Abstract
We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form \({\mathrm {div}}\gamma (\nabla \cdot )\), where \(\gamma \) is a maximal monotone graph in \(\mathbb {R}^n \times \mathbb {R}^n\) obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that \(\gamma \) is single-valued, a solution with better integrability and regularity properties is constructed. The proof given here, however, is self-contained.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bourbaki, N.: Espaces vectoriels topologiques. Chapitres 1 à 5, new edn. Masson, Paris (1981). MR 633754
Haase, M.: Convexity inequalities for positive operators. Positivity 11(1), 57–68 (2007). MR 2297322 (2008d:39034)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001). MR 1865628 (2002i:90002)
Krylov, N.V., Rozovskiĭ, B.L.: Current problems in mathematics. Stochastic Evolution Equations, vol. 14 (Russian), pp. 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979). MR MR570795 (81m:60116)
Liu, W., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer, Cham (2015). MR 3410409
Marinelli, C., Röckner, M.: On the maximal inequalities of Burkholder, Davis and Gundy. Expo. Math. 34(1), 1–26 (2016). MR 3463679
Marinelli, C., Scarpa, L.: Strong solutions to SPDEs with monotone drift in divergence form. arXiv:1612.08260
Marinelli, C., Scarpa, L.: A note on doubly nonlinear SPDEs with singular drift in divergence form. arXiv:1712.05595
Pardoux, E.: Equations aux derivées partielles stochastiques nonlinéaires monotones, Ph.D. thesis, Université Paris XI (1975)
Scarpa, L.: Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type. J. Differ. Eqn. 263(4), 2113–2156 (2017)
Strauss, W.A.: On continuity of functions with values in various Banach spaces. Pacific J. Math. 19, 543–551 (1966). MR 0205121 (34 #4956)
Acknowledgements
The authors are partially supported by The Royal Society through its International Exchange Scheme. Parts of this chapter were written while the first-named author was visiting the Interdisziplinäres Zentrum für Komplexe Systeme at the University of Bonn, hosted by Prof. S. Albeverio.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Marinelli, C., Scarpa, L. (2018). On the Well-Posedness of SPDEs with Singular Drift in Divergence Form. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-74929-7_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74928-0
Online ISBN: 978-3-319-74929-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)