zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Barles, G., Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. var. partial differential equations, 30, 4, 449-466, (2007) · Zbl 1136.35004
[2] Barles, G., Solution de viscosité des équations de Hamilton-Jacobi, (1994), Springer-Verlag Paris · Zbl 0819.35002
[3] Boccardo, L.; Murat, F., Remarques sur lʼhomogénéisation de certains problèmes quasi-linéaires, Portugal. math., 41, 1-4, 535-562, (1982) · Zbl 0524.35042
[4] Braun, O.M.; Kivshar, Y.-S., The Frenkel-Kontorova models, concepts, methods and applications, (2004), Springer-Verlag · Zbl 1140.82001
[5] Crandall, M.-G.; Ishii, H.; Lions, P.-L., Userʼs guide to viscosity solutions of second order partial differential equations, Bull. amer. math. soc. (N.S.), 27, 2, 1-67, (1992) · Zbl 0755.35015
[6] Evans, L.-C., The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. roy. soc. Edinburgh sect. A, 111, 3-4, 359-375, (1989) · Zbl 0679.35001
[7] Floria, L.-M.; Mazo, J.-J., Dissipative dynamics of the Frenkel-Kontorova models, Adv. phys., 45, 6, 505-598, (1996)
[8] Forcadel, N.; Imbert, C.; Monneau, R., Homogenization of fully overdamped Frenkel-Kontorova models, J. differential equations, 246, 3, 1057-1097, (2009) · Zbl 1171.49023
[9] Forcadel, N.; Imbert, C.; Monneau, R., Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete contin. dyn. syst., 23, 3, 785-826, (2009) · Zbl 1154.35306
[10] Imbert, C.; Monneau, R., Homogenization of first-order equations with \((u / \varepsilon)\)-periodic Hamiltonians. part I: local equations, Arch. ration. mech. anal., 187, 1, 49-89, (2008) · Zbl 1127.70009
[11] Jerrard, R., Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials, Adv. differential equations, 2, 1, 1-38, (1997) · Zbl 1023.35526
[12] Kontorova, T.; Frenkel, Y.-I., On the theory of plastic deformation and doubling, Zh. eksp. and teor. fiz., 8, (1938), 89-., 1340-., 1349-. (in Russian) · Zbl 0021.08502
[13] P.-L. Lions, G.-C. Papanicolau, S.-R.-S. Varadhan, Homogenisation of Hamilton-Jacobi equations, unpublished preprint, 1986.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.