48 problems

Long-time equilibrium conjecture. The one-dimensional solution should satisfy

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, with initial data $\rho_0$ satisfying the paper's initi…

The particle system is given by Systems and, while the corresponding kinetic model is given by Systems and. Mean-field convergence conjecture. The particle system converges to the…

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 $N\in\mathbb{N}$ and let $X_N$ be the finite-dimensional polynomial space with orthogonal projection $\Pi_{X_N}$. Let $\Omega$, $\partial_x$, and $\partial_x^*$ be the operator…

Convergence conjecture. The solution $f(t)$ converges to the global equilibrium $f_{\infty}$ as $t\to\infty$; moreover, the convergence rate should be estimated.

Let $n$ denote the spatial dimension. The paper's macroscopic limit is established for $n\in\{2,3\}$, using stable equilibria from Lemma. Higher-dimensional macroscopic-limit conje…

Micro-resolution conjecture. Resolving $g$ is at least as difficult as capturing $f$, and may be more difficult because $g$ is more oscillatory in velocity.

HOLO non-injectivity conjecture. The observed lack of consistency of HOLO is caused by $Delta t$ becoming sufficiently large that $\mathcal{E}^*$ is no longer injective.

Small-micro-density conjecture. When $utotext{large}$, $n{g^tau}$ should be sufficiently small that the convergence rate of the MM-L method is not harmed.

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, and let $\rho_0$ be initial data satisfying the paper's…

Consider the granular kinetic equation with interaction exponent $\gamma>2$ and a solution with the initial conditions studied in the paper. Absence-of-blow-up conjecture. The solu…

Let $f^N_{(k)}$ denote the empirical measure of a genetic-algorithm particle system with population size $N$ at iteration $k$, and let $f_{(k)}$ denote the corresponding kinetic ev…

Let $W_k=W_k(t,x)$ denote the energy spectrum, let $\bm{k}=(k_1,k_2,k_3)\in\mathbb{R}^{3d}$, and define … Let $\Omega(\bm{k})$ be the resonance factor defined in the source, and le…

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…

Kinetic integro-differential Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension, $\lambda$, and $\Lambda$, such that

Kinetic Krylov–Safonov conjecture. There exist constants $\alpha>0$ and $C$, depending only on the dimension and $\lambda,\Lambda$, such that

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…

Angular-momentum asymptotic-information conjecture. The distribution of the angular momentum is the only microscopic information retained in the time-asymptotic limit as…

Let $f$ and \text{\boldmath u} be a large weak solution of the kinetic Cucker–Smale model coupled with the three-dimensional incompressible Navier–Stokes equations, with initia…

Three-dimensional Gibbsian tessellation conjecture. If the rate $\hat f$ satisfies a system of kinetic equations analogous to (1.21), then the outlined strategy should yield a cons…

Keras-overhead conjecture. The overhead in calling the Keras function texttt{model.predict} is the dominant cost in network evaluation, especially when the number of moments to be…

Moment-reconstruction-error conjecture. The moment reconstruction error $E_{\bw}$ would be a more reliable measure for closure accuracy than the multiplier error $E_{\bsalpha}$. Th…