×

The unique existence of weak solution and the optimal control for time-fractional third grade fluid system. (English) Zbl 1407.76005

Summary: The paper concerns the third grade fluid system with the time-fractional derivative of the order \(\alpha \in(0,1)\). We first establish unique existence criterion of weak solutions in the case that the dimension \(n=3\). Then we prove the sufficient condition of optimal pairs.

MSC:

76A05 Non-Newtonian fluids
35Q35 PDEs in connection with fluid mechanics
49J20 Existence theories for optimal control problems involving partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rivlin, R. S.; Ericksen, J. L., Stress-deformation relations for isotropic materials, Journal of Rational Mechanics and Analysis, 4, 323-425 (1955) · Zbl 0064.42004
[2] Ladyzhenskaya, O. A., New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 102, 85-104 (1967) · Zbl 0202.37802
[3] Fosdick, R. L.; Rajagopal, K. R., Anomalous features in the model of ‘second order fluids’, Archive for Rational Mechanics and Analysis, 70, 2, 145-152 (1979) · Zbl 0427.76006 · doi:10.1007/BF00250351
[4] Amrouche, C.; Cioranescu, D., On a class of fluids of grade 3, International Journal of Non-Linear Mechanics, 32, 1, 73-88 (1997) · Zbl 0887.76007 · doi:10.1016/0020-7462(95)00072-0
[5] Sequeira, A.; Videman, J., Global existence of classical solutions for the equations of third grade fluids, Journal of Mathematical and Physical Sciences, 29, 2, 47-69 (1995) · Zbl 0839.76005
[6] Busuioc, V.; Iftimie, D., Global existence and uniqueness of solutions for the equations of third grade fluids, International Journal of Non-Linear Mechanics, 39, 1, 1-12 (2004) · Zbl 1169.76319 · doi:10.1016/S0020-7462(02)00121-X
[7] Paicu, M., Global existence in the energy space of the solutions of a non-Newtonian fluid, Physica D: Nonlinear Phenomena, 237, 10-12, 1676-1686 (2008) · Zbl 1143.76359 · doi:10.1016/j.physd.2008.03.019
[8] Hamza, M.; Paicu, M., Global existence and uniqueness result of a class of third-grade fluids equations, Nonlinearity, 20, 5, 1095-1114 (2007) · Zbl 1114.76004 · doi:10.1088/0951-7715/20/5/003
[9] Zhao, C.; Liang, Y.; Zhao, M., Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations, Nonlinear Analysis: Real World Applications, 15, 229-238 (2014) · Zbl 1297.35194 · doi:10.1016/j.nonrwa.2013.08.001
[10] Chai, X.; Chen, Z.; Niu, W., Large time behavior of a third grade fluid system, Acta Mathematica Scientia B, 36, 6, 1590-1608 (2016) · Zbl 1374.35072 · doi:10.1016/S0252-9602(16)30092-3
[11] Herrmann, R., Fractional Calculus: An Introduction for Physicists (2014), Singapore: World Scientific, Singapore · Zbl 1293.26001 · doi:10.1142/8934
[12] Hilfer, R., Applications of Fractional Calculus in Physics (2000), Singapore: World Scientific, Singapore · Zbl 0998.26002 · doi:10.1142/9789812817747
[13] El-Shahed, M.; Salem, A., On the generalized Navier-Stokes equations, Applied Mathematics and Computation, 156, 1, 287-293 (2004) · Zbl 1134.76323 · doi:10.1016/j.amc.2003.07.022
[14] Ganji, Z. Z.; Ganji, D. D.; Ganji, A. D.; Rostamian, M., Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method, Numerical Methods for Partial Differential Equations, 26, 1, 117-124 (2010) · Zbl 1423.35396 · doi:10.1002/num.20420
[15] Peng, L.; Zhou, Y.; Ahmad, B.; Alsaedi, A., The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos, Solitons & Fractals, 102, 218-228 (2017) · Zbl 1374.35428 · doi:10.1016/j.chaos.2017.02.011
[16] Momani, S.; Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Applied Mathematics and Computation, 177, 2, 488-494 (2006) · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025
[17] de Carvalho-Neto, P. M.; Planas, G., Mild solutions to the time fractional Navier-Stokes equations in RN, Journal of Differential Equations, 259, 7, 2948-2980 (2015) · Zbl 1436.35316 · doi:10.1016/j.jde.2015.04.008
