December  2021, 14(12): 4383-4408. doi: 10.3934/dcdss.2021124

Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

* Corresponding author: Chunyou Sun

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

In this paper, for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^{2} $ as the index $ \alpha $ of the general dissipative operator $ (-\Delta)^{\alpha} $ belongs to $ (0,\frac{1}{2}] $, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the $ L^{\infty} $ bounds given in Caffarelli et al. [4], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [12] still holds under a slightly weaker conditions $ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $ and $ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $ with some $ p>2 $.

Citation: Yang Liu, Chunyou Sun. Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4383-4408. doi: 10.3934/dcdss.2021124
References:
[1]

L. C. Berselli and D. Catania, A note on the Euler-Voigt system in a 3D bounded domain: Propagation of singularities and absence of the boundary layer, AIMS Math., 4 (2019), 1-11.  doi: 10.3934/Math.2019.1.1.

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[3]

A. Bronzi and R. Rosa, On the convergence of statistical solutions of the 3D Navier-Stokes- $\alpha$ model as $\alpha$ vanishes, Discrete Contin. Dyn. Syst., 34 (2014), 19-49.  doi: 10.3934/dcds.2014.34.19.

[4]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[5]

V. V. ChepyzhovM. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407.  doi: 10.1016/j.matpur.2011.04.007.

[6]

A. Cheskidov, Global attractors of evolutionary systems, J. Dyn. Differ. Equ., 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.

[7]

A. Cheskidov and M. Dai, The existence of a global attractor for critical quasi-geostrophic equation in $L^{2}$, J. Math. Fluid Mech., 20 (2018), 213-225.  doi: 10.1007/s00021-017-0324-7.

[8]

A. Cheskidov and S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.

[9]

J. W. Cholewa and T. Dlotko, A note on the 3-D Navier-Stokes equations, Topol. Methods Nonlinear Anal., 52 (2018), 195-212.  doi: 10.12775/TMNA.2017.049.

[10]

J. W. Cholewa and T. Dlotko, Fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B., 23 (2018), 2967-2988.  doi: 10.3934/dcdsb.2017149.

[11]

P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^{2}$, Comm. Math. Phys., 275 (2007), 529-551.  doi: 10.1007/s00220-007-0310-7.

[12]

P. ConstantinA. Tarfulea and V. Vicol, Absence of anomalous dissipation of energy in forced two dimensional fluid equations, Arch. Ration. Mech. Anal., 212 (2014), 875-903.  doi: 10.1007/s00205-013-0708-7.

[13]

T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.  doi: 10.1007/s00245-016-9368-y.

[14]

R. Farwig and R. Kanamaru, Optimality of Serrin type extension criteria to the Navier-Stokes equations, Adv. Nonlinear Anal., 10 (2021), 1071-1085.  doi: 10.1515/anona-2020-0130.

[15]

C. Foiaş, Statistical study of Navier-Stokes equations Ⅰ, Rend. Sem. Mat. Univ. Padova., 48 (1972), 219-348. 

[16]

C. Foiaş, Statistical study of Navier-Stokes equations Ⅱ, Rend. Sem. Mat. Univ. Padova., 49 (1973), 9-123. 

[17]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83, Cambridge University Press, Cambridge, 2001. doi: 10.1063/1.1522171.

[18]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.

[19]

L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics, 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[20]

B. GuoD. HuangQ. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^{2}$, Adv. Nonlinear Stud., 16 (2016), 249-272.  doi: 10.1515/ans-2015-5018.

[21]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Doklady AN SSSR., 30 (1941), 299–303. (Reprinted in Proceedings of the Royal Society of London. Series A, 434 (1991), 9–13). doi: 10.1098/rspa.1991.0075.

[22]

P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical Q-spaces, J. Funct. Anal., 259 (2010), 2457-2519.  doi: 10.1016/j.jfa.2010.07.013.

[23]

J.-L. Lions, Quelques Méthods de Résolution des Problémes aux Limites Nonlinéaires, Dunod, Paris, 1969.

[24]

C. Miao and L. Xue, Global wellposedness for a modified critical dissipative quasi-geostrophic equation, J. Differential Equations., 252 (2012), 792-818.  doi: 10.1016/j.jde.2011.08.018.

[25]

M. Paddick, The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 2673-2709.  doi: 10.3934/dcds.2016.36.2673.

[26]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, Ph.D. Thesis, The University of Chicago, 1995. Dynamical Problems in Non-Linear Advective Partial Differential Equations, Ph.D. Thesis, The University of Chicago, 1995.

[27]

O. G. Smolyanov and S. V. Fomin, Measures on linear topological spaces, Russ. Math. Surv., 31 (1976), 3-56.  doi: 10.1070/rm1976v031n04abeh001553.

[28]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521.

[29]

H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428.  doi: 10.1016/j.aml.2011.09.028.

[30]

J. Wu, Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.

[31]

J. Wu, The generalized incompressible Navier Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.  doi: 10.4310/DPDE.2004.v1.n4.a2.

[32]

J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263 (2006), 803-831.  doi: 10.1007/s00220-005-1483-6.

[33]

X.-G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1395-1418.  doi: 10.3934/era.2020074.

[34]

V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1407-1456. 

[35]

T. Zhang and D. Fang, Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech., 14 (2012), 311-324.  doi: 10.1007/s00021-011-0069-7.

show all references

References:
[1]

L. C. Berselli and D. Catania, A note on the Euler-Voigt system in a 3D bounded domain: Propagation of singularities and absence of the boundary layer, AIMS Math., 4 (2019), 1-11.  doi: 10.3934/Math.2019.1.1.

