doi: 10.3934/dcdsb.2018312

Mild solutions to the time fractional Navier-Stokes delay differential inclusions

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

* Corresponding author

Received  March 2018 Revised  July 2018 Published  October 2018

Fund Project: This work was supported by NSF of China (Grants No. 41875084, 11571153), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-ot03 and lzujbky-2018-it58

In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order $\alpha\in(0,1)$. We first prove the local and global existence, decay and regularity properties of mild solutions when the initial data belongs to $C([-h,0];D(A_r^\varepsilon))$. The fractional resolvent operator theory and some techniques of measure of noncompactness are successfully applied to obtain the results.

Citation: Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018312
References:
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R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

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T. Caraballo and X. Y. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101. doi: 10.3934/dcdss.2015.8.1079.

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T. Caraballo and X. Y. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359. doi: 10.4310/DPDE.2014.v11.n4.a3.

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T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.

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T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.

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P. M. Carvalho-Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de São Paulo, São Carlos, 2013.

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P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^{N}$, J. Differential Equations, 259 (2015), 2948-2980. doi: 10.1016/j.jde.2015.04.008.

[9]

Y. K. Chen and C. H. Wei, Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 36 (2016), 5309-5322. doi: 10.3934/dcds.2016033.

[10]

P. Y. Chen and Y. X. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728. doi: 10.1007/s00033-013-0351-z.

[11]

P. Y. ChenX. P. Zhang and Y. X. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803. doi: 10.1016/j.camwa.2017.01.009.

[12]

P. Y. ChenX. P. Zhang and Y. X. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992. doi: 10.3934/cpaa.2018094.

[13]

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.

[14]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69. doi: 10.1007/s00021-015-0234-5.

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M. El-Shahed and A. Salem, On the generalized Navier-Stokes equations, Appl. Math. Comput., 156 (2004), 287-293. doi: 10.1016/j.amc.2003.07.022.

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L. Ferreira and E. Villamizar-Roa, Fractional Navier-Stokes equations and a Hölder-type inequality in a sum of singular spaces, Nonlinear Anal., 74 (2011), 5618-5630. doi: 10.1016/j.na.2011.05.047.

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J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603.

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J. García-LuengoP. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201. doi: 10.3934/dcds.2014.34.181.

[22]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. doi: 10.1515/ans-2013-0205.

[23]

M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057.

[24]

X. L. Guo and Y. Y. Men, On partial regularity of suitable weak solutions to the stationary fractional Navier-Stokes equations in dimension four and five, Acta Math. Sin. (Engl. Ser.), 33 (2017), 1632-1646. doi: 10.1007/s10114-017-7125-z.

[25]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225.

[26]

Q. S. Jiu and Y. Q. Wang, On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 321-343. doi: 10.4310/DPDE.2014.v11.n4.a2.

[27]

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[28]

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[29]

T. D. Ke and D. Lan, Global attractor for a class of functional differential inclusions with Hille-Yosida operators, Nonlinear Anal., 103 (2014), 72-86. doi: 10.1016/j.na.2014.03.006.

[30]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[31]

P. E. KloedenJ. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.

[32]

P. E. KloedenP. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.

[33]

P. E. Kloeden and J. Valero, The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 28 (2010), 161-179. doi: 10.3934/dcds.2010.28.161.

[34]

M. LiC. M. Huang and F. Z. Jiang, Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes, Appl. Anal., 96 (2017), 1269-1284. doi: 10.1080/00036811.2016.1186271.

[35]

X. C. LiX. Y. Yang and Y. H. Zhang, Error estimates of mixed finite element methods for time-fractional Navier-Stokes equations, J. Sci. Comput., 70 (2017), 500-515. doi: 10.1007/s10915-016-0252-3.

[36]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994), 246-251.

[37]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.

[38]

P. Marín-RubioJ. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008.

[39]

S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494. doi: 10.1016/j.amc.2005.11.025.

[40]

C. J. Niche and G. Planas, Existence and decay of solutions in full space to Navier-Stokes equations with delays, Nonlinear Anal., 74 (2011), 244-256. doi: 10.1016/j.na.2010.08.038.

