11 problems

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 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…

Landau equation conditional regularity conjecture. As long as the three density functions $\rho$, $E$, and $H$ are “under control”, solutions of the Landau equation remain smooth,…

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…

Entropic-gap conjecture. With a bit more work, there is a constant $K>0$ such that

Let $p<3/2$. The $L^p\to L^\infty$ regularization rate of the heat equation is said to be the rate obtained by heat-flow smoothing from $L^p$ data. Landau regularization conjecture…

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…

Let $f_{\rm in}$ be the initial datum in the setting of the paper's Landau weak-strong uniqueness theorem, and let $L^{\infty,k}$ denote the weighted bounded space used there. The…

Let $Q_1$ and $Q_{1/2}$ denote the kinetic cylinders appearing in the equation, let $a$, $c$, $f$, $g$, and $\bar a$, $\bar c$ be the corresponding coefficients and functions, and…

The bounded ancient-solution Liouville conjecture. The solution $f$ must be a stationary Maxwellian. This is intended to rule out non-Maxwellian bounded ancient solutions arising f…

Let $f:[0,T]\times\mathbb R^3\to[0,\infty)$ be a classical solution of the space-homogeneous Landau equation with the Landau-Coulomb operator $Q$, and let $f(0,v)=f_0(v)$. The Land…