doi: 10.3934/dcdsb.2022087
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Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise

School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001 China

* Corresponding author: Dongmei Xu, xudongmei@126.com

Received  August 2021 Revised  February 2022 Early access May 2022

Both sufficient and necessary criteria for the existence of a bi-parametric attractor which attaches with forward compactness is established. Meanwhile, we prove that, under certain conditions, the components of the random attractor of a non-autonomous dynamical system can converge in time to those of the random attractor of the limiting autonomous dynamical system. As an application of the abstract theory, we show that the non-autonomous stochastic $ g $-Navier-Stokes (g-NS) equation possesses a forward compact random attractor such that it is forward asymptotically autonomous to a random attractor of the autonomous g-NS equation.

Citation: Fuzhi Li, Dongmei Xu. Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022087
References:
[1]

C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous $g$-Navier-Stokes equations, Ann. Polon. Math., 103 (2012), 277-302.  doi: 10.4064/ap103-3-5.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

H.-O. Bae and J. Roh, Existence of solutions of the $g$-Navier-Stokes equations, Taiwanese J. M., 8 (2004), 85-102.  doi: 10.11650/twjm/1500558459.

[4]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[5]

Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Łukaszewicz and J. Realb, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919.  doi: 10.1016/j.jde.2013.07.043.

[6]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.

[7]

T. CaraballoJ. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Disrete Contin. Dyn. Syst., 6 (2000), 875-892.  doi: 10.3934/dcds.2000.6.875.

[8]

T. CaraballoG. Łukaszewiczd and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonau-tonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

[10]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49 American Mathematical Society, Providence, RI, 2002. doi: 10.1051/cocv:2002056.

[11]

I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[14]

H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

[15]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[16]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.

[17]

J. Duan and B. Schmalfuß, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.

[18]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stochastic Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.

[19]

J. Jiang and Y. Hou, The global attractor of $g$-Navier-Stokes equations with linear dampness on $\mathbb R^2$, Appl. Math. Comput., 215 (2009), 1068-1076.  doi: 10.1016/j.amc.2009.06.035.

[20]

J. JiangY. Hou and X. Wang, Pullback attractor of 2D nonautonomous $g$-Navier-Stokes equations with linear dampness, Appl. Math. Mech. (English Ed.), 32 (2011), 151-166.  doi: 10.1007/s10483-011-1402-x.

[21]

J.-J. Jiang and X.-X. Wang, Global attractor of 2D autonomous $g$-Navier-Stokes equations, Appl. Math. Mech. (English Ed.), 34 (2013), 385-394.  doi: 10.1007/s10483-013-1678-7.

[22]

D. Iftimie and G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations, 169 (2001), 281-331.  doi: 10.1006/jdeq.2000.3900.

[23]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[24]

P. E. Kloeden and J. Simen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[25]

P. E. KloedenJ. Simen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[26]

M. KwakH. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier-Stokes equations, J. Math. Anal. Appl., 315 (2006), 436-461.  doi: 10.1016/j.jmaa.2005.04.050.

[27]

J. A. LangaG. Łukaszewiczd and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749.  doi: 10.1016/j.na.2005.12.017.

[28]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[29]

F. LiD. Xu and J. Yu, Regular measurable backward compact random attractor for $g$-Navier-Stokes equation, Commun. Pure Appl. Anal., 19 (2020), 3137-3157.  doi: 10.3934/cpaa.2020136.

[30]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[31]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[32]

Y. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.

[33]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.1090/s0894-0347-1993-1179539-4.

[34]

J. Roh, $g$-Navier-Stokes Equations, Thesis, University of Minnesota, (2001).

[35]

J. Roh, Dynamics of the $g$-Navier-Stokes equations, J. Differential Equations, 211 (2005), 452-484.  doi: 10.1016/j.jde.2004.08.016.

[36]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.

[37]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[38]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[39]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[40]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[41]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.

[42]

R. Wang and Y. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[43]

R. Wang and Y. Li, Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise, J. Appl. Anal. Comput., 10 (2020), 1199-1222.  doi: 10.11948/20180145.

[44]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[45]

J. YinY. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.

[46]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.

