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August  2019, 24(8): 4021-4030. doi: 10.3934/dcdsb.2019048

A remark on global solutions to random 3D vorticity equations for small initial data

Michael Röckner c, , Rongchan Zhu a,c, and  Xiangchan Zhu b,c,d,,

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

* Corresponding author

Received  March 2018 Revised  May 2018 Published  August 2019 Early access  February 2019

Fund Project: Supported in part by NSFC (11671035, 11771037). Financial support by the DFG through the CRC 1283"Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is acknowledged.

In this paper, we prove that the solution constructed in [2] satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in [8]. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see [10]).

Keywords: Stochastic vorticity equations, controlled rough path, small initial data.
Mathematics Subject Classification: 60H15, 82C28.
Citation: Michael Röckner, Rongchan Zhu, Xiangchan Zhu. A remark on global solutions to random 3D vorticity equations for small initial data. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4021-4030. doi: 10.3934/dcdsb.2019048
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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