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Trajectory and global attractors for generalized processes
A remark on global solutions to random 3D vorticity equations for small initial data
a. | Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China |
b. | School of Science, Beijing Jiaotong University, Beijing 100044, China |
c. | Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany |
d. | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
In this paper, we prove that the solution constructed in [
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
V. Barbu and M. Röckner,
Global solutions to random 3D vorticity equations for small initial data, Journal of Differential Equations, 263 (2017), 5395-5411.
doi: 10.1016/j.jde.2017.06.020. |
[3] |
R. Buckdahn,
Linear Skorohod stochastic differential equations, Probab. Th. Rel. Fields, 90 (1991), 223-240.
doi: 10.1007/BF01192163. |
[4] |
J.-Y. Chemin, Perfect Infompressible Fluids, Oxford Lecture Series in Mathematics and its applications, 14, Oxford Science Publications, 1998. |
[5] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 45 of Encyclopedia of mathematics and its applications, Cambridge University Press, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[6] |
J. Diehl, P. Friz and W. Stannat.,
Stochastic partial differential equations: A rough path view on weak solutions via Feynman-Kac, Ann. Fac. Sci. Toulouse Math. (6), 26 (2017), 911-947.
doi: 10.5802/afst.1556. |
[7] |
P. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014.
doi: 10.1007/978-3-319-08332-2. |
[8] |
M. Gubinelli,
Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.
doi: 10.1016/j.jfa.2004.01.002. |
[9] |
M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), e6, 75 pp.
doi: 10.1017/fmp.2015.2. |
[10] |
T. Kato and H. Fujita,
On the nonstationary Navier-Stokes system, Rend. Sem. mat. Univ. Padova, 32 (1962), 243-260.
|
[11] |
J. Mourrat and H. Weber,
Global well-posedness of the dynamic $ \Phi^4 $ model in the plane, The Annals of Probability, 45 (2017), 2398-2476.
doi: 10.1214/16-AOP1116. |
[12] |
D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications (New York), Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-1-4757-2437-0. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
V. Barbu and M. Röckner,
Global solutions to random 3D vorticity equations for small initial data, Journal of Differential Equations, 263 (2017), 5395-5411.
doi: 10.1016/j.jde.2017.06.020. |
[3] |
R. Buckdahn,
Linear Skorohod stochastic differential equations, Probab. Th. Rel. Fields, 90 (1991), 223-240.
doi: 10.1007/BF01192163. |
[4] |
J.-Y. Chemin, Perfect Infompressible Fluids, Oxford Lecture Series in Mathematics and its applications, 14, Oxford Science Publications, 1998. |
[5] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 45 of Encyclopedia of mathematics and its applications, Cambridge University Press, 1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[6] |
J. Diehl, P. Friz and W. Stannat.,
Stochastic partial differential equations: A rough path view on weak solutions via Feynman-Kac, Ann. Fac. Sci. Toulouse Math. (6), 26 (2017), 911-947.
doi: 10.5802/afst.1556. |
[7] |
P. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014.
doi: 10.1007/978-3-319-08332-2. |
[8] |
M. Gubinelli,
Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.
doi: 10.1016/j.jfa.2004.01.002. |
[9] |
M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), e6, 75 pp.
doi: 10.1017/fmp.2015.2. |
[10] |
T. Kato and H. Fujita,
On the nonstationary Navier-Stokes system, Rend. Sem. mat. Univ. Padova, 32 (1962), 243-260.
|
[11] |
J. Mourrat and H. Weber,
Global well-posedness of the dynamic $ \Phi^4 $ model in the plane, The Annals of Probability, 45 (2017), 2398-2476.
doi: 10.1214/16-AOP1116. |
[12] |
D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications (New York), Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-1-4757-2437-0. |
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