Abstract
In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:
Here, \({(A(t))_{t\in [0,T]}}\) are unbounded operators with domains \({(D(A(t)))_{t\in [0,T]}}\) which may be time dependent. We assume that \({(A(t))_{t\in [0,T]}}\) satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and \({u_0\in E}\) is the initial value. W H is a cylindrical Brownian motion on a separable Hilbert space H. We assume that the Banach space E is a UMD space with type 2. Under locally Lipschitz conditions we show that there exists a unique local mild solution of (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapień, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second-order stochastic partial differential equation which has been studied by Sanz-Solé and Vuillermot. This leads to several improvements of their result.
Article PDF
Avoid common mistakes on your manuscript.
References
Acquistapace P.: Evolution operators and strong solutions of abstract linear parabolic equations. Differential Integral Equations 1(4), 433–457 (1988)
Acquistapace P., Terreni B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78, 47–107 (1987)
Acquistapace P., Terreni B.: Regularity properties of the evolution operator for abstract linear parabolic equations. Differential Integral Equations 5(5), 1151–1184 (1992)
D. Albrecht, X. Duong, and A. McIntosh, Operator theory and harmonic analysis, Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., Canberra, 1996, pp. 77–136.
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126.
H. Amann, Linear and quasilinear parabolic problems. Vol. I, Abstract linear theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995.
V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998.
Brzeźniak Z.: Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4(1), 1–45 (1995)
Brzeźniak Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61(3-4), 245–295 (1997)
Brzeźniak Z., Maslowski B., Seidler J.: Stochastic nonlinear beam equations. Probab. Theory Related Fields 132(1), 119–149 (2005)
Brzeźniak Z., van Neerven J.M.A.M.: Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise. J. Math. Kyoto Univ. 43(2), 261–303 (2003)
D. L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 233–269.
Cerrai S.: Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125(2), 271–304 (2003)
Da Prato G., Kwapień S., Zabczyk J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23(1), 1–23 (1987)
Da Prato G., Zabczyk J.: A note on stochastic convolution. Stochastic Anal. Appl. 10(2), 143–153 (1992)
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992.
Dawson D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5, 1–52 (1975)
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)
Dettweiler J., van Neerven J.M.A.M., Weis L.W.: Space-time regularity of solutions of parabolic stochastic evolution equations. Stoch. Anal. Appl. 24, 843–869 (2006)
J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995.
M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006.
N. J. Kalton and L. W. Weis, The H ∞-calculus and square function estimates, Preprint, 2004.
Kato T.: Remarks on pseudo-resolvents and infinitesimal generators of semi-groups. Proc. Japan Acad. 35, 467–468 (1959)
N. V. Krylov, An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 185–242.
N. V. Krylov and B. L. Rozovski, Stochastic evolution equations, Current problems in mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 71–147, 256.
P. C. Kunstmann and L. W. Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311.
Le Merdy C.: The Weiss conjecture for bounded analytic semigroups. J. London Math. Soc. (2) 67(3), 715–738 (2003)
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969.
Lunardi A.: On the evolution operator for abstract parabolic equations. Israel J. Math. 60(3), 281–314 (1987)
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.
Maniar L., Schnaubelt R.: The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions. J. Differential Equations 235(1), 308–339 (2007)
Maniar L., Schnaubelt R.: Robustness of Fredholm properties of parabolic evolution equations under boundary perturbations. J. Lond. Math. Soc. (2) 77(3), 558–580 (2008)
R. Manthey and T. Zausinger, Stochastic evolution equations in \({L^{2\nu}_\rho}\), Stochastics Stochastics Rep. 66 (1999), no. 1-2, 37–85.
A. McIntosh, Operators which have an H ∞ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231.
A. McIntosh and A. Yagi, Operators of type ω without a bounded H ∞ functional calculus, Miniconference on Operators in Analysis (Sydney, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1990, pp. 159–172.
van Neerven J.M.A.M., Veraar M.C., Weis L.: Stochastic integration in UMD Banach spaces. Ann. Probab. 35(4), 1438–1478 (2007)
van Neerven J.M.A.M., Veraar M.C., Weis L.W.: Stochastic evolution equations in UMD Banach spaces. J. Functional Anal. 255, 940–993 (2008)
van Neerven J.M.A.M., Weis L.W.: Stochastic integration of functions with values in a Banach space. Studia Math. 166(2), 131–170 (2005)
G. Nickel, On evolution semigroups and wellposedness of nonautonomous Cauchy problems. Ph.D. thesis, Tübingen: Univ. Tübingen, Math. Fak. 91 S., 1996.
Nickel G.: Evolution semigroups for nonautonomous Cauchy problems. Abstr. Appl. Anal. 2(1-2), 73–95 (1997)
E. Pardoux, Équations aux dérivées partielles stochastiques nonlinéares monotones: étude de solutions fortes de type Itô., Ph.D. thesis, Université Paris-Orsay, 1975.
Pardoux E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(2), 127–167 (1979)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
Pisier G.: Martingales with values in uniformly convex spaces. Israel J. Math. 20(3-4), 326–350 (1975)
B.L. Rozovski, Stochastic evolution systems, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer Academic Publishers Group, Dordrecht, 1990, Linear theory and applications to nonlinear filtering, Translated from the Russian by A. Yarkho.
Sanz-Solé M., Vuillermot P.-A.: Hölder-Sobolev regularity of solutions to a class of SPDE’s driven by a spatially colored noise. C. R. Math. Acad. Sci. Paris 334(10), 869–874 (2002)
Sanz-Solé M., Vuillermot P.-A.: Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist. 39(4), 703–742 (2003)
R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 401–472.
R. Schnaubelt and M.C. Veraar, Stochastic equations with boundary noise, In preparation., 2008.
R. Seeley, Interpolation in L p with boundary conditions, Studia Math. 44 (1972), 47–60, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I.
Seidler J.: Da Prato-Zabczyk’s maximal inequality revisited. I. Math. Bohem. 118(1), 67–106 (1993)
H. Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass., 1979.
H. Tanabe, Functional analytic methods for partial differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 204, Marcel Dekker Inc., New York, 1997.
Triebel H.: Interpolation theory, function spaces, differential operators, second ed. Johann Ambrosius Barth, Heidelberg (1995)
M.C. Veraar, Stochastic Integration in Banach spaces and Applications to Parabolic Evolution Equations, Ph.D. thesis, Delft University of Technology, 2006, http://fa.its.tudelft.nl/~veraar/.
Veraar M.C., Zimmerschied J.: Non-autonomous stochastic Cauchy problems in Banach spaces. Studia Math. 185(1), 1–34 (2008)
Yagi A.: Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. II. Funkcial. Ekvac. 33(1), 139–150 (1990)
Yagi A.: Abstract quasilinear evolution equations of parabolic type in Banach spaces. Boll. Un. Mat. Ital. B (7) 5(2), 341–368 (1991)
K. Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the sixth (1980) edition.
Zhang X.: L p-theory of semi-linear SPDEs on general measure spaces and applications. J. Funct. Anal. 239(1), 44–75 (2006)
Acknowledgements
The author is grateful to Roland Schnaubelt and Lutz Weis for helpful discussions. Moreover, he thanks the anonymous referees for carefully reading the manuscript and for giving many useful comments.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by the Alexander von Humboldt foundation.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Veraar, M.C. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations. J. Evol. Equ. 10, 85–127 (2010). https://doi.org/10.1007/s00028-009-0041-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-009-0041-7