Advertisement

Potential Analysis

, Volume 50, Issue 2, pp 279–326 | Cite as

Quasilinear Parabolic Stochastic Evolution Equations Via Maximal Lp-Regularity

  • Luca HornungEmail author
Article
  • 91 Downloads

Abstract

We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterized by a blow-up alternative. The key idea is an iterative application of the theory about maximal Lp- regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions and to a generalized Navier-Stokes equation describing non-Newtonian fluids. In the first example, we can even show that the solution exists globally.

Keywords

Quasilinear stochastic equations Stochastic maximal Lp-regularity Stochastic evolution equations in Banach spaces Blow-up alternative Functional calculus Stochastic reaction diffusion equation Non-Newtonian fluids 

Mathematics Subject Classification (2010)

60H15 60H30 35K59 65J08 58D25 60H15 76A05 35Q35 35K57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

I gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Moreover, I thank my advisor Lutz Weis and Roland Schnaubelt for many useful discussions and for pointing out references on the subject. I am also grateful, that Moritz Egert and Robert Haller-Dintelmann showed me the precise dependence of the H-calculus for divergence-form operators with mixed boundary conditions on the coefficients in Lemma 5.4. Last but not least, I want to mention the help of Fabian Hornung and Christine Grathwohl. They read the article carefully and gave many useful comments.

