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November  2019, 18(6): 3121-3135. doi: 10.3934/cpaa.2019140

Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

School of Mathematical Science, and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China

* Corresponding author

Received  November 2018 Revised  January 2019 Published  May 2019

This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

Citation: Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
References:
[1]

J. M. Burgers, The Nonlinear Diffusion Equation, 1st edition, Springer Science Business Media, Dordrecht, 1974. Google Scholar

[2]

Z. BrzezniakL. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Glob. Stoch. Anal., 1 (2011), 149-174.   Google Scholar

[3]

L. BertiniN. Cancrini and G. Lasinio, The stochastic Burgers equation, Commun. Math. Phys., 165 (1994), 211-232.   Google Scholar

[4]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225-236.  doi: 10.1090/qam/42889.  Google Scholar

[5] P. Constantin and C. Poias, Navier-Stokes Equations, edition, University of Chicago press, Chicago, 1988.   Google Scholar
[6]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers equation, NODEA-Nonlinear Diff., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[7]

G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, 1st edition, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[9]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.  Google Scholar

[10]

W. N. EJ. C. Mattingly and Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Commun. Math. Phys., 224 (2001), 83-106.  doi: 10.1007/s002201224083.  Google Scholar

[11]

W. N. EK. KhaninA. Mazel and Y. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math., 151 (2000), 877-960.  doi: 10.2307/121126.  Google Scholar

[12]

B. Goldys and B. Maslowskib, Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations, J. Funct. Anal., 226 (2005), 230-255.  doi: 10.1016/j.jfa.2004.12.009.  Google Scholar

[13]

M. Gourcy, Large deviation principle of occupation measure for stochastic Burgers equation, Ann. I. H. Poincaré-PR, 43 (2007), 441-459.  doi: 10.1016/j.anihpb.2006.07.003.  Google Scholar

[14]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.  Google Scholar

[15]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distance and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.  Google Scholar

[16]

H. HajaiejL. MolinetT. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations, in Harmonic Analysis and Nonlinear Partial Differential Equations, Res. Inst. Math. Sci., 5 (2011), 159-175.   Google Scholar

[17]

E. Hopf, The partial differential equation $ u_t + uu_x = \mu u_xx $, Commun. Pure Appl. Math., 3 (1950), 201-230.  doi: 10.1002/cpa.3160030302.  Google Scholar

[18]

M. Kwa$\acute{s}$nicki, Eigenvalues of fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.  doi: 10.1016/j.jfa.2011.12.004.  Google Scholar

[19]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[20]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization analyticity, J. Statist. Phys., 108 (2002), 1157-1179.  doi: 10.1023/A:1019799700126.  Google Scholar

[21]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

show all references

References:
[1]

J. M. Burgers, The Nonlinear Diffusion Equation, 1st edition, Springer Science Business Media, Dordrecht, 1974. Google Scholar

[2]

Z. BrzezniakL. Debbi and B. Goldys, Ergodic properties of fractional stochastic Burgers equation, Glob. Stoch. Anal., 1 (2011), 149-174.   Google Scholar

[3]

L. BertiniN. Cancrini and G. Lasinio, The stochastic Burgers equation, Commun. Math. Phys., 165 (1994), 211-232.   Google Scholar

[4]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225-236.  doi: 10.1090/qam/42889.  Google Scholar

[5] P. Constantin and C. Poias, Navier-Stokes Equations, edition, University of Chicago press, Chicago, 1988.   Google Scholar
[6]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers equation, NODEA-Nonlinear Diff., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[7]

G. Da Parto and J. Zabcyzk, Ergodicity for Infinite Dimensional Systems, 1st edition, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.  Google Scholar

[9]

Q. DuM. GunzburgerR. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.  doi: 10.1142/S0218202512500546.  Google Scholar

[10]

W. N. EJ. C. Mattingly and Ya. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Commun. Math. Phys., 224 (2001), 83-106.  doi: 10.1007/s002201224083.  Google Scholar

[11]

W. N. EK. KhaninA. Mazel and Y. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math., 151 (2000), 877-960.  doi: 10.2307/121126.  Google Scholar

[12]

B. Goldys and B. Maslowskib, Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations, J. Funct. Anal., 226 (2005), 230-255.  doi: 10.1016/j.jfa.2004.12.009.  Google Scholar

[13]

M. Gourcy, Large deviation principle of occupation measure for stochastic Burgers equation, Ann. I. H. Poincaré-PR, 43 (2007), 441-459.  doi: 10.1016/j.anihpb.2006.07.003.  Google Scholar

[14]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math., 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.  Google Scholar

[15]

M. Hairer and J. C. Mattingly, Spectral gaps in Wasserstein distance and the 2D stochastic Navier-Stokes equations, Ann. Probab., 36 (2008), 2050-2091.  doi: 10.1214/08-AOP392.  Google Scholar

[16]

H. HajaiejL. MolinetT. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations, in Harmonic Analysis and Nonlinear Partial Differential Equations, Res. Inst. Math. Sci., 5 (2011), 159-175.   Google Scholar

[17]

E. Hopf, The partial differential equation $ u_t + uu_x = \mu u_xx $, Commun. Pure Appl. Math., 3 (1950), 201-230.  doi: 10.1002/cpa.3160030302.  Google Scholar

[18]

M. Kwa$\acute{s}$nicki, Eigenvalues of fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.  doi: 10.1016/j.jfa.2011.12.004.  Google Scholar

[19]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[20]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization analyticity, J. Statist. Phys., 108 (2002), 1157-1179.  doi: 10.1023/A:1019799700126.  Google Scholar

[21]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer-Verlag, Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

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