September  2014, 13(5): 1789-1797. doi: 10.3934/cpaa.2014.13.1789

Some results for pathwise uniqueness in Hilbert spaces

1. 

Scuola Normale Superiore, Piazza dei Cavalieri 6, 56126 Pisa, Italy

2. 

Dipartimento di Matematica Applicata, "U.Dini" Università di Pisa, V.le B. Pisano 26/b, 56126 Pisa

Received  September 2013 Revised  March 2014 Published  June 2014

An abstract evolution equation in Hilbert spaces with Hölder continuous drift is considered. By proceeding as in [3], we transform the equation in another equation with Lipschitz continuous coefficients.Then we prove existence and uniqueness of this equation by a fixed point argument.
Citation: Giuseppe Da Prato, Franco Flandoli. Some results for pathwise uniqueness in Hilbert spaces. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1789-1797. doi: 10.3934/cpaa.2014.13.1789
References:
[1]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups,, Semigroup Forum, 49 (1994), 349. doi: 10.1007/BF02573496. Google Scholar

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces,, London Mathematical Society, 293 (2002). doi: 10.1017/CBO9780511543210. Google Scholar

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications,, \emph{J. Funct. Anal.}, 259 (2010), 243. doi: 10.1016/j.jfa.2009.11.019. Google Scholar

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models,, Lecture Notes in Mathematics, (2015). doi: 10.1007/978-3-642-18231-0. Google Scholar

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces,, \emph{Israel J. Math.}, 55 (1986), 257. doi: 10.1007/BF02765025. Google Scholar

[6]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications,, Dunod, (1968). Google Scholar

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces,, Dissertationes Math. (Rozprawy Mat.), (2004). doi: 10.4064/dm426-0-1. Google Scholar

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions,, \emph{Comm. Mat. Phys.}, 11 (2008), 247. Google Scholar

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar

show all references

References:
[1]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups,, Semigroup Forum, 49 (1994), 349. doi: 10.1007/BF02573496. Google Scholar

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces,, London Mathematical Society, 293 (2002). doi: 10.1017/CBO9780511543210. Google Scholar

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications,, \emph{J. Funct. Anal.}, 259 (2010), 243. doi: 10.1016/j.jfa.2009.11.019. Google Scholar

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models,, Lecture Notes in Mathematics, (2015). doi: 10.1007/978-3-642-18231-0. Google Scholar

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces,, \emph{Israel J. Math.}, 55 (1986), 257. doi: 10.1007/BF02765025. Google Scholar

[6]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications,, Dunod, (1968). Google Scholar

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces,, Dissertationes Math. (Rozprawy Mat.), (2004). doi: 10.4064/dm426-0-1. Google Scholar

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions,, \emph{Comm. Mat. Phys.}, 11 (2008), 247. Google Scholar

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar

[1]

Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79

[2]

Walter Allegretto, Yanping Lin, Shuqing Ma. Hölder continuous solutions of an obstacle thermistor problem. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 983-997. doi: 10.3934/dcdsb.2004.4.983

[3]

Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180

[4]

Lucio Boccardo, Alessio Porretta. Uniqueness for elliptic problems with Hölder--type dependence on the solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1569-1585. doi: 10.3934/cpaa.2013.12.1569

[5]

Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks & Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191

[6]

Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018

[7]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093

[8]

Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations & Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679

[9]

Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79

[10]

Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315

[11]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[12]

Luis Barreira, Claudia Valls. Hölder Grobman-Hartman linearization. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 187-197. doi: 10.3934/dcds.2007.18.187

[13]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157

[14]

Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169

[15]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19

[16]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87

[17]

Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669

[18]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197

[19]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375

[20]

Paul-Eric Chaudru De Raynal. Weak regularization by stochastic drift : Result and counter example. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1269-1291. doi: 10.3934/dcds.2018052

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]