[18] Zhou, Y.; Peng, L., On the time-fractional Navier—Stokes equations, Computers & Mathematics with Applications, 73, 6, 874-891 (2017) · Zbl 1409.76027 · doi:10.1016/j.camwa.2016.03.026
[19] Zhou, Y.; Peng, L., Weak solutions of the time-fractional Navier—Stokes equations and optimal control, Computers & Mathematics with Applications, 73, 6, 1016-1027 (2017) · Zbl 1412.35233 · doi:10.1016/j.camwa.2016.07.007
[20] Zhou, Y.; Peng, L.; Ahmad, B.; Alsaedi, A., Energy methods for fractional Navier-Stokes equations, Chaos, Solitons & Fractals, 102, 78-85 (2017) · Zbl 1374.35432 · doi:10.1016/j.chaos.2017.03.053
[21] Al-Mdallal, Q.; Abro, K.; Khan, I., Analytical Solutions of Fractional Walter’s B Fluid with Applications, Complexity, 2018 (2018) · Zbl 1398.76211 · doi:10.1155/2018/8131329
[22] Maraaba, T. A.; Jarad, F.; Baleanu, D., On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science China Mathematics, 51, 10, 1775-1786 (2008) · Zbl 1179.26024 · doi:10.1007/s11425-008-0068-1
[23] Maraaba, T.; Baleanu, D.; Jarad, F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49, 8 (2008) · Zbl 1152.81550 · doi:10.1063/1.2970709
[24] Adjabi, Y.; Jarad, F.; Baleanu, D.; Abdeljawad, T., On Cauchy problems with CAPuto Hadamard fractional derivatives, Journal of Computational Analysis and Applications, 21, 4, 661-681 (2016) · Zbl 1336.34010
[25] Gambo, Y. Y.; Ameen, R.; Jarad, F.; Abdeljawad, T., Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives, Advances in Difference Equations (2018) · Zbl 1445.34013 · doi:10.1186/s13662-018-1594-y
[26] Temam, R., Navier-Stokes Equations (1984), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0572.35083
[27] Temam, R., Navier-Stokes equations and nonlinear functional analysis. Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66 (1995), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA · Zbl 0833.35110 · doi:10.1137/1.9781611970050
[28] Robinson, J. C., Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative parabolic PDEs and the Theory Of Global Attractors. Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative parabolic PDEs and the Theory Of Global Attractors, Cambridge Texts in Applied Mathematics (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0980.35001
[29] Zhou, Y., Basic Theory of Fractional Differential Equations (2014), Singapore: World Scientific, Singapore · Zbl 1336.34001 · doi:10.1142/9069
[30] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, New York, NY, USA (2006), Elsevier · Zbl 1092.45003
[31] Agrawal, O. P., Fractional variational calculus in terms of Riesz fractional derivatives, Journal of Physics A: Mathematical and Theoretical, 40, 24, 6287-6303 (2007) · Zbl 1125.26007 · doi:10.1088/1751-8113/40/24/003
[32] Alikhanov, A. A., A priori estimates for solutions of boundary value problems for equations of fractional order, Differentsial’nye Uravneniya, 46, 5, 658-664 (2010) · Zbl 1208.35161 · doi:10.1134/S0012266110050058
[33] Bellout, H.; Bloom, F., Incompressible bipolar and non-Newtonian viscous fluid flow. Incompressible bipolar and non-Newtonian viscous fluid flow, Advances in Mathematical Fluid Mechanics (2014), Cham, switzerland: Birkhäuser/ Springer, Cham, switzerland · Zbl 1291.76001 · doi:10.1007/978-3-319-00891-2
[34] Ladyzhenskaya, O. A., Mathematical problems of the dynamics of a viscous incompressible liquid. Mathematical problems of the dynamics of a viscous incompressible liquid, Second revised and supplemented edition (1970), Moscow, Russia: Nauka, Moscow, Russia · Zbl 0215.29004
[35] Busuioc, A. V.; Iftimie, D.; Paicu, M., On steady third grade fluids equations, Nonlinearity, 21, 7, 1621-1635 (2008) · Zbl 1148.35061 · doi:10.1088/0951-7715/21/7/013
[36] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Paris, Farnce: Dunod, Paris, Farnce · Zbl 0189.40603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.