Skip to main content
SpringerLink
Log in

Homogeneous Sobolev global-in-time maximal regularity and related trace estimates

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper, we prove global-in-time \(\dot{\textrm{H}}^{\alpha ,q}\)-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X, provided \(q\in (1,+\infty )\), \(\alpha \in (-1+1/q,1/q)\). We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space. In order to put our result in perspective, we also provide a short review on \(\textrm{L}^q\)-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation theory by Danchin, Hieber, Mucha and Tolksdorf. This theory will be used to build the appropriate trace space, from which we want to choose the initial data, and the solution of our abstract Cauchy problem to fall in.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

The manuscript has no associated data. No new data were created during the study.

Notes

  1. To see it, one can perform the change of variable \(x=\frac{t-u}{\tau +(t-s)}\), in order to obtain \(\frac{1}{\tau +(t-s)}\int _{0}^{\frac{t-s}{\tau +(t-s)}} \frac{x^{\alpha -1}}{(1-x)^{1+\alpha }}\textrm{d}x\).

  2. Again, no need of completeness here, since the involved limits are already constructed.

References

  1. W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander. Vector-valued Laplace transforms and Cauchy problems, volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2\({}^{\text{nd}}\) edition, 2011.

  2. A. Agresti, N. Lindemulder, and M.C. Veraar. On the trace embedding and its applications to evolution equations. Math. Nachr., 00:1–32, 2023.

    MathSciNet  Google Scholar 

  3. H. Amann. Linear and quasilinear parabolic problems, Vol. I: Abstract linear theory, volume 89 of Monographs in Mathematics. Birkhäuser Verlag Basel, 1995.

  4. H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011.

  5. L. Brandolese and S. Monniaux. Well-posedness for the Boussinesq system in critical spaces via maximal regularity. Ann. Inst. Fourier (Grenoble), 2021.

    Google Scholar 

  6. J. Bourgain. Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat., 21:163–168, 1983.

    Article  MathSciNet  Google Scholar 

  7. D. L. Burkholder. A geometric condition that implies the existence of certain singular integrals of banach-space-valued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, (Chicago, Ill. , 1981), volume I, II of Wadsworth Math. Ser., pages 270–286, Wadsworth, Belmont, CA, 1983.

  8. T. Coulhon and D. Lamberton. Régularité \({\rm L}^p\) pour les équations d’évolution. Publ. Math. Univ. Paris VII, 1(1):155–165, 1986. Séminaire d’Analyse Fonctionnelle 1984/1985.

  9. R. Danchin, M. Hieber, P. B. Mucha, and P. Tolksdorf. Free Boundary Problems via Da Prato-Grisvard Theory. arXiv e-prints, November 2021. arXiv:2011.07918v2.

  10. R. Denk, M. Hieber, and J. Prüss. R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 166(788):xvii+114, 2003.

  11. R. Danchin and P. B. Mucha. A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space. J. Funct. Anal., 256(3):881–927, 2009.

    Article  MathSciNet  Google Scholar 

  12. R. Danchin and P. B. Mucha. Critical functional framework and maximal regularity in action on systems of incompressible flows. Mémoires de la Société Mathématique de France, 143, 2015.

  13. G. Da Prato and P. Grisvard. Sommes d’opérateurs linéaires et équations différentielles opérationnelles. J. Math. Pures Appl. (9), 54(3):305–387, 1975.

  14. L. de Simon. Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rendiconti del Seminario Matematico della Università di Padova, 34:205–223, 1964.

    MathSciNet  Google Scholar 

  15. G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z., 196(2):189–201, 1987.

    Article  MathSciNet  Google Scholar 

  16. A. Gaudin. On homogeneous Sobolev and Besov spaces on the whole and the half space. arXiv e-prints, page arXiv:2211.07707v2, November 2022.

  17. S. Guerre-Delabrière. \(L_ p\)-regularity of the Cauchy problem and the geometry of Banach spaces. Ill. J. Math., 39(4):556–566, 1995.