[41]

L. PengY. ZhouB. Ahmad and A. Alsaedi, The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos Solitons Fractals, 102 (2017), 218-228. doi: 10.1016/j.chaos.2017.02.011.

[42]

I. Podlubny, Fractional Difierential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.

[43]

H. Singh, A new stable algorithm for fractional Navier-Stokes equation in polar coordinate, Int. J. Appl. Comput. Math., 3 (2017), 3705-3722. doi: 10.1007/s40819-017-0323-7.

[44]

L. Tang and Y. Yu, Partial Hölder regularity of the steady fractional Navier-Stokes equations Calc. Var. Partial Differential Equations, 55 (2016), Art. 31, 18 pp. doi: 10.1007/s00526-016-0967-x.

[45]

H. Y. XuX. Y. Jiang and B. Yu, Numerical analysis of the space fractional Navier-Stokes equations, Appl. Math. Lett., 69 (2017), 94-100. doi: 10.1016/j.aml.2017.02.006.

[46]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048.

[47]

F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Ration. Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.

[48]

Z. C. Zhai, Some Regularity Estimates for Mild Solutions to Fractional Heat-Type and Navier-Stokes Equations, Thesis (Ph.D.)-Memorial University of Newfoundland (Canada)., 2009.

[49]

Y. ZhouL. PengB. Ahmad and A. Alsaedi, Energy methods for fractional Navier-Stokes equations, Chaos Solitons Fractals, 102 (2017), 78-85. doi: 10.1016/j.chaos.2017.03.053.

[50]

Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl., 73 (2017), 1016-1027. doi: 10.1016/j.camwa.2016.07.007.

[51]

Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl., 73 (2017), 874-891. doi: 10.1016/j.camwa.2016.03.026.

show all references

References:
[1]

R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, Verlag, Basel, 1992. doi: 10.1007/978-3-0348-5727-7.

[2]

B. De AndradeA. N. CarvalhoP. M. Carvalho-Neto and P. Marín-Rubio, Semilinear fractional differential equations: Global solutions, critical nonlinearities and comparison results, Topol. Methods Nonlinear Anal., 45 (2015), 439-467. doi: 10.12775/TMNA.2015.022.

[3]

T. Caraballo and X. Y. Han, A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 1079-1101. doi: 10.3934/dcdss.2015.8.1079.

[4]

T. Caraballo and X. Y. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359. doi: 10.4310/DPDE.2014.v11.n4.a3.

[5]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.

[6]

T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.

[7]

P. M. Carvalho-Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de São Paulo, São Carlos, 2013.

[8]

P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^{N}$, J. Differential Equations, 259 (2015), 2948-2980. doi: 10.1016/j.jde.2015.04.008.

[9]

Y. K. Chen and C. H. Wei, Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 36 (2016), 5309-5322. doi: 10.3934/dcds.2016033.

[10]

P. Y. Chen and Y. X. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728. doi: 10.1007/s00033-013-0351-z.

[11]

P. Y. ChenX. P. Zhang and Y. X. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803. doi: 10.1016/j.camwa.2017.01.009.

[12]

P. Y. ChenX. P. Zhang and Y. X. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992. doi: 10.3934/cpaa.2018094.

[13]

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.

[14]

L. Debbi, Well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on the torus and on bounded domains, J. Math. Fluid Mech., 18 (2016), 25-69. doi: 10.1007/s00021-015-0234-5.

[15]

K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.

[16]

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

[17]

M. El-Shahed and A. Salem, On the generalized Navier-Stokes equations, Appl. Math. Comput., 156 (2004), 287-293. doi: 10.1016/j.amc.2003.07.022.

[18]

L. Ferreira and E. Villamizar-Roa, Fractional Navier-Stokes equations and a Hölder-type inequality in a sum of singular spaces, Nonlinear Anal., 74 (2011), 5618-5630. doi: 10.1016/j.na.2011.05.047.

[19]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.

