On the homogenization of conservation laws with resonant oscillatory source

Article type: Research Article

Authors: Amadori, Debora

Affiliations: Dip. di Matematica Pura e Applicata, Università degli Studi dell'Aquila, Via Vetoio, loc. Coppito, 67100 L'Aquila, Italy E-mail: amadori@univaq.it

Abstract: We consider the scalar conservation law with oscillatory, periodic source term \[u^{\varepsilon}_{t}+f(u^{\varepsilon})_{x}=\dfrac{1}{\varepsilon}V'\big(\dfrac{x}{\varepsilon}\big),\quad x\in \mathbb{R},\ t>0,\ \varepsilon>0,\] and with initial data \[u^{\varepsilon}(x,0)=u_{o}\big(x,\dfrac{x}{\varepsilon}\big).\] For possibly resonant initial data, we prove a corrector-type result for this problem, extending a previous one by E and Serre [Asymptotic Anal. 5 (1992), 311–316]: an asymptotic representation \[$U(x,t,\tfrac{x}{\varepsilon})$ is identified for the sequence uε(x,t), and the strong convergence of the asymptotic expansion is shown.

Keywords: conservation laws, periodic source term, oscillations, homogenization

Journal: Asymptotic Analysis, vol. 46, no. 1, pp. 53-79, 2006