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An \(L^{p}\)-Approach to the Well-Posedness of Transport Equations Associated with a Regular Field: Part I

  • L. Arlotti
  • B. LodsEmail author
Article
  • 26 Downloads

Abstract

Transport equations associated with a Lipschitz field \(\mathscr {F}\) on some subspace of \({\mathbb {R}}^N\) endowed with some general measure \(\mu \) are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the \(L^{1}\)-context to \(L^{p}\)-spaces \(1< p <\infty \). This is the first part of a two-part contribution (see in Arlotti and Lods An \(L^{p}\)-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.

Keywords

Transport equation boundary conditions \(C_0\)-semigroups characteristic curves 

Mathematics Subject Classification

47D06 47D05 47N55 35F05 82C40 

Notes

References

  1. 1.
    Arlotti, L.: Explicit transport semigroup associated to abstract boundary conditions. In: Discrete Contin. Dyn. Syst. A, Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference. Suppl. vol. I, pp. 102–111 (2011)Google Scholar
  2. 2.
    Arlotti, L., Lods, B.: An \(L^{p}\)-approach to the well-posedness of transport equations associated to a regular field—part II. Meditter. J. Math. 16, 145 (2019).  https://doi.org/10.1007/s00009-019-1426-7
  3. 3.
    Arlotti, L., Banasiak, J., Lods, B.: A new approach to transport equations associated to a regular field: trace results and well-posedness. Mediterr. J. Math. 6, 367–402 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arlotti, L., Banasiak, J., Lods, B.: On general transport equations with abstract boundary conditions. The case of divergence free force field. Mediterr. J. Math. 8, 1–35 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Banasiak, J., Falkiewicz, A., Namayanja, P.: Semigroup approach to diffusion and transport problems on networks. Semigroup Forum 93, 427–443 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bardos, C.: Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport. Ann. Sci. École Nrm. Sup. 3, 185–233 (1970)CrossRefGoogle Scholar
  7. 7.
    Batkai, A., Kramar Fijavž, M., Rhandi, A.: Positive operator semigroups. In: From Finite to Infinite Dimensions. Operator Theory: Advances and Applications, vol. 257. Birkhauser, Cham (2017)Google Scholar
  8. 8.
    Beals, R., Protopopescu, V.: Abstract time-dependent transport equations. J. Math. Anal. Appl. 121, 370–405 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cessenat, M.: Théorèmes de traces \(L_p\) pour les espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris Ser. I 299, 831–834 (1984)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cessenat, M.: Théorèmes de traces pour les espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris Ser. I 300, 89–92 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dautray, R., Lions, J. L.: Mathematical analysis and numerical methods for science and technology. In: Evolution problems II, vol. 6. Springer, Berlin (2000)Google Scholar
  12. 12.
    Engel, K.-J., Kramar Fijavž, M.: Exact and positive controllability of boundary control systems. Netw. Heterog. Media 12, 319–337 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Halmos, P.R.: Measure Theory, 3rd edn. Van Nostrand, Toronto (1954)Google Scholar
  14. 14.
    Kramar, M., Sikolya, E.: Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249, 139–162 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Università di UdineUdineItaly
  2. 2.Department ESOMASUniversità degli Studi di Torino and Collegio Carlo AlbertoTorinoItaly

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