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

Yejuan Wang; Tongtong Liang

doi:10.3934/dcdsb.2018312

Article Contents

2019, Volume 24, Issue 8: 3713-3740. Doi: 10.3934/dcdsb.2018312

This issue Previous Article Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient Next Article The Vlasov-Navier-Stokes equations as a mean field limit

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

Yejuan Wang^, and
Tongtong Liang

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

* Corresponding author
* Corresponding author

Received: March 2018

Revised: July 2018

Early access: October 2018

Published: August 2019

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

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Abstract

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.

Keywords:

Fractional delay differential inclusions,
Navier-Stokes equations,
mild solution,
singular initial data,
measure of noncompactness,
upper semi-continuity.

Mathematics Subject Classification: Primary: 33E12, 34K37, 35Q30, 35R11, 35B65.

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References

	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.
	B. De Andrade , A. N. Carvalho , P. 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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	P. Y. Chen , X. 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.
	P. Y. Chen , X. 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.
	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.
	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.
	K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.
	T. Dlotko , Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018) , 99-128. doi: 10.1007/s00245-016-9368-y.
	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.
	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.
	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.
	J. García-Luengo , P. 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.
	J. García-Luengo , P. 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.
	J. García-Luengo , P. 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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	P. E. Kloeden , J. 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.
	P. E. Kloeden , P. 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.
	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.
	M. Li , C. 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.
	X. C. Li , X. 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.
	F. Mainardi , On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994) , 246-251.
	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.
	P. Marín-Rubio , J. 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.
	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.
	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.
	L. Peng , Y. Zhou , B. 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.
	I. Podlubny, Fractional Difierential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.
	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.
	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.
	H. Y. Xu , X. 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.
	R. N. Wang , D. 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.
	F. B. Weissler , The Navier-Stokes initial value problem in $L^p$, Arch. Ration. Mech. Anal., 74 (1980) , 219-230. doi: 10.1007/BF00280539.
	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.
	Y. Zhou , L. Peng , B. 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.
	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.
	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.