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: February 28, 2018

Revised: June 30, 2018

Early access: October 2018

Published: July 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

Abstract Full Text(HTML) Related Papers Cited by

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.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.