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Existence Results for Nonlinear Mono-energetic Singular Transport Equations: \(L_1\)-Spaces

  • Khalid Latrach
  • Hssaine Oummi
  • Ahmed ZeghalEmail author
Article
  • 11 Downloads

Abstract

We prove some results regarding the existence of solutions on \(L^1\)-spaces to a nonlinear mono-energetic singular transport equation (i.e., transport equation with unbounded collision frequency and unbounded collision operator) in slab geometry. Our approach consists in rewriting the problem as a fixed point one involving a nonlinear ws-compact operator and we use a recent fixed point theorem for this class of operators (see Theorem 2.10) to derive existence results. We present a detailed analysis for the case where the boundary conditions are modeled by specular reflections, while the problem with periodic boundary conditions is discussed succinctly, because, except some minor modifications, the arguments of proofs are almost the same.

Keywords

Nonlinear transport equation fixed point theorems existence results weak compactness 

Mathematics Subject Classification

35F20 45K05 47H10 

Notes

Acknowledgements

The authors would like to thank the referees for the useful comments. H. Oummi is supported by the National Center for Scientific and Technical Research (Grant No. 18USMS2016).

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Authors and Affiliations

  1. 1.Université Clermont Auvergne CNRS, LMBPClermont-FerrandFrance
  2. 2.Laboratoire de Mathématiques et ApplicationsUniversité Sultan Moulay Slimane Faculté des Sciences et TechniquesBéni-MellalMorocco
  3. 3.Faculté des Sciences et TechniquesUniversité Abdelmalek EssaadiTangerMorocco

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