[20]

J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621. doi: 10.3934/cpaa.2015.14.1603.

[21]

J. García-LuengoP. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201. doi: 10.3934/dcds.2014.34.181.

[22]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. doi: 10.1515/ans-2013-0205.

[23]

M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057.

[24]

X. L. Guo and Y. Y. Men, On partial regularity of suitable weak solutions to the stationary fractional Navier-Stokes equations in dimension four and five, Acta Math. Sin. (Engl. Ser.), 33 (2017), 1632-1646. doi: 10.1007/s10114-017-7125-z.

[25]

S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225.

[26]

Q. S. Jiu and Y. Q. Wang, On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 321-343. doi: 10.4310/DPDE.2014.v11.n4.a2.

[27]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, 2001. doi: 10.1515/9783110870893.

[28]

T. Kato, Strong $L_{p}$-solutions of the Navier-Stokes equation in $\mathbb{R}^{m}$, with applications to weak solution, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.

[29]

T. D. Ke and D. Lan, Global attractor for a class of functional differential inclusions with Hille-Yosida operators, Nonlinear Anal., 103 (2014), 72-86. doi: 10.1016/j.na.2014.03.006.

[30]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[31]

P. E. KloedenJ. A. Langa and J. Real, Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.

[32]

P. E. KloedenP. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.

[33]

P. E. Kloeden and J. Valero, The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations, Discrete Contin. Dyn. Syst., 28 (2010), 161-179. doi: 10.3934/dcds.2010.28.161.

[34]

M. LiC. M. Huang and F. Z. Jiang, Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes, Appl. Anal., 96 (2017), 1269-1284. doi: 10.1080/00036811.2016.1186271.

[35]

X. C. LiX. Y. Yang and Y. H. Zhang, Error estimates of mixed finite element methods for time-fractional Navier-Stokes equations, J. Sci. Comput., 70 (2017), 500-515. doi: 10.1007/s10915-016-0252-3.

[36]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994), 246-251.

[37]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.

[38]

P. Marín-RubioJ. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008.

[39]

S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494. doi: 10.1016/j.amc.2005.11.025.

[40]

C. J. Niche and G. Planas, Existence and decay of solutions in full space to Navier-Stokes equations with delays, Nonlinear Anal., 74 (2011), 244-256. doi: 10.1016/j.na.2010.08.038.

[41]

L. PengY. ZhouB. Ahmad and A. Alsaedi, The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos Solitons Fractals, 102 (2017), 218-228. doi: 10.1016/j.chaos.2017.02.011.

[42]

I. Podlubny, Fractional Difierential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.

[43]

H. Singh, A new stable algorithm for fractional Navier-Stokes equation in polar coordinate, Int. J. Appl. Comput. Math., 3 (2017), 3705-3722. doi: 10.1007/s40819-017-0323-7.

[44]

L. Tang and Y. Yu, Partial Hölder regularity of the steady fractional Navier-Stokes equations Calc. Var. Partial Differential Equations, 55 (2016), Art. 31, 18 pp. doi: 10.1007/s00526-016-0967-x.

[45]

H. Y. XuX. Y. Jiang and B. Yu, Numerical analysis of the space fractional Navier-Stokes equations, Appl. Math. Lett., 69 (2017), 94-100. doi: 10.1016/j.aml.2017.02.006.

[46]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048.

[47]

F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Ration. Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.

[48]

Z. C. Zhai, Some Regularity Estimates for Mild Solutions to Fractional Heat-Type and Navier-Stokes Equations, Thesis (Ph.D.)-Memorial University of Newfoundland (Canada)., 2009.

[49]

Y. ZhouL. PengB. Ahmad and A. Alsaedi, Energy methods for fractional Navier-Stokes equations, Chaos Solitons Fractals, 102 (2017), 78-85. doi: 10.1016/j.chaos.2017.03.053.

[50]

Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl., 73 (2017), 1016-1027. doi: 10.1016/j.camwa.2016.07.007.

[51]

Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl., 73 (2017), 874-891. doi: 10.1016/j.camwa.2016.03.026.

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