Potential Analysis

, Volume 46, Issue 3, pp 527–567 | Cite as

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

  • Chiara Gallarati
  • Mark Veraar
Open Access


In this paper we study maximal L p -regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the L p -boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an L p (L q )-theory for such equations for \(p,q\in (1, \infty )\). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.


Singular integrals Maximal Lp-regularity Evolution equations Functional calculus Elliptic operators Ap-weights \(\mathcal {R}\)-boundedness Extrapolation Quasi-linear PDE 

Mathematics Subject Classification (2010)

Primary: 42B20 42B37; Secondary: 34G10 35B65 42B15 47D06 35K90 34G20 35K55 


  1. 1.
    Abels, H., Terasawa, Y.: On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Acquistapace, P., Terreni, B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78, 47–107 (1987)MathSciNetMATHGoogle Scholar
  3. 3.
    Albrecht, D., Duong, X.T., McIntosh, A.: Operator theory and harmonic analysis. In: Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Volume 34 of Proc. Centre Math. Appl. Austral. Nat. Univ., pp. 77–136. Canberra, Austral. Nat. Univ. (1996)Google Scholar
  4. 4.
    Amann, H.: Linear and Quasilinear Parabolic Problems. Vol. I, Abstract Linear Theory, Volume 89 of Monographs in Mathematics. Birkhäuser Boston Inc., Boston (1995)CrossRefGoogle Scholar
  5. 5.
    Amann, H.: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4(4), 417–430 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Amann, H.: Maximal regularity and quasilinear parabolic boundary value problems. In: Recent Advances in Elliptic and Parabolic Problems, pp. 1–17. World Sci. Publ., Hackensack (2005)Google Scholar
  7. 7.
    Arendt, W., Chill, R., Fornaro, S., Poupaud, C.: L p-maximal regularity for non-autonomous evolution equations. J. Differ. Equ. 237(1), 1–26 (2007)Google Scholar
  8. 8.
    Arendt, W., Dier, D., Laasri, H., Ouhabaz, E. M.: Maximal regularity for evolution equations governed by non-autonomous forms. Adv. Differential Equations 19(11-12), 1043–1066 (2014)MathSciNetMATHGoogle Scholar
  9. 9.
    Auscher, P., McIntosh, A., Nahmod, A.: Holomorphic functional calculi of operators, quadratic estimates and interpolation. Indiana Univ. Math. J. 46(2), 375–403 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bastero, J., Milman, M., Ruiz, F.J.: On the connection between weighted norm inequalities, commutators and real interpolation. Mem. Amer. Math. Soc. 154 (731), viii+80 (2001)MathSciNetMATHGoogle Scholar
  11. 11.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976). Grundlehren der Mathematischen Wissenschaften, No. 223Google Scholar
  12. 12.
    Brudnyı̆, Y.A., Krugljak, N.Y.: Interpolation functors and interpolation spaces. Vol. I, volume 47 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1991). Translated from the Russian by Natalie Wadhwa, With a preface by Jaak PeetreGoogle Scholar
  13. 13.
    Chill, R., Fiorenza, A.: Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces. J. Evol. Equ., 1–34 (2014)Google Scholar
  14. 14.
    Clément, P., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3(Special Issue), 17–32 (Unknown Month 1993)Google Scholar
  15. 15.
    Clément, P., de Pagter, B., Sukochev, F.A., Witvliet, H.: Schauder decompositions and multiplier theorems. Studia Math. 138(2), 135–163 (2000)MathSciNetMATHGoogle Scholar
  16. 16.
    Clément, P., Prüss, J.: Global existence for a semilinear parabolic Volterra equation. Math. Z. 209(1), 17–26 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \(h^{\infty }\) functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996)Google Scholar
  18. 18.
    Cruz-Uribe, D. V., Martell, J. M., Pérez, C.: Weights, Extrapolation and the Theory of Rubio De Francia, Volume 215 of Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel (2011)MATHGoogle Scholar
  19. 19.
