Advertisement

An Application of the S-Functional Calculus to Fractional Diffusion Processes

  • Fabrizio Colombo
  • Jonathan Gantner
Article

Abstract

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the \({H^\infty}\) functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form
$$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$
where \({e_{\ell}, {\ell} = 1, 2, 3}\) are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables \({x = (x_{1}, x_{2}, x_{3})}\) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version \({T^{\alpha}, {\rm for} \alpha \in (0, 1)}\), of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

Mathematics Subject Classification (2010)

Primary 47A10 47A60 

Keywords

H functional calculus for quaternionic operators fractional powers of vector operators S-spectrum fractional diffusion and fractional evolution processes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Adler, Quaternionic Quantum Mechanics and Quaternionic Quantum Fields, Volume 88 of International Series of Monographs on Physics. Oxford University Press, New York. 1995.Google Scholar
  2. 2.
    Alpay D., Colombo F., Gantner J., Kimsey D.P.: Functions of the infinitesimal generator of a strongly continuous quaternionic group . Anal. Appl. (Singap.) 15, 279–311 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Alpay, F. Colombo, J. Gantner, I. Sabadini, A new resolvent equation for the Sfunctional calculus, J. Geom. Anal. 25 no. 3 (2015), 1939–1968.Google Scholar
  4. 4.
    Alpay D., Colombo F., Kimsey D.P.: The spectral theorem for for quaternionic unbounded normal operators based on the S-spectrum. J. Math. Phys. 57, 023503 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Alpay D., Colombo F., Kimsey D.P., Sabadini I.: The spectral theorem for unitary operators based on the S-spectrum. Milan J. Math. 84, 41–61 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Alpay, F. Colombo, I. Lewkowicz, I Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, Milan J. Math. 83 no. 1 (2015), 91–144.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alpay D., Colombo F., Qian T., Sabadini I.: The H functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators. J. Funct. Anal. 271, 1544–1584 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Alpay, F. Colombo, I. Sabadini, Perturbation of the generator of a quaternionic evolution operator, Anal. Appl. (Singap.) 13 no. 4 (2015), 347–370.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Alpay, F. Colombo, I. Sabadini, Slice Hyperholomorphic Schur Analysis, Volume 256 of Operator Theory: Advances and Applications. Birkhäuser, Basel, 2017.Google Scholar
  10. 10.
    Balakrishnan A.V.: Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10, 419–437 (1960)MathSciNetCrossRefGoogle Scholar
  11. 11.
    G. Birkhoff, J. von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 no. 4 (1936), 823–843.MathSciNetCrossRefGoogle Scholar
  12. 12.
    C. Bucur, E. Valdinoci, Nonlocal diffusion and applications, Volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, Cham; Unione Matematica Italiana, Bologna, 2016.Google Scholar
  13. 13.
    Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32, 1245–1260 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Caffarelli L., Soria F., Vazquez J.L.: Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. (JEMS) 15, 1701–1746 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caffarelli L., Vazquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Colombo, J. Gantner, Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity. Trans. Amer. Math. Soc. 370 no. 2 (2018), 1045–1100.MathSciNetCrossRefGoogle Scholar
  17. 17.
    F. Colombo, J. Gantner, Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space, J. Phys. A 51 (2018), 305201 (25pp).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Colombo F., Gantner J.: On power series expansions of the S-resolvent operator and the Taylor formula. J. Geom. Phys. 110, 154–175 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. Colombo, J. Gantner, D.P. Kimsey, Spectral Theory on the S-spectrum for quaternionic operators, Volume 270 of Operator Theory: Advances and Applications, to appear in 2019.Google Scholar
  20. 20.
    F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 no. 3 (2009), 601–627.MathSciNetCrossRefGoogle Scholar
  21. 21.
    F. Colombo, I. Sabadini, On the formulations of the quaternionic functional calculus, J. Geom. Phys. 60 no. 10 (2010), 1490–1508.MathSciNetCrossRefGoogle Scholar