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[3]

A. Bronzi and R. Rosa, On the convergence of statistical solutions of the 3D Navier-Stokes- $\alpha$ model as $\alpha$ vanishes, Discrete Contin. Dyn. Syst., 34 (2014), 19-49.  doi: 10.3934/dcds.2014.34.19.

[4]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[5]

V. V. ChepyzhovM. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407.  doi: 10.1016/j.matpur.2011.04.007.

[6]

A. Cheskidov, Global attractors of evolutionary systems, J. Dyn. Differ. Equ., 21 (2009), 249-268.  doi: 10.1007/s10884-009-9133-x.

[7]

A. Cheskidov and M. Dai, The existence of a global attractor for critical quasi-geostrophic equation in $L^{2}$, J. Math. Fluid Mech., 20 (2018), 213-225.  doi: 10.1007/s00021-017-0324-7.

[8]

A. Cheskidov and S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306.  doi: 10.1016/j.aim.2014.09.005.

[9]

J. W. Cholewa and T. Dlotko, A note on the 3-D Navier-Stokes equations, Topol. Methods Nonlinear Anal., 52 (2018), 195-212.  doi: 10.12775/TMNA.2017.049.

[10]

J. W. Cholewa and T. Dlotko, Fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B., 23 (2018), 2967-2988.  doi: 10.3934/dcdsb.2017149.

[11]

P. Constantin and F. Ramos, Inviscid limit for damped and driven incompressible Navier-Stokes equations in $\mathbb{R}^{2}$, Comm. Math. Phys., 275 (2007), 529-551.  doi: 10.1007/s00220-007-0310-7.

[12]

P. ConstantinA. Tarfulea and V. Vicol, Absence of anomalous dissipation of energy in forced two dimensional fluid equations, Arch. Ration. Mech. Anal., 212 (2014), 875-903.  doi: 10.1007/s00205-013-0708-7.

[13]

T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.  doi: 10.1007/s00245-016-9368-y.

[14]

R. Farwig and R. Kanamaru, Optimality of Serrin type extension criteria to the Navier-Stokes equations, Adv. Nonlinear Anal., 10 (2021), 1071-1085.  doi: 10.1515/anona-2020-0130.

[15]

C. Foiaş, Statistical study of Navier-Stokes equations Ⅰ, Rend. Sem. Mat. Univ. Padova., 48 (1972), 219-348. 

[16]

C. Foiaş, Statistical study of Navier-Stokes equations Ⅱ, Rend. Sem. Mat. Univ. Padova., 49 (1973), 9-123. 

[17]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83, Cambridge University Press, Cambridge, 2001. doi: 10.1063/1.1522171.

[18]

U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.

[19]

L. Grafakos, Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics, 250, Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2.

[20]

B. GuoD. HuangQ. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^{2}$, Adv. Nonlinear Stud., 16 (2016), 249-272.  doi: 10.1515/ans-2015-5018.

[21]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Doklady AN SSSR., 30 (1941), 299–303. (Reprinted in Proceedings of the Royal Society of London. Series A, 434 (1991), 9–13). doi: 10.1098/rspa.1991.0075.

[22]

P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier-Stokes equations in some critical Q-spaces, J. Funct. Anal., 259 (2010), 2457-2519.  doi: 10.1016/j.jfa.2010.07.013.

[23]

J.-L. Lions, Quelques Méthods de Résolution des Problémes aux Limites Nonlinéaires, Dunod, Paris, 1969.

[24]

C. Miao and L. Xue, Global wellposedness for a modified critical dissipative quasi-geostrophic equation, J. Differential Equations., 252 (2012), 792-818.  doi: 10.1016/j.jde.2011.08.018.

[25]

M. Paddick, The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 2673-2709.  doi: 10.3934/dcds.2016.36.2673.

[26]

S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, Ph.D. Thesis, The University of Chicago, 1995. Dynamical Problems in Non-Linear Advective Partial Differential Equations, Ph.D. Thesis, The University of Chicago, 1995.

[27]

O. G. Smolyanov and S. V. Fomin, Measures on linear topological spaces, Russ. Math. Surv., 31 (1976), 3-56.  doi: 10.1070/rm1976v031n04abeh001553.

[28]

X. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst., 23 (2009), 521-540.  doi: 10.3934/dcds.2009.23.521.

[29]

H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428.  doi: 10.1016/j.aml.2011.09.028.

[30]

J. Wu, Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.

[31]

J. Wu, The generalized incompressible Navier Stokes equations in Besov spaces, Dyn. Partial Differ. Equ., 1 (2004), 381-400.  doi: 10.4310/DPDE.2004.v1.n4.a2.

[32]

J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys., 263 (2006), 803-831.  doi: 10.1007/s00220-005-1483-6.

[33]

X.-G. YangL. LiX. Yan and L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1395-1418.  doi: 10.3934/era.2020074.

[34]

V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1407-1456. 

[35]

T. Zhang and D. Fang, Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech., 14 (2012), 311-324.  doi: 10.1007/s00021-011-0069-7.

[1]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[2]

Claude W. Bardos, Trinh T. Nguyen, Toan T. Nguyen, Edriss S. Titi. The inviscid limit for the 2D Navier-Stokes equations in bounded domains. Kinetic and Related Models, 2022, 15 (3) : 317-340. doi: 10.3934/krm.2022004

[3]

Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95

[4]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[5]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545

[6]

Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375

[7]

Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221

[8]

Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic and Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021

[9]

Alexei Ilyin, Kavita Patni, Sergey Zelik. Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2085-2102. doi: 10.3934/dcds.2016.36.2085

[10]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611

[11]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[12]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[13]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[14]

Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178

[15]

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

[16]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[17]

Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315

[18]

Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39

[19]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[20]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (137)
  • HTML views (121)
  • Cited by (0)

Other articles
by authors

[Back to Top]