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Diffusion as a singular homogenization of the Frenkel-Kontorova model. (English) Zbl 1222.35018
The purpose of the paper is to describe the asymptotic behaviour of a system of particles which solves a fully overdamped Frenkel-Kontorova model for which the velocity is proportional to the force acting on these particles. The system is written as
$\frac{dU_i}{d\tau}(\tau)= \varepsilon^{2(\alpha-1)} F\Bigg(\bigg[ \frac{U_{i+j}(\tau)-U_i(\tau)}{\varepsilon^{2(\alpha-1)}}\bigg]_j^m,U_i(\tau)\Bigg)$
for $$\tau\in (0,\infty)$$ and $$i\in\mathbb Z$$. The initial condition $$U_i(0)=\frac{1}{\varepsilon}u_0 (i\varepsilon^\alpha)$$ is added. Here $$[V_j]_j^m=(V_{-m},V_{-m+1},\dots, V_{m-1},V_m)$$ and $$F\in C^1(\mathbb R^{2m}\times\mathbb R)$$ and satisfies further hypotheses, among which is a 1-periodicity condition with respect to the second variable. This intends to generalize the original Frenkel-Kontorova model for which $$F(V_{-1},V_1,V_0)= V_{-1}+V_1+\sin(2\pi V_0)$$.
Introducing the rescaling
$U^\varepsilon(t,x)= \varepsilon U_{\big\lfloor\frac{x}{\varepsilon^\alpha}\big\rfloor} \bigg(\frac{t}{\varepsilon^{2\alpha}}\bigg),$
where $$\lfloor\cdot\rfloor$$ denotes the floor integer part, the authors end with the problem
$U_t^\varepsilon(t,x)= \frac{1}{\varepsilon}F \Bigg(\bigg[\frac{U^\varepsilon (t,x+j\varepsilon^\alpha)- U^\varepsilon(t,x)}{\varepsilon^{2\alpha -1}}\bigg]_j^m, \frac{U^\varepsilon(t,x)}{\varepsilon}\Bigg),$
with the initial condition $$U^\varepsilon(0,x)= u_0(\lfloor \frac{x}{\varepsilon^\alpha}\rfloor \varepsilon^\alpha)$$.
The main result of the paper proves that, if $$\alpha >2$$, the unique viscosity solution of this parabolic problem uniformly converges on compacts sets of $$(0,\infty)\times\mathbb R$$ to the viscosity solution $$u^0$$ of $$u_t^0=G(u_x^0)u_{xx}^0$$ in $$(0,\infty)\times\mathbb R$$ with the initial condition $$u^0(0,x)= u_0(x)$$. Here $$G$$ is a positive and continuous function from $$(0,+\infty)$$ to $$\mathbb R$$ such that $$\lim_{p\rightarrow 0}G(p)=0$$ and $$\lim_{p\rightarrow \infty}G(p)>0$$. The main step of the proof of this convergence result consists to prove that $$U_\varepsilon$$ is close to an expression involving a so-called hull function $$h$$ (ansatz). Then, the authors give the expression of $$G$$ in terms of this hull function $$h$$. The keys of the proof are the properties of the different terms of the ansatz. Assuming that $$\alpha =2$$, the limit problem is slightly different.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B10 Periodic solutions to PDEs 35Q70 PDEs in connection with mechanics of particles and systems of particles 70H20 Hamilton-Jacobi equations in mechanics 35F21 Hamilton-Jacobi equations 35D40 Viscosity solutions to PDEs
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