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 and 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), 349-367. doi: 10.1007/BF02573496.

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society, Lecture Notes 293, Cambridge University Press, 2002. doi: 10.1017/CBO9780511543210.

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. doi: 10.1016/j.jfa.2009.11.019.

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, 2015, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266. doi: 10.1007/BF02765025.

[6]

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

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces, Dissertationes Math. (Rozprawy Mat.), no. 426, 2004. doi: 10.4064/dm426-0-1.

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Comm. Mat. Phys., 11 (2008), 247-259.

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.

show all references

References:
[1]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49 (1994), 349-367. doi: 10.1007/BF02573496.

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society, Lecture Notes 293, Cambridge University Press, 2002. doi: 10.1017/CBO9780511543210.

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. doi: 10.1016/j.jfa.2009.11.019.

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, 2015, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266. doi: 10.1007/BF02765025.

[6]

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

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces, Dissertationes Math. (Rozprawy Mat.), no. 426, 2004. doi: 10.4064/dm426-0-1.

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Comm. Mat. Phys., 11 (2008), 247-259.

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.

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