On the almost periodic homogenization of non-linear scalar conservation laws

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• 作者：Jean Silva
• 关键词：Primary 35B40 ; 35B35 ; Secondary 35L65 ; 35K55
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：54
• 期：4
• 页码：3623-3641
• 全文大小：534 KB
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• 作者单位：Jean Silva (1)
  
  1. Instituto de Matem谩tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

The paper deals with the homogenization problem of non-linear scalar conservations laws \(\partial _t u_\varepsilon + \nabla _x\cdot (a(\frac{x}{\varepsilon })f(u_\varepsilon ))=0\). The vector field a is assumed to be incompressible and its components are assumed to be almost periodic. We prove that the weak limit of the family \(\{u_{\varepsilon }{\}}_{\varepsilon >0}\) is the mean value of a function, which we call U(z, x, t), that is the entropy solution of a similar conservation law in the macroscopic variables, but with coefficients depending on the microscopic variables and has the property that the function \(z\mapsto \int _{\mathbb {R}_+^{n+1}}U(z,x,t)\varphi (x,t)\,dx\,dt\) is also almost periodic for any \(\varphi \in C_c(\mathbb {R}_+^{n+1})\). Mathematics Subject Classification Primary 35B40 35B35 Secondary 35L65 35K55