American Institute of Mathematical Sciences

Advanced
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  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 & 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. Google Scholar

[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. Google Scholar

[3]

R. Buckdahn, Linear Skorohod stochastic differential equations, Probab. Th. Rel. Fields, 90 (1991), 223-240. doi: 10.1007/BF01192163. Google Scholar

[4]

J.-Y. Chemin, Perfect Infompressible Fluids, Oxford Lecture Series in Mathematics and its applications, 14, Oxford Science Publications, 1998. Google Scholar

[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. Google Scholar
[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. Google Scholar

[7]

P. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014. doi: 10.1007/978-3-319-08332-2. Google Scholar

[8]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002. Google Scholar

[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. Google Scholar

[10]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. mat. Univ. Padova, 32 (1962), 243-260. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

R. Buckdahn, Linear Skorohod stochastic differential equations, Probab. Th. Rel. Fields, 90 (1991), 223-240. doi: 10.1007/BF01192163. Google Scholar

[4]

J.-Y. Chemin, Perfect Infompressible Fluids, Oxford Lecture Series in Mathematics and its applications, 14, Oxford Science Publications, 1998. Google Scholar

[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. Google Scholar
[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. Google Scholar

[7]

P. Friz and M. Hairer, A Course on Rough Paths, Springer, 2014. doi: 10.1007/978-3-319-08332-2. Google Scholar

[8]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002. Google Scholar

[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. Google Scholar

[10]

T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. mat. Univ. Padova, 32 (1962), 243-260. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[2]

Dung Le. Strong positivity of continuous supersolutions to parabolic equations with rough boundary data. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1521-1530. doi: 10.3934/dcds.2015.35.1521
[3]

Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87
[4]

Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305
[5]

Brian Smith and Gilbert Weinstein. On the connectedness of the space of initial data for the Einstein equations. Electronic Research Announcements, 2000, 6: 52-63.

[6]

Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631
[7]

Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501
[8]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[9]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021
[10]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[11]

Marcello D'Abbicco. Small data solutions for semilinear wave equations with effective damping. Conference Publications, 2013, 2013 (special) : 183-191. doi: 10.3934/proc.2013.2013.183
[12]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
[13]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447
[14]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[15]

Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
[16]

Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759
[17]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[18]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817
[19]

Elena Cordero, Fabio Nicola, Luigi Rodino. Schrödinger equations with rough Hamiltonians. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4805-4821. doi: 10.3934/dcds.2015.35.4805
[20]

Cheng Wang. The primitive equations formulated in mean vorticity. Conference Publications, 2003, 2003 (Special) : 880-887. doi: 10.3934/proc.2003.2003.880

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (35)
  • HTML views (272)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences