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Abstract We prove the rigidity ofrectifiableboundaries with constantdistributionalmean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds).As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant.The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong $C^k$C^{k}-norms.Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres. 

Abstract This paper focuses on dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. More precisely, the paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a gas consisting of hard spheres, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time. We expect that this paper can serve as a guideline for deriving a generalized Boltzmann equation that incorporates higher-order interactions among particles. 

We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the flux $f(u) = u^p (2 ≤ p ≤ 4)$ in the regime of its strict convexity, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $$a$$-contraction method with time-dependent shift and suitable weight function, which answers a question in [Blochas and Cheng, arXiv2501.01537, 2025]. Additionally, if the initial perturbation belongs to $L^1$ , then $L^2$ time-asymptotic decay rate $$t^{−1/4}$$ can be obtained. 

In this paper, we study stability properties of solutions to scalar conservation laws with a class of nonconvex fluxes. Using the theory of $a$-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations and give estimates on the weight coefficient $a$in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2\times <\#comment/>2$system setting by Chen, Golding, Krupa, and Vasseur [Arch. Ration. Mech. Anal. 246 (2022), no. 1, 299–332 and J. Hyperbolic Differ. Equ. 20 (2023), no. 3, 541–602].