Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-10T03:22:23.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Well-posedness for nonlinear SPDEs with strongly continuous perturbation

Part of: Stochastic analysis Parabolic equations and systems

Published online by Cambridge University Press:  11 March 2020

Guy Vallet  and
Aleksandra Zimmermann
Guy Vallet
Affiliation:
LMAP UMR CNRS 5142, IPRA BP 1155, 64013 Pau Cedex, France (guy.vallet@univ-pau.fr)
Aleksandra Zimmermann
Affiliation:
Faculty of Mathematics, Thea-Leymann-Str. 9, 45127 Essen, Germany (aleksandra.zimmermann@uni-due.de)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

Keywords

Nonlinear SPDEstrongly continuous perturbationadditive stochastic forcingpseudomonotone SPDE

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35K92: Quasilinear parabolic equations with $p$-Laplacian 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 265 - 295
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

References

1Appell, J. and Váth, M., The space of Carathéodory functions. In Nonlinear analysis and related problems (Russian), vol. 2 of Tr. Inst. Mat. (Minsk), pp. 39–43. Natl. Akad. Nauk Belarusi, Inst. Mat. (Minsk, 1999).Google Scholar
2Billingsley, P.. Convergence of probability measures (New York: Wiley, 1999).CrossRefGoogle Scholar
3Da Prato, G. and Zabczyk, J., Stochastic equations in infinite dimensions. Vol. 152 of Encyclopedia of Mathematics and its Applications, 2nd edn (Cambridge: Cambridge University Press, 2014).CrossRefGoogle Scholar
4Debussche, A., Glatt-Holtz, N. and Temam, R.. Local martingale and pathwise solutions for an abstract fluids model. Physica D 240 (2011), 1123–1144.CrossRefGoogle Scholar
5Edwards, R. E.. Functional analysis (New York: Dover Publications Inc., 1995).Google Scholar
6Flandoli, F. and Gatarek, D.. Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102 (1995), 367–391.CrossRefGoogle Scholar
7Garsia, A., Rodemich, E. and Rumsey, H.. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 (1970/71), 565–578.CrossRefGoogle Scholar
8Gyöngy, I. and Krylov, N.. Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Theory Relat. Fields 105 (1996), 143–158.CrossRefGoogle Scholar
9Kallenberg, O.. Foundations of modern probability (New York: Springer, 2002).CrossRefGoogle Scholar
10Liu, W. and Róckner, M.. Stochastic partial differential equations: an introduction (Cham: Springer, 2015).Google Scholar
11Pardoux, E., Équations aux dérivées partielles stochastiques non linéaires monotones, Ph.D. Thesis, University of Paris Sud, 1975.Google Scholar
12Roubíček, T.. Nonlinear partial differential equations with applications (Basel: Springer, 2013).CrossRefGoogle Scholar
13Simon, J.. Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. 146 (1987), 65–96.CrossRefGoogle Scholar
14Stroock, D. W. and Varadhan, S. R. S.. Multidimensional diffusion processes (Berlin/New York: Springer, 1979).Google Scholar
15Temam, R.. Navier–Stokes equations. Theory and numerical analysis (Amsterdam/New York/Oxford: North-Holland Publishing Co., 1977).Google Scholar
16Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. A.. Probability distributions on Banach spaces (Dordrecht/Boston: D. Reidel Publishing Company, 1987).CrossRefGoogle Scholar
17Vallet, G. and Zimmermann, A.. Well-posedness for a pseudomonotone evolution problem with multiplicative noise. J. Evol. Equ. 19 (2019), 153–202.Google Scholar
18Yosida, K., Functional analysis. Classics in Mathematics, vol. 123 (Berlin/Heidelberg: Springer, 1995).CrossRefGoogle Scholar