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

Yang Liu; Chunyou Sun

doi:10.3934/dcdss.2021124

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2021, Volume 14, Issue 12: 4383-4408. Doi: 10.3934/dcdss.2021124

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

Yang Liu and
Chunyou Sun^,

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

^* Corresponding author: Chunyou Sun
^* Corresponding author: Chunyou Sun

Received: August 2021

Revised: September 2021

Early access: October 2021

Published: December 2021

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Abstract

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 $.

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Mathematics Subject Classification: Primary: 35K05, 35Q30, 76D05.

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[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. Chepyzhov, M. 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. Constantin, A. 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. Guo, D. Huang, Q. 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. Yang, L. Li, X. 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.