjecture. There is an algebraic tail in space after the sound wave.

Kinetic conjecture. In a leading Feynman diagram, the sum of these phases vanishes for every choice of internal wave numbers, with the cancellation holding for $m=0,\ldots,n$.

Kinetic conjecture. In a leading Feynman diagram, the Kirchhoff rule never forces an identification of the form $\delta(k_j)$ with some wave vector $k_j$. In addition, the sum of t…

Let the Landau equation be the kinetic equation under consideration, and say that a solution decays to equilibrium slower than exponentially if its convergence to equilibrium is no…

For the inelastic Maxwell model in self-similar scaling, consider solutions with power-like high-energy tails. Ernst–Brito conjecture. Such solutions determine the universal long-t…

H-theorem conjecture. A form of the H-theorem remains valid even when $mathcal{A}$ is not transitive.

Conjecture. Recovering the complete Leslie viscous stress tensor is possible with an anisotropic BGK collision operator involving several relaxation timescales.