    MathSciNet  Google Scholar 

  18. M. Haase. The Functional Calculus for Sectorial Operators, volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006.

  19. B. H. Haak, M. Haase, and P. C. Kunstmann. Perturbation, interpolation, and maximal regularity. Adv. Diff. Eq., 11(2):201–240, 2006.

    MathSciNet  Google Scholar 

  20. T. Hytönen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach Spaces : Volume I, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer, Cham, 2016.

  21. N. J. Kalton and G. Lancien. A solution to the problem of \(L^p\)-maximal regularity. Math. Z., 235:559–568, 2000.

    Article  MathSciNet  Google Scholar 

  22. M. Khöne, J. Prüss, and M. Wilke. On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces. J. Evol. Equ., 10:443–463, 2010.

    Article  MathSciNet  Google Scholar 

  23. P. C. Kunstmann and L. Weis. Maximal \({L}^p\)-regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^{\infty }\)-functional Calculus, volume 1855 of Lecture Notes in Mathematics, pages 65–311. Springer Berlin Heidelberg, 2004.

  24. N. Lindemulder, M. Meyries, and M. Veraar. Complex interpolation with Dirichlet boundary conditions on the half line. Math. Nachr., 291(16):2435–2456, 2018.

    Article  MathSciNet  Google Scholar 

  25. J. LeCrone, J. Prüss, and M. Wilke. On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces II. J. Evol. Equ., 14:509–533, 2014.

    Article  MathSciNet  Google Scholar 

  26. J. Lindenstrauss and L. Tzafriri. Sequence spaces, volume 92 of Ergeb. Math. Grenzgeb. Springer-Verlag, Berlin, 1977.

    Google Scholar 

  27. A. Lunardi. Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 3\({}^{\text{ rd }}\) edition, 2018.

  28. M. Meyries and M.C. Veraar. Sharp embedding results for spaces of smooth functions with power weights. Studia Math., 208(3):257–293, 2012.

    Article  MathSciNet  Google Scholar 

  29. M. Meyries and M.C. Veraar. Traces and embeddings of anisotropic function spaces. Math. Ann., 360:571–606, 2014.

    Article  MathSciNet  Google Scholar 

  30. J. Prüss. Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in \({L}_p\)-spaces. Mathematica Bohemica, 127:311–327, 01 2002.

  31. J. Prüss and G. Simonett. Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics. Birkhäuser Cham, 2016.

    Book  Google Scholar 

  32. M.-H. Ri and R. Farwig. Maximal \(L^1\)-regularity of generators for bounded analytic semigroups in Banach spaces. Acta Math. Sci., Ser. B, Engl. Ed., 42(4):1261–1272, 2022.

  33. Y. Sawano. Theory of Besov spaces, volume 56 of Developments in Mathematics. Springer, Singapore, 2018.

  34. B. Scharf, H.-J. Schmeißer, and W. Sickel. Traces of vector-valued Sobolev spaces. Math. Nachr., 285(8-9):1082–1106, 2012.

    Article  MathSciNet  Google Scholar 

  35. L. Weis. Operator-Valued Multiplier Theorems and Maximal \(L^p\)-Regularity. Math. Ann., 319(4):735–758, 2001.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the ANR Project RAGE ANR-18-CE40-0012. The author expresses here his most sincere thanks to the anonymous referee, for the careful reading and good remarks that greatly helped to improve the overall quality of the manuscript. The author would like to thank Sylvie Monniaux and Pascal Auscher for their useful remarks during earlier presentations of the current work. The author would also like to thank Bernhard H. Haak for pointing out the article [19].

Author information

Authors and Affiliations

  1. Aix-Marseille Université, CNRS, I2M, Marseille, France

    Anatole Gaudin

Authors
  1. Anatole Gaudin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatole Gaudin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gaudin, A. Homogeneous Sobolev global-in-time maximal regularity and related trace estimates. J. Evol. Equ. 24, 15 (2024). https://doi.org/10.1007/s00028-024-00946-x

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00028-024-00946-x

Keywords

Mathematics Subject Classification

Search

Navigation