    David, G., Journé, J.-L.: A boundedness criterion for generalized calderón-Zygmund operators. Ann. of Math. (2) 120(2), 371–397 (1984)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    David, G., Journé, J.-L., Semmes, S.: Opérateurs de calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoamericana 1(4), 1–56 (1985)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Denk, R., Geissert, M., Hieber, M., Saal, J., Sawada, O.: The spin-coating process: analysis of the free boundary value problem. Comm. Partial Differential Equations 36(7), 1145–1192 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Denk, R., Hieber, M., PrĂĽss, J.: R-boundedness, fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788) (2003)Google Scholar
  23. 23.
    Dier, D.: Non-autonomous maximal regularity for forms of bounded variation. J. Math. Anal. Appl. 425(1), 33–54 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Dier, D., Zacher, R.: Non-autonomous maximal regularity in Hilbert spaces. Online first in J. Evol Equ. (2016)Google Scholar
  25. 25.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators, Volume 43 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  26. 26.
    Dong, H., Kim, D.: On the L p-solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Ration. Mech. Anal. 199(3), 889–941 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Dore, G.: Maximal regularity in L p spaces for an abstract Cauchy problem. Adv. Differential Equations 5(1–3), 293–322 (2000)MathSciNetMATHGoogle Scholar
  28. 28.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Volume 194 of Graduate Texts in Mathematics. Springer, New York (2000)Google Scholar
  29. 29.
    Fackler, S.: The Kalton-Lancien theorem revisited: maximal regularity does not extrapolate. J. Funct. Anal. 266(1), 121–138 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Fackler, S.: J.-L. Lions’ problem concerning maximal regularity of equations governed by non-autonomous forms. To appear in Ann. Inst. H. Poincaré Anal Non linéaire (2016)Google Scholar
  31. 31.
    Fröhlich, A.: The Stokes operator in weighted L q-spaces. II. Weighted resolvent estimates and maximal L p-regularity. Math. Ann. 339(2), 287–316 (2007)Google Scholar
  32. 32.
    Gallarati, C., Lorist, E., Veraar, M.C.: On the â„“ s-boundedness of a family of integral operators. to appear in Revista MatemĂ tica Iberoamericana. arXiv:1410.6657 (2015)
  33. 33.
    Gallarati, C., Veraar, M.C.: Evolution families and maximal regularity for systems of parabolic equations. to appear in Advances in Differential Equations. See arxiv preprint server, arXiv:1510.07643
  34. 34.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1985). Notas de Matemática [Mathematical Notes], 104MATHGoogle Scholar
  35. 35.
    Geissert, M., Hess, M., Hieber, M., Schwarz, C., Stavrakidis, K.: Maximal L p- L q-estimates for the Stokes equation: a short proof of Solonnikov’s theorem. J. Math. Fluid Mech. 12(1), 47–60 (2010)Google Scholar
  36. 36.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionMATHGoogle Scholar
  37. 37.
    Grafakos, L.: Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2008)Google Scholar
  38. 38.
    Grafakos, L.: Modern Fourier Analysis, Volume 250 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2009)Google Scholar
  39. 39.
    Grisvard, P.: Espaces intermédiaires entre espaces de Sobolev avec poids. Ann. Scuola Norm. Sup. Pisa (3) 17, 255–296 (1963)MathSciNetMATHGoogle Scholar
  40. 40.
    Haak, B. H., Haase, M.: Square function estimates and functional calculi. arXiv:1311.0453 (2013)
  41. 41.
    Haak, B.H., Ouhabaz, E.M.: Maximal regularity for non-autonomous evolution equations. Math. Ann. 363(3–4), 1117–1145 (2015)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Haase, M.H.A.: The Functional Calculus for Sectorial Operators, Volume 169 of Operator Theory: Advances and Applications. Basel, Birkhäuser (2006)CrossRefGoogle Scholar
  43. 43.
    Haller, R., Heck, H., Hieber, M.: Muckenhoupt weights and maximal L p-regularity. Arch. Math. (Basel) 81(4), 422–430 (2003)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Haller-Dintelmann, R., Rehberg, J.: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equ. 247(5), 1354–1396 (2009)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Hänninen, T.S., Hytönen, T.P.: The A 2 theorem and the local oscillation decomposition for Banach space valued functions. J. Operator Theory 72(1), 193–218 (2014)Google Scholar
  46. 46.
