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Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient
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 |
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.
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 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. |
[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. 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. |
[12] |
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. |
[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-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. |
[21] |
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. |
[22] |
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. |
[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. 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. |
[32] |
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. |
[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. 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. |
[35] |
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. |
[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-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. |
[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. 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. |
[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. 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. |
[46] |
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. |
[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. 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. |
[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 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. |
[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. 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. |
[12] |
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. |
[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-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. |
[21] |
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. |
[22] |
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. |
[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. 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. |
[32] |
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. |
[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. 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. |
[35] |
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. |
[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-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. |
[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. 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. |
[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. 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. |
[46] |
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. |
[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. 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. |
[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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