[47]

W. Zhao, Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2499-2526.  doi: 10.3934/dcdsb.2018065.

show all references

References:
[1]

C. T. Anh and D. T. Quyet, Long-time behavior for 2D non-autonomous $g$-Navier-Stokes equations, Ann. Polon. Math., 103 (2012), 277-302.  doi: 10.4064/ap103-3-5.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

H.-O. Bae and J. Roh, Existence of solutions of the $g$-Navier-Stokes equations, Taiwanese J. M., 8 (2004), 85-102.  doi: 10.11650/twjm/1500558459.

[4]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[5]

Z. BrzeźniakT. CaraballoJ. A. LangaY. LiG. Łukaszewicz and J. Realb, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differential Equations, 255 (2013), 3897-3919.  doi: 10.1016/j.jde.2013.07.043.

[6]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.

[7]

T. CaraballoJ. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Disrete Contin. Dyn. Syst., 6 (2000), 875-892.  doi: 10.3934/dcds.2000.6.875.

[8]

T. CaraballoG. Łukaszewiczd and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonau-tonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076.  doi: 10.1512/iumj.1993.42.42049.

[10]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49 American Mathematical Society, Providence, RI, 2002. doi: 10.1051/cocv:2002056.

[11]

I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[14]

H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

[15]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[16]

H. CuiY. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.

[17]

J. Duan and B. Schmalfuß, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.

[18]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stochastic Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.

[19]

J. Jiang and Y. Hou, The global attractor of $g$-Navier-Stokes equations with linear dampness on $\mathbb R^2$, Appl. Math. Comput., 215 (2009), 1068-1076.  doi: 10.1016/j.amc.2009.06.035.

[20]

J. JiangY. Hou and X. Wang, Pullback attractor of 2D nonautonomous $g$-Navier-Stokes equations with linear dampness, Appl. Math. Mech. (English Ed.), 32 (2011), 151-166.  doi: 10.1007/s10483-011-1402-x.

[21]

J.-J. Jiang and X.-X. Wang, Global attractor of 2D autonomous $g$-Navier-Stokes equations, Appl. Math. Mech. (English Ed.), 34 (2013), 385-394.  doi: 10.1007/s10483-013-1678-7.

[22]

D. Iftimie and G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations, 169 (2001), 281-331.  doi: 10.1006/jdeq.2000.3900.

[23]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[24]

P. E. Kloeden and J. Simen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[25]

P. E. KloedenJ. Simen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[26]

M. KwakH. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier-Stokes equations, J. Math. Anal. Appl., 315 (2006), 436-461.  doi: 10.1016/j.jmaa.2005.04.050.

[27]

J. A. LangaG. Łukaszewiczd and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749.  doi: 10.1016/j.na.2005.12.017.

[28]

F. LiY. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.

[29]

F. LiD. Xu and J. Yu, Regular measurable backward compact random attractor for $g$-Navier-Stokes equation, Commun. Pure Appl. Anal., 19 (2020), 3137-3157.  doi: 10.3934/cpaa.2020136.

[30]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[31]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[32]

Y. LiR. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.

[33]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568.  doi: 10.1090/s0894-0347-1993-1179539-4.

[34]

J. Roh, $g$-Navier-Stokes Equations, Thesis, University of Minnesota, (2001).

[35]

J. Roh, Dynamics of the $g$-Navier-Stokes equations, J. Differential Equations, 211 (2005), 452-484.  doi: 10.1016/j.jde.2004.08.016.

[36]

X. SongC. Sun and L. Yang, Pullback attractors for 2D Navier-Stokes equations on time-varying domains, Nonlinear Anal. Real World Appl., 45 (2019), 437-460.  doi: 10.1016/j.nonrwa.2018.07.013.

[37]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[38]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[39]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[40]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[41]

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356.

[42]

R. Wang and Y. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[43]

R. Wang and Y. Li, Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise, J. Appl. Anal. Comput., 10 (2020), 1199-1222.  doi: 10.11948/20180145.

[44]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[45]

J. YinY. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.

[46]

W. Zhao, $H^1$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noises, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.

[47]

W. Zhao, Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2499-2526.  doi: 10.3934/dcdsb.2018065.

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