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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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