  22. 22.
    F. Colombo, I. Sabadini, The F-spectrum and the SC-functional calculus, Proc. Roy. Soc. Edinburgh Sect. A 142 no. 3 (2012), 479–500.Google Scholar
  23. 23.
    F. Colombo, I. Sabadini, The quaternionic evolution operator, Adv. Math. 227 no. 5 (2011), 1772–1805.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Colombo F., Sabadini I., Struppa D.C.: A new functional calculus for noncommuting operators. J. Funct. Anal. 254, 2255–2274 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Colombo, I. Sabadini, D.C. Struppa, Entire Slice Regular Functions. Volume of SpringerBriefs in Mathematics. Springer International Publishing, 2016.Google Scholar
  26. 26.
    F. Colombo, I. Sabadini, D.C. Struppa, Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions, Volume 289 of Progress in Mathematics, Birkhäuser, Basel, 2011.CrossRefGoogle Scholar
  27. 27.
    K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics, Springer, New York. 2000.Google Scholar
  28. 28.
    Farenick D.R., Pidkowich B.A.F.: The spectral theorem in quaternions. Linear Algebra Appl. 371, 75–102 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    S.G. Gal, I. Sabadini, Quaternionic Approximation with application to slice regular functions, Volume of Frontiers in Mathematics, Birkäuser, to appear in 2019.Google Scholar
  30. 30.
    J. Gantner, A direct approach to the S-functional calculus for closed operators, J. Operator Theory 77 no. 2 (2017), 101–145.MathSciNetCrossRefGoogle Scholar
  31. 31.
    J. Gantner, On the equivalence of complex and quaternionic quantum mechanics, Quantum Stud. Math. Found. 5 no. 2 (2018), 357–390.MathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Gantner, Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic SFunctional Calculus and Spectral Operators, Mem. Amer. Math. Soc., to appear.Google Scholar
  33. 33.
    Gentili G., Stoppato C., Struppa D.C.: Regular functions of a quaternionic variable, Volume of Springer Monographs in Mathematics. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  34. 34.
    R. Ghiloni, V. Recupero, Semigroups over real alternative *-algebras: generation theorems and spherical sectorial operators, Trans. Amer. Math. Soc. 368 no. 4 (2016), 2645–2678.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Grillo G., Muratori M., Punzo F.: Fractional porous media equations: existence and uniqueness of weak solutions with measure data. Calc. Var. Partial Differential Equations 54, 3303–3335 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Guzman, Growth properties of semigroups generated by fractional powers of certain linear operators, J. Funct. Anal. 23 no. 4 (1976), 331–352.MathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Haase, Spectral mapping theorems for holomorphic functional calculi, J. London Math. Soc. (2), 71 no. 3 (2005), 723–739.Google Scholar
  38. 38.
    Haase M.: The functional calculus for sectorial operators. Volume 169 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2006)CrossRefGoogle Scholar
  39. 39.
    Kato T.: Note on fractional powers of linear operators. Proc. Japan Acad. 36, 94–96 (1960)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Komatsu H.: Fractional powers of operators. Pacific J. Math. 19, 285–346 (1966)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Mazon J., Rossi J., Toledo J.: Fractional p-Laplacian evolution equations. J. Math. Pures Appl. 105, 810–844 (2016)MathSciNetCrossRefGoogle Scholar
  42. 42.
    A. McIntosh, Operators which have an H functional calculus, In: Miniconference on operator theory and partial differential equations (North Ryde, 1986), pp. 210–231. Volume 14 of Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 1986.Google Scholar
  43. 43.
    C.-K. Ng, On quaternionic functional analysis,Math. Proc. Cambridge Philos. Soc. 143 no. 2 (2007), 391–406.Google Scholar
  44. 44.
    J.L. Vazquez, The porous medium equation. Mathematical theory. Volume of Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.Google Scholar
  45. 45.
    Watanabe J.: On some properties of fractional powers of linear operators. Proc. Japan Acad. 37, 273–275 (1961)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Yosida K.: Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them. Proc. Japan Acad. 36, 86–89 (1960)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Politecnico di Milano Dipartimento di MatematicaMilanoItaly

Personalised recommendations