References

  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm Pure Appl. Math. 17, 35–92 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amann, H.: quasilinear parabolic problems. Vol. I, volume 89 of Monographs in Mathematics. Abstract linear theory, Boston (1995)CrossRefGoogle Scholar
  3. 3.
    Amann, H.: Quasilinear parabolic problems via maximal regularity. Adv. Diff. Equat. 10(10), 1081–1110 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Amann, H., Hieber, M., Simonett, G.: Bounded H -calculus for elliptic operators. Differ. Integr. Equat. 7(3-4), 613–653 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Amann, H., Hieber, M., Simonett, G.: Bounded H -calculus for elliptic operators. Differ. Integr. Equat. 7(3-4), 613–653 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Auscher, P., Badr, N., Haller-Dintelmann, R., Rehberg, J.: The square root problem for second-order, divergence form operators with mixed boundary conditions on L p. J. Evol. Equ. 15(1), 165–208 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Barbu, V., Da Prato, G., Röckner, M.: Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37(2), 428–452,03 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids (1977)Google Scholar
  9. 9.
    Bothe, D., Prüss, J.: L P-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brzeźniak, Z.: On stochastic convolution in Banach spaces and applications. Stochast. Stochast. Rep. 61(3-4), 245–295 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brzeźniak, Z., Motyl, E.: Existence of a martingale solution of the stochastic navier–stokes equations in unbounded 2d and 3d domains. J. Diff. Equat. 254(4), 1627–1685 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Brzeźniak, Z., Peszat, S.: Space-time continuous solutions to SPDE’s driven by a homogeneous Wiener process. Stud. Math. 137(3), 261–299 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Brzeźniak, Z., Peszat, S.: Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces. In: Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), volume 28 of CMS Conference Proceedings, pp. 55–64. American Mathematics Society, Providence (2000)Google Scholar
  14. 14.
    Brzeźniak, Z., Peszat, S.: Stochastic two dimensional Euler equations. Ann. Probab. 29(4), 1796–1832 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Brzeźniak, Z.a., Carroll, A.: Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces. In: Séminaire de Probabilités XXXVII, volume 1832 of Lecture Notes in Math., pp. 251–289. Springer, Berlin (2003)Google Scholar
  16. 16.
    Brzeźniak, Z.a., Debbi, L.: On stochastic Burgers equation driven by a fractional Laplacian and space-time white noise. In: Stochastic differential equations: theory and applications, volume 2 of Interdisciplinary Mathematical Sciences, pp. 135–167. World Scientific Publications, Hackensack (2007)Google Scholar
  17. 17.
    Carroll, A.: The stochastic nonlinear heat equation. PhD thesis. University of Hull, Hull (1999)Google Scholar
  18. 18.
    Clément, P., de Pagter, B., Sukochev, F.A., Witvliet, H.: Schauder decomposition and multiplier theorems. Stud. Math. 138(2), 135–163 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Clément, P., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3(Special Issue), 17–32 (1993/94)Google Scholar
  20. 20.
    Debussche, A., de Moor, S., Hofmanová, M.: A regularity result for quasilinear stochastic partial differential equations of parabolic type. SIAM J. Math Anal. 47(2), 1590–1614 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. 44(3), 1916–1955 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R.T. Seeley. Math. Ann. 328(4), 545–583 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Denk, R., Hieber, M., Prüss, J.: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788), viii+ 114 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Disser, K., Kaiser, H.-C., Rehberg, J.: Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems. SIAM J. Math. Anal. 47(3), 1719–1746 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Disser, K., Elst, t., Rehberg, J.: On maximal parabolic regularity for non-autonomous parabolic operators. Preprint, arXiv:1604.05850 (2016)
  26. 26.
    Duong, X.T., McIntosh, A.: Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients. J. Geom Anal. 6(2), 181–205 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Duong, X.T., Robinson, D.W.: Semigroup kernels, Poisson bounds, and holomorphic functional calculus. J. Funct. Anal. 142(1), 89–128 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Duong, X.T., Simonett, G.: H -calculus for elliptic operators with nonsmooth coefficients. Differ. Integr. Equat. 10(2), 201–217 (1997)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Duong, X.T., Yan, L.X.: Bounded holomorphic functional calculus for non-divergence form differential operators. Differ. Integr. Equat. 15(6), 709–730 (2002)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Egert, M.: Lp-estimates for the square root of elliptic systems with mixed boundary conditions. 42, eprint arXiv:1712.09851 (2017)
  31. 31.
    Egert, M., Haller-Dintelmann, R., Tolksdorf, P.: The Kato square root problem for mixed boundary conditions. J. Funct. Anal. 267(5), 1419–1461 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Elschner, J., Rehberg, J., Schmidt, G.: Optimal regularity for elliptic transmission problems including C 1 interfaces. Interfaces Free Bound. 9(2), 233–252 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Elworthy, K.D.: Stochastic differential equations on manifolds, volume 70 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1982)CrossRefGoogle Scholar
  34. 34.
    Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Funct. Anal. 109(2), 257–288 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Giuseppe Da Prato, M.R., Barbu, V.: Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math J. 57, 187–212 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Grafakos, L.: Classical fourier analysis. Graduate texts in mathematics, 3rd edn., vol. 249. Springer, New York (2014)Google Scholar
  37. 37.
