59 problems

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…

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…

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…

Gevrey regularity conjecture. Any finite-regularity solution of the spatially inhomogeneous Boltzmann equation belongs to the Gevrey class

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 $B(\lvert v-v_*\rvert,\sigma)=\Phi(\lvert v-v_*\rvert)b(\cos\theta)$,…

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.

Global-equilibrium convergence conjecture. The solution $f(t)$ converges to $f_{infty}$ as $t\to\infty$. The corresponding fundamental problem is to prove or disprove this converge…

Let $\mathfrak{q}\in[0,1]$, and consider the massless Boltzmann equation with scattering kernel … on an FLRW spacetime, with Maxwell--Jüttner equilibrium $J(t^{2\mathfrak{q}}p)$ an…

Let $H(t)$ denote the entropy along a solution of the space-homogeneous Boltzmann equation. McKean's conjecture. McKean's first conjecture asserts that … This paper disproves the c…

Let $H(f_t)$ denote the entropy of a solution $f_t$ of the Boltzmann equation, and consider its consecutive higher-order time derivatives. McKean's higher-order monotonicity conjec…

Let $f_t$ be a solution of the Boltzmann equation, and let its Fisher information be denoted by $I(f_t)$. McKean's conjecture. The Fisher information $I(f_t)$ is monotonically decr…

Higher-order TRMC convergence conjecture. Convergence results of type can be extended to these higher-order time-relaxed Monte Carlo methods by iterating the same proof argument fo…

Strong-solution existence conjecture. The existence result can be extended to strong measure solutions, or even strong density solutions.

Let $T>0$ and let $b1'$ be such that $1-2Tb1'>0$. An exponential weight is a factor of the form $e^{b2|v|^2}$ in the velocity variable $v$. Critical exponential-weight conjecture.…

Let the Boltzmann equation describe the evolution of a particle distribution, and let the entropy dissipation denote the dissipation functional associated with its entropy along th…

Half-plus regularity propagation conjecture. The solution carries the $\frac12+$ regularity of the initial data up to time $T_0(s)$: for every $T\in(0,T_0(s))$,

Consider interaction potentials satisfying … where $s\in[0,1]$, and let the Boltzmann equation be subjected to the scaling referred to as the weak-collision scaling. Landau-limit c…

The Boltzmann–Grad limit of the Lorentz gas describes the dynamics of a tagged particle as the radius of the fixed scatterers tends to zero. Lorentz's conjecture. The limiting syst…

In the frameworks of the works by Djehiche and colleagues and by Rezakhanlou and colleagues, the large-deviation lower bound is currently established only for a class of good paths…