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 & Continuous Dynamical Systems - S, 2021, 14 (12) : 4383-4408. doi: 10.3934/dcdss.2021124
References:
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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.  Google Scholar

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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

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