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This paper deals with the existence of solutions of a class of contact mean-field game systems of first order consisting of a contact Hamilton-Jacobi equation and a continuity equation. Evans found a connection between Hamilton-Jacobi equations and continuity equations from the weak KAM point of view, where the coupling term is zero. Inspired by his work, we prove the main existence result by analyzing the properties of the Mather set for contact Hamiltonian systems.

We present an innovative process for directly transforming poly(ethylene terephthalate) (PET), a polymer extensively used in food and beverage packaging, into trans‐isomer‐enriched 1,4‐cyclohexanedimethanol (CHDM), a key ingredient in advanced specialty polymers. Our approach leverages a dual‐catalyst system featuring palladium on reduced graphene oxide (Pd/r‐GO) and oxalate‐gel‐derived copper‐zinc oxide (og‐CuZn), utilizing hydrogenation/hydrogenolysis relay catalysis. This method efficiently transforms PET into polyethylene‐1,4‐cyclohexanedicarboxylate (PECHD), which is then converted into CHDM with an impressive overall yield of 95 % in a two‐stage process. Our process effectively handles various post‐consumer PET plastics, converting them into CHDM with yields between 78 % and 89 % across different substrates. Additionally, we demonstrate the applicability and scalability of this approach through a temperature‐programmed three‐stage relay process on a 10‐gram scale, which results in purified CHDM with an isolated yield of 87 % and a notably higher trans/cis ratio of up to 4.09/1, far exceeding that of commercially available CHDM. This research not only provides a viable route for repurposing PET waste but also enhances the control of selectivity patterns in multistage relay catalysis. [ABSTRACT FROM AUTHOR]

Here we present a hybrid catalytic pathway for efficient synthesis of 5-aminomethyl-2-furancarboxylic acid (AMFC), a bio-based nylon-6 analogue monomer, from 5-hydroxymethylfurfural (HMF). This method combines homogeneous-catalyzed selective oxidation of HMF to 5-formyl-2-furancarboxylic acid (FFCA) with heterogeneous-catalyzed reductive amination using ammonia as the nitrogen source. Through this relay strategy, we achieve significant enhancements in overall efficiency, resulting in isolation yields of up to 92% for highly selective and scalable AMFC production from HMF. [ABSTRACT FROM AUTHOR]

The efficient and environmentally friendly removal of halogen substituents from organic substances is crucial for reducing the ecological and health hazards posed by persistent halogenated compounds. In this regard, we introduce CoN3P1@NP-PC, a novel asymmetric CoN3P1 single-atom catalyst, which demonstrates outstanding efficiency in transfer hydrodehalogenation (HDH) using just two equivalents of HCOONH4 as a benign and effective reducing agent. This catalyst was synthesized through a simple post-implantation method, capitalizing on the high porosity and large surface area of ZIF-8 precursors. Extensive characterization via a range of spectroscopic and microscopic techniques confirmed CoN3P1@NP-PC's superior atomic dispersion and asymmetric structural integrity, which was more favourable for HCOONH4 activation allowing facile generation of hydrogen radicals along with electron transfer. In our comprehensive evaluation, CoN3P1@NP-PC displayed outstanding versatility for the transformation of a diverse array of halogenated substrates, efficiently processing compounds with varied functional groups and complex molecular structures. Notably, CoN3P1@NP-PC proved particularly effective in breaking down brominated flame retardants such as 2,4,6-tribromophenol and tetrabromobisphenol A, reinforcing its potential as a powerful tool for ecological restoration and pollution control. [ABSTRACT FROM AUTHOR]