    Hardy, G.H., Littlewood, J.E., PĂłlya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 editionMATHGoogle Scholar
  47. 47.
    Hytönen, T.P.: An operator-valued T b theorem. J. Funct. Anal. 234(2), 420–463 (2006)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Hytönen, T.P.: The vector-valued nonhomogeneous Tb theorem. Int. Math. Res. Not. IMRN 2, 451–511 (2014)MathSciNetMATHGoogle Scholar
  49. 49.
    Hytönen, T.P., Weis, L.W.: A T1 theorem for integral transformations with operator-valued kernel. J. Reine Angew. Math. 599, 155–200 (2006)MathSciNetMATHGoogle Scholar
  50. 50.
    Hytönen, T.P., Weis, L.W.: Singular convolution integrals with operator-valued kernel. Math. Z. 255(2), 393–425 (2007)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Kaljabin, G.A.: Generalized method of traces in the theory of the interpolation of Banach spaces. Mat. Sb. (N.S.), 106(148)(1), 85–93, 144 (1978)Google Scholar
  52. 52.
    Kalton, N.J., Kucherenko, T.: Operators with an absolute functional calculus. Math. Ann. 346(2), 259–306 (2010)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Kalton, N.J., Lancien, G.: A solution to the problem of L p-maximal regularity. Math. Z. 235(3), 559–568 (2000)Google Scholar
  54. 54.
    Kalton, N.J., Lancien, G.: L p-maximal regularity on Banach spaces with a Schauder basis. Arch. Math. (Basel) 78(5), 397–408 (2002)Google Scholar
  55. 55.
    Kalton, N.J., Weis, L., The \(h^{\infty }\)-calculus and square function estimates (2014). arXiv:1411.0472
  56. 56.
    Kalton, N. J., Weis, L. W.: The \(h^{\infty }\)-calculus and sums of closed operators. Math. Ann. 321(2), 319–345 (2001)Google Scholar
  57. 57.
    Kim, D.: Elliptic and parabolic equations with measurable coefficients in L p-spaces with mixed norms. Methods Appl. Anal. 15(4), 437–467 (2008)Google Scholar
  58. 58.
    Kim, D.: Parabolic equations with partially BMO coefficients and boundary value problems in Sobolev spaces with mixed norms. Potential Anal. 33(1), 17–46 (2010)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Köhne, M., Prüss, J., Wilke, M.: On quasilinear parabolic evolution equations in weighted L p-spaces. J. Evol. Equ. 10(2), 443–463 (2010)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Krylov, N.V.: The heat equation in L q((0,T),L p)-spaces with weights. SIAM J. Math. Anal. 32(5), 1117–1141 (2001)Google Scholar
  61. 61.
    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Volume 96 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  62. 62.
    Kunstmann, P.C., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and \(h^{\infty }\)-functional calculus. In: Functional Analytic Methods for Evolution Equations, Volume 1855 of Lecture Notes in Math, pp. 65–311. Springer, Berlin (2004)Google Scholar
  63. 63.
    Latushkin, Y., Prüss, J., Schnaubelt, R.: Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 9(3-4), 595–633 (2008)MathSciNetMATHGoogle Scholar
  64. 64.
    Le Merdy, C.: On square functions associated to sectorial operators. Bull. Soc. Math. France 132(1), 137–156 (2004)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Lions, J.-L.: Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren Der Mathematischen Wissenschaften Bd. 111. Springer, Berlin (1961)Google Scholar
  66. 66.
    Lions, J.-L.: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Avant Propos de P. Lelong, Dunod, Paris (1968)MATHGoogle Scholar
  67. 67.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems Progress in Nonlinear Differential Equations and their Applications, vol. 16. Basel, Birkhäuser (1995)Google Scholar
  68. 68.
    McIntosh, A.: Operators Which have an \(H_{\infty }\) Functional Calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), Volume 14 of Proc. Centre Math. Anal. Austral. Nat. Univ., pp. 210–231. Austral. Nat. Univ., Canberra (1986)Google Scholar
  69. 69.
