11 problems

The kinetic Schauder estimate. The displayed estimate asserts a uniform interior Hölder bound for the velocity Hessian of smooth solutions in terms of the solution, the coefficient…

Let $k\geq 1$, $d\geq 1$, and let $u=u(t,x_1,\dots,x_k,v)$ be a nonnegative weak subsolution of … Set … … and define…

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…

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

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

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

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…

Constrained-problem construction conjecture. First solve the unconstrained problem using the formula above. Then, whenever $\min \tilde u\leq 0$, replace $\tilde u$ by $v^2/2$ on t…