Homogenization of accelerated Frenkel-Kontorova models with \(n\) types of particles (Q2844836)

The authors describe a homogenization result for systems of ODEs which describe the dynamics of particles. They consider a generalized Frenkel-Kontorova (FK) model written as NEWLINE\[NEWLINE\begin{cases} \left( u_{j}\right) _{\tau } = & \alpha _{0}(\xi _{j}-u_{j}), \\ \left( \xi _{j}\right) _{\tau } = & 2F_{j}(\tau ,\left[ u(\tau ,\cdot ) \right] _{j,m})+\alpha _{0}(u_{j}-\xi _{j}), \end{cases} NEWLINE\]NEWLINE with NEWLINE\[NEWLINE\begin{cases} u_{j+n}(\tau ,y) = & u_{j}(\tau ,y+1), \\ \xi _{j+n}(\tau ,y) = & \xi _{j}(\tau ,y+1), \end{cases}NEWLINE\]NEWLINE for all \((\tau ,y)\in (0,\infty )\times \mathbb{R}\) and all \(j\in \mathbb{Z}\) . Here \(\left[ u(\tau ,y)\right] _{j,m}=(u_{j-m}(\tau ,y),\ldots ,u_{j+m}(\tau ,y))\) and the functions \(F_{j}\) satisfy regularity, monotonicity, periodicity and ordering properties. The authors prove that \( F_{j}(\tau ,V_{-1},V_{0},V_{1})=\theta _{j+1}(V_{1}-V_{0})-\theta _{j}(V_{0}-V_{1})+\sin (2\pi V_{0})+L\) satisfies these hypotheses. Defining \( u_{j}^{\varepsilon }(t,x)=\varepsilon u_{j}(t/\varepsilon ,x/\varepsilon )\) and \(\xi _{j}^{\varepsilon }(t,x)=\varepsilon \xi _{j}(t/\varepsilon ,x/\varepsilon )\) the authors end up with a rescaled generalized FK model. The first main result of the paper proves the existence of a unique viscosity solution of the following Cauchy problem NEWLINE\[NEWLINE\begin{cases} \left( u_{j}\right) _{\tau } = & \alpha _{0}(\xi _{j}-u_{j}), \\ \left( \xi _{j}\right) _{\tau } = & G_{j}(\tau ,\left[ u(\tau ,\cdot ) \right] _{j,m},\xi _{j},\inf_{y^{\prime }\in \mathbb{R}}(\xi _{j}(\tau ,y^{\prime }-py^{\prime })+py-\xi _{j}(\tau ,y),(\xi _{j})_{y}), \end{cases}NEWLINE\]NEWLINE called effective Cauchy problem, where \(G_{j}\) can be expressed in terms of \( F_{j}\) and of other data of the problem. For the proof of this existence and uniqueness result, the authors first define the notion of viscosity solution for such systems, through that of viscosity sub- and super-solutions. They also prove a comparison principle within this context. Moreover \(u_{j}\) and \( \xi _{j}\) are proved to be continuous and to satisfy some ordering property. The second main result establishes the asymptotic behaviour of the rescaled generalized FK model. For that purpose, the authors essentially build appropriate test-functions. The authors then prove further properties of the previous Cauchy problem. They introduce the notion of hull function. They prove an ergodicity property of this Cauchy problem and they build such hull functions for general Hamiltonians. In the last part of the paper, they build Lipschitz continuous approximate hull functions and they prove some further properties of the effective Hamiltonian.