Developing practical and scalable catalytic approaches for directly converting concentrated 5-hydroxymethylfurfural (HMF) into value-added bio-based chemicals represents a major challenge in efforts to make the lignocellulosic biorefinery a sustainable and ecologically viable reality. We present here Ir/Na-ZSM-5 as a robust catalyst for efficiently producing 2,5-bis(methoxymethyl)furan (BMMF), a crucial compound for advanced biorefineries, through the reductive etherification of HMF. Benefitting from its unique structural features as well as the cooperative interaction between selective hydrogenation enabled by ultrasmall Ir nanoparticles and appropriate surface acidity resulting from alkali exchange, Ir/Na-ZSM-5 achieves impressive BMMF yields (91%) and maintains over 98% carbon balance from concentrated HMF (up to 22.6 wt%) under mild conditions (50 °C, 1 MPa H2). Moreover, the catalyst displays outstanding stability with an unmatched turnover number during extended flow-reaction processes, underscoring its applicability for large-scale production. [ABSTRACT FROM AUTHOR]

Assume $ H = H(x, p, u) $ with $ (x, p)\in T^*M $ and $ u\in\mathbb{S} $, is smooth and satisfies Tonelli conditions in $ p $, Lipschitz continuity condition in $ u $, where $ M $ is a compact connected smooth manifold without boundary. We find a compact interval $ \mathcal C $ such that equation \begin{document}$ H(x, \partial_xu(x), u(x)) = c $\end{document} has solutions if and only if $ c\in \mathcal C $. We also study the long-time behavior of the unique viscosity solution $ u^c $ of$ \partial_t u(x, t)+H(x, \partial_xu(x, t), u(x, t)) = c, \quad u(x, 0) = \varphi(x)\in C(M, \mathbb{R}). $If $ c\in \mathcal C $, $ u^c $ is bounded by a constant independent of $ c $ and Lipschitz with respect to the argument $ x $ with a Lipschitz constant independent of $ c $ and $ \varphi $. If $ c\notin \mathcal C $, then the long-time average of $ u^c $ can be characterized by a function $ c\mapsto\rho(c) $ which admits a modulus of continuity. We obtain these results by analyzing properties of a kind of one-parameter semigroups of operators. All the aforementioned results show the fundamental difference between Hamilton-Jacobi equations with Hamiltonians $ H(x, p, u) $ and $ \bar H(x, p) $. [ABSTRACT FROM AUTHOR]

Benzimidazoles possess a series of applications for industrial chemistry and biomedicine. However, the complicated synthetic steps and harsh reaction conditions limit its further development. Herein, we reported an efficient, environmentally friendly, and stable bimetallic CoMoO4@C catalyst, which used low‐cost cobalt and molybdenum as the basic raw material. In base‐free conditions, it has been proved that the yield could achieve to 99.9% for the synthesis of benzimidazoles with liberating water as the sole byproduct, and the catalyst remains stable and efficient even after five successive cycle tests. Additionally, experiments and characterizations confirm the good oxidation activity of the catalyst benefits from the numerous oxygen vacancies provided by the high concentration of low‐valent cobalt (Co2+) and the doping of carbon improves the intermolecular transport of electrons. Furthermore, this strategy could potentially be applied in the industrial production of benzimidazoles. [ABSTRACT FROM AUTHOR]

This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations \begin{equation*} w_t+H(x,w,w_x)=0, \end{equation*} where H(x,u,p) is strictly decreasing in u and satisfies Tonelli conditions in p. We show that each viscosity solution of the ergodic contact Hamilton-Jacobi equation H(x,u,u_x)=0 can be reached by many different viscosity solutions of the above evolutionary equation in a finite time. [ABSTRACT FROM AUTHOR]

We study the Hamilton–Jacobi equations H (x , D u , u) = 0 in M and ∂ u / ∂ t + H (x , D x u , u) = 0 in M × (0 , ∞) , where the Hamiltonian H = H (x , p , u) depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem. [ABSTRACT FROM AUTHOR]

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems H (x , u , p) H (x , u , p) with certain dependence on the contact variable u u. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set S ~ s S ~ s consists of strongly static orbits, which coincides with the Aubry set A ~ A ~ in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show S ~ s ⫋ A ~ S ~ s ⫋ A ~ in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of H H on u u fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones. [ABSTRACT FROM AUTHOR]