    Griepentrog, J.A., Gröger, K., Kaiser, H.-C., Rehberg, J.: Interpolation for function spaces related to mixed boundary value problems. Math Nachr. 241, 110–120 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Haase, M.: The functional calculus for sectorial operators Operator Theory: Advances and Applications, vol. 169. Basel, Birkhäuser (2006)CrossRefGoogle Scholar
  39. 39.
    Haller-Dintelmann, K., Rehberg, J.: Elliptic and parabolic reguarity for second order divergence operators with mixed boundary conditions. Preprint, arXiv:1310.3679 (2013)
  40. 40.
    Hofmanova, M., Zhang, T.: Quasilinear parabolic stochastic partial differential equations: existence, uniqueness Preprint, arXiv:1501.00548 (2015)
  41. 41.
    Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kalton, N., Kunstmann, P., Weis, L.: Perturbation and interpolation theorems for the H -calculus with applications to differential operators. Math. Ann. 336(4), 747–801 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Kalton, N., Kunstmann, P., Weis, L.: Perturbation and interpolation theorems for the H -calculus with applications to differential operators. Math. Ann. 336(4), 747–801 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Kalton, N.J., Weis, L.: The H -calculus and sums of closed operators. Math. Ann. 321(2), 319–345 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Karatzas, I., Shreve, S.E.: Methods of mathematical finance Applications of mathematics, vol. 39. Springer, New York (2010). [nachdr.] editionGoogle Scholar
  46. 46.
    Klenke, A.: Probability theory: a comprehensive course. Universitext, 2nd edn. Springer, London (2014)zbMATHCrossRefGoogle Scholar
  47. 47.
    Krylov, N.V., Rozovskiı̆, B.L.: Stochastic evolution equations. In: Current problems in mathematics, Vol. 14 (Russian), pp. 71–147. 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)Google Scholar
  48. 48.
    Kunstmann, P.C., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus. In: Functional analytic methods for evolution equations, volume 1855 of Lecture Notes in Mathematics, pp. 65–311. Springer, Berlin (2004)Google Scholar
  49. 49.
    Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasi-linear equations of parabolic type. Translations of mathematical monographs, vol. 23. American Mathematical Society, Providence, 5 [pr.] edition (1998)Google Scholar
  50. 50.
    Liu, W., Röckner, M.: SPDE in Hilbert space with locally monotone coefficients. J. Funct Anal. 259(11), 2902–2922 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Lunardi, A.: Abstract quasilinear parabolic equations. Math. Ann. 267(3), 395–415 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    McIntosh, A.: Operators which have an H functional calculus. In: Miniconference on operator theory and partial differential equations (North Ryde, 1986), volume 14 of Proceedings of the Centre Mathematical Analysis and Australian Naturalium Universitatis, pp. 210–231. Australian Naturalium Universitatis, Canberra (1986)Google Scholar
  53. 53.
    Mikulevicius, R., Rozovskii, B.L.: Global L 2-solutions of stochastic Navier-Stokes equations. Ann. Probab. 33(1), 137–176 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Morrey, Jr., C.B.: Multiple integrals in the calculus of variations. Classics in Mathematics. Springer-Verlag, Berlin (2008). Reprint of the 1966 edition [MR0202511]zbMATHCrossRefGoogle Scholar
  55. 55.
    Pisier, G., Letta, G., Pratelli, M.: Probabilistic methods in the geometry of Banach spaces, volume 14 of Proceedings of the Centre Mathematical Analysis and Australian Naturalium Universitatis, pp. 167–241. Springer, Berlin Heidelberg (1986)Google Scholar
  56. 56.
    Prévôt, C., Röckner, M.: A concise course on stochastic partial differential equations. Lecture notes in mathematics, p. 2007. Springer, Berlin Heidelberg (1905). [u.a.]Google Scholar
  57. 57.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Prüss, J., Simonett, G.: Moving interfaces and quasilinear parabolic evolution equations (2016)Google Scholar
  59. 59.
    Seidler, J.: Da Prato-Zabczyk’s maximal inequality revisited. I. Math. Bohem. 118(1), 67–106 (1993)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Showalter, R.E.: Monotone operators in Banach space and nonlinear partial differential equations Mathematical surveys and monographs, vol. 49. American Mathematical Society, Providence (1997)Google Scholar
  61. 61.
    Ṡneı̆berg, I.J.: Spectral properties of linear operators in interpolation families of Banach spaces. Mat. Issled. 9(2(32)), 214–229, 254–255 (1974)MathSciNetGoogle Scholar
  62. 62.
    Triebel, H.: Interpolation theory, function spaces, differential operators. Barth, Heidelberg (1995). 2., rev. and enl. ed editionzbMATHGoogle Scholar
  63. 63.
    van Neerven, J.: γ-radonifying operators-a survey. In: The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, volume 44 of Proceedings of the Centre Mathematical Application and Australian Naturalium Universitatis, pp. 1–61. Australian Naturalium Universitatis, Canberra (2010)Google Scholar
  64. 64.
    van Neerven, J., Veraar, M., Weis, L.: Stochastic evolution equations in UMD Banach spaces. J. Funct Anal. 255(4), 940–993 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    van Neerven, J., Veraar, M., Weis, L.: Maximal L p-regularity for stochastic evolution equations. SIAM J. Math. Anal. 44(3), 1372–1414 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    van Neerven, J., Veraar, M., Weis, L.: Stochastic maximal L p-regularity. Ann. Probab. 40(2), 788–812 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    van Neerven, J., Veraar, M.C., Weis, L.: Stochastic integration in UMD Banach spaces. Ann Probab. 35(4), 1438–1478 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Vignati, A.T., Vignati, M.: Spectral theory and complex interpolation. J. Funct. Anal. 80(2), 383–397 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Yagi, A.: Abstract parabolic evolution equations and their applications. Springer monographs in mathematics. Springer, Berlin (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations