American Institute of Mathematical Sciences

Advanced
July  2006, 6(4): 835-866. doi: 10.3934/dcdsb.2006.6.835

On the stochastic Burgers equation with a polynomial nonlinearity in the real line

Jong Uhn Kim 1,

1. 

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, United States

Received  January 2005 Revised  September 2005 Published  April 2006

We discuss the Cauchy problem for the stochastic Burgers equation with a nonlinear term of polynomial growth in the whole real line. We also establish the existence of an invariant measure when the equation has an additional zero order dissipation. Many authors have discussed similar issues for the stochastic Burgers equation in various different contexts. But our results for the whole real line are new. Also, our method is different from those of the previous works on the stochastic Burgers equation. In particular, our result on the existence of an invariant measure relies on the author's recent work on a certain class of stochastic evolution equations.
Keywords: Burgers equation, invariant measure., Cauchy problem, Brownian motion.
Mathematics Subject Classification: Primary: 35L65, 35R60, 60H1.
Citation: Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835
[1]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[2]

Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245
[3]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[4]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[5]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
[6]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012
[7]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149
[8]

Bernadette N. Hahn, Melina-Loren Kienle Garrido, Christian Klingenberg, Sandra Warnecke. Using the Navier-Cauchy equation for motion estimation in dynamic imaging. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022018
[9]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[10]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085
[11]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643
[12]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[13]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[14]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[15]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131
[16]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[17]

Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084
[18]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039
[19]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic and Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001
[20]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy