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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.14411 |
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| _version_ | 1866912540284420096 |
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| author | Gao, Yuan Stephan, Artur |
| author_facet | Gao, Yuan Stephan, Artur |
| contents | Microscopic behaviors of chemical reactions can be described by a random time-changed Poisson process, whose large-volume limit determines the macroscopic behaviors of species concentrations, including both typical and non-typical trajectories. When the reaction intensities (or fluxes) exhibit a separation of fast-slow scales, the macroscopic typical trajectory is governed by a system of $\varepsilon$-dependent nonlinear reaction rate equations (RRE), while the non-typical trajectories deviating from the typical ones are characterized by an $\varepsilon$-dependent exponentially nonlinear Hamilton-Jacobi equation (HJE). In this paper, for general chemical reactions, we study the fast-slow limit as $\varepsilon\to 0$ for the viscosity solutions of the associated HJE with a state-constrained boundary condition. We identify the limiting effective HJE on a slow manifold, along with an effective variational representation for the solution. Through the uniform convergence of the viscosity solutions and the $Γ$-convergence of the variational solution representations, we rigorously show that all non-typical (and also typical) trajectories are concentrated on the slow manifold and the effective macroscopic dynamics are described by the coarse-grained RRE and HJE, respectively. This approach for studying the fast-slow limit is applicable to, but not limited to, reversible chemical reactions described by gradient flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast-slow chemical reactions: convergence of Hamilton-Jacobi equation and variational representation Gao, Yuan Stephan, Artur Analysis of PDEs Microscopic behaviors of chemical reactions can be described by a random time-changed Poisson process, whose large-volume limit determines the macroscopic behaviors of species concentrations, including both typical and non-typical trajectories. When the reaction intensities (or fluxes) exhibit a separation of fast-slow scales, the macroscopic typical trajectory is governed by a system of $\varepsilon$-dependent nonlinear reaction rate equations (RRE), while the non-typical trajectories deviating from the typical ones are characterized by an $\varepsilon$-dependent exponentially nonlinear Hamilton-Jacobi equation (HJE). In this paper, for general chemical reactions, we study the fast-slow limit as $\varepsilon\to 0$ for the viscosity solutions of the associated HJE with a state-constrained boundary condition. We identify the limiting effective HJE on a slow manifold, along with an effective variational representation for the solution. Through the uniform convergence of the viscosity solutions and the $Γ$-convergence of the variational solution representations, we rigorously show that all non-typical (and also typical) trajectories are concentrated on the slow manifold and the effective macroscopic dynamics are described by the coarse-grained RRE and HJE, respectively. This approach for studying the fast-slow limit is applicable to, but not limited to, reversible chemical reactions described by gradient flows. |
| title | Fast-slow chemical reactions: convergence of Hamilton-Jacobi equation and variational representation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.14411 |