On the homogenization of conservation laws with resonant oscillatory source

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 u^ε(x,t), and the strong convergence of the asymptotic expansion is shown.