    Mei, T.: Notes on matrix valued paraproducts. Indiana Univ. Math. J. 55(2), 747–760 (2006)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Meyer, S., Wilke, M.: Optimal regularity and long-time behavior of solutions for the Westervelt equation. Appl. Math. Optim. 64(2), 257–271 (2011)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Meyries, M.: Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions. Nonlinear Anal. 75(5), 2922–2935 (2012)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Meyries, M., Schnaubelt, R.: Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights. J. Funct. Anal. 262(3), 1200–1229 (2012)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Meyries, M., Schnaubelt, R.: Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions. Math. Nachr. 285(8-9), 1032–1051 (2012)MathSciNetCrossRefMATHGoogle Scholar
  74. 74.
    van Neerven, J.M.A.M.: γ-radonifying operators—a survey. In: The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Volume 44 of Proc. Centre Math. Appl. Austral. Nat. Univ., pp 1–61. Austral. Nat. Univ., Canberra (2010)Google Scholar
  75. 75.
    van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Maximal L p-regularity for stochastic evolution equations. SIAM J. Math. Anal. 44(3), 1372–1414 (2012)Google Scholar
  76. 76.
    van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: Stochastic maximal L p-regularity. Ann. Probab. 40(2), 788–812 (2012)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    van Neerven, J.M.A.M., Veraar, M.C., Weis, L.W.: On the R-boundedness of stochastic convolution operators. Positivity 19(2), 355–384 (2015)Google Scholar
  78. 78.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Volume 44 of Applied Mathematical Sciences. Springer, New York (1983)Google Scholar
  79. 79.
    Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)MathSciNetCrossRefMATHGoogle Scholar
  80. 80.
    Portal, P., Štrkalj, Ž.: Pseudodifferential operators on Bochner spaces and an application. Math. Z. 253(4), 805–819 (2006)Google Scholar
  81. 81.
    Prüss, J.: Maximal regularity for evolution equations in L p-spaces. Conf. Semin. Mat. Univ. Bari 285, 1–39 (2003) (2002)Google Scholar
  82. 82.
    Prüss, J., Schnaubelt, R.: Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl. 256(2), 405–430 (2001)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted L p-spaces. Arch. Math. 82(5), 415–431 (2004)Google Scholar
  84. 84.
    Prüss, J., Vergara, V., Zacher, R.: Well-posedness and long-time behaviour for the non-isothermal cahn-Hilliard equation with memory. Discrete Contin. Dyn. Syst. 26(2), 625–647 (2010)MathSciNetMATHGoogle Scholar
  85. 85.
    Saal, J.: Wellposedness of the tornado-hurricane equations. Discrete Contin. Dyn. Syst. 26(2), 649–664 (2010)MathSciNetCrossRefMATHGoogle Scholar
  86. 86.
    Schnaubelt, R.: Asymptotic behaviour of parabolic nonautonomous evolution equations. In: Functional Analytic Methods for Evolution Equations, Volume 1855 of Lecture Notes in Math, pp. 401–472. Springer, Berlin (2004)Google Scholar
  87. 87.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton (1993). With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, IIIGoogle Scholar
  88. 88.
    Tanabe, H.: Equations of Evolution, Volume 6 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1979)Google Scholar
  89. 89.
    Tanabe, H.: Functional Analytic Methods for Partial Differential Equations, Volume 204 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (1997)Google Scholar
  90. 90.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar
  91. 91.
    Weis, L.: A new approach to maximal L p-regularity. In: Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Volume 215 of Lecture Notes in Pure and Appl. Math., pp. 195–214. Dekker, New York (2001)Google Scholar
  92. 92.
    Weis, L.: Operator-valued Fourier multiplier theorems and maximal L p-regularity. Math Ann. 319(4), 735–758 (2001)Google Scholar
  93. 93.
    Weis, L.: The \(h^{\infty }\) holomorphic functional calculus for sectorial operators—a survey. In: Partial Differential Equations and Functional Analysis, Volume 168 of Oper. Theory Adv. Appl., pp. 263–294. Basel, Birkhäuser (2006)Google Scholar
  94. 94.
    Yagi, A.: Abstract quasilinear evolution equations of parabolic type in Banach spaces. Boll. Un. Mat. Ital. B (7) 5(2), 341–368 (1991)MathSciNetMATHGoogle Scholar
  95. 95.
    Yoshikawa, S., Pawlow, I., Zaja̧czkowski, W.M.: Quasi-linear thermoelasticity system arising in shape memory materials. SIAM J. Math. Anal. 38(6), 1733–1759 (2007). (electronic)Google Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

Personalised recommendations