Applications of Mathematics, first online, pp. 1-25
Existence of regular time-periodic solutions for a class of non-Newtonian double-diffusive convection system
Qiong Wu, Changjia Wang
Received November 29, 2024. Published online June 9, 2025.
Abstract: We investigate a system of partial differential equations that models the motion of an incompressible double-diffusion convection fluid. The additional stress tensor is generated by a potential with $p$-structure. In a three-dimensional periodic setting and $p\in[\frac53,2)$, we employ a regularized approximation scheme in conjunction with the Galerkin method to establish the existence of regular solutions, provided that the forcing term is properly small. Furthermore, we demonstrate the existence of periodic regular solutions with period $T$ when the external force exhibits periodicity in time with the same period $T$.
References: [1] A. Abbatiello, P. Maremonti: Existence of regular time-periodic solutions to shear-thinning fluids. J. Math. Fluid Mech. 21 (2019), Article ID 29, 14 pages. DOI 10.1007/s00021-019-0435-4 | MR 3938375 | Zbl 1416.35177
[2] R. A. Adams, J. J. F. Fournier: Sobolev Spaces. Pure and Applied Mathematics 140. Academic Press, Amsterdam (2003). DOI 10.1016/s0079-8169(03)x8001-0 | MR 2424078 | Zbl 1098.46001
[3] G. Astarita, G. Marrucci: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London (1974).
[4] V. Y. Belov, B. V. Kapitonov: A hydrodynamic model of a chemically active fluid. Sib. Math. J. 24 (1983), 823-833. DOI 10.1007/bf00970307 | MR 0731038 | Zbl 0599.35129
[5] F. Chen, B. Guo: The suitable weak solution for the Cauchy problem of the double-diffusive convection system. Appl. Anal. 98 (2019), 1724-1740. DOI 10.1080/00036811.2018.1441995 | MR 3955790 | Zbl 1417.35114
[6] F. Chen, B. Guo, L. Zeng: The well-posedness of the double-diffusive convection system in a bounded domain. Math. Methods Appl. Sci. 41 (2018), 4327-4336. DOI 10.1002/mma.4895 | MR 3824560 | Zbl 1397.35201
[7] F. Chen, B. Guo, L. Zeng: The well-posedness for the Cauchy problem of the double-diffusive convection system. J. Math. Phys. 60 (2019), Article ID 011511, 14 pages. DOI 10.1063/1.5052668 | MR 3903541 | Zbl 1406.76078
[8] L. Diening, C. Ebmeyer, M. Růžička: Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure. SIAM J. Numer. Anal. 45 (2007), 457-472. DOI 10.1137/05064120x | MR 2300281 | Zbl 1140.65060
[9] D. D. Joseph: Stability of Fluid Motions. I. Springer Tracts in Natural Philosophy 27. Springer, Berlin (1976). DOI 10.1007/978-3-642-80991-0 | MR 0449147 | Zbl 0345.76022
[10] S. A. Lorca, M. A. Rojas-Medar: Weak solutions for the viscous incompressible chemically active fluids. Rev. Mat. Estat. 14 (1996), 183-199. MR 1465922 | Zbl 0894.76091
[11] J. Málek, K. R. Rajagopal: Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. Evolutionary Equations II. Handbook of Differential Equations. Elsevier/North-Holland, Amsterdam (2005), 371-459. DOI 10.1016/s1874-5717(06)80008-3 | MR 2182831 | Zbl 1095.35027
[12] A. C. Moretti, M. A. Rojas-Medar, M. D. Rojas-Medarm: The equations of a viscous incompressible chemically active fluid: Existence and uniqueness of strong solutions in an unbounded domain. Comput. Math. Appl. 44 (2002), 287-299. DOI 10.1016/s0898-1221(02)00148-7 | MR 1912828 | Zbl 1179.76015
[13] J. Nečas, J. Málek, M. Rokyta, M. Růžička: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation 13. Chapman & Hall, London (1996). DOI 10.1007/978-1-4899-6824-1 | MR 1409366 | Zbl 0851.35002
[14] J. Pedlosky: Geophysical Fluid Dynamics. Springer, New York (1979). DOI 10.1007/978-1-4684-0071-7 | Zbl 0429.76001
[15] T. Radko: Double-Diffusive Convection. Cambridge University Press, Cambridge (2013). DOI 10.1017/cbo9781139034173 | MR 3183649 | Zbl 1301.76001
[16] M. A. Ragusa, F. Wu: Global regularity and stability of solutions to the 3D double-diffusive convection system with Navier boundary conditions. Adv. Differ. Equ. 26 (2021), 281-304. DOI 10.57262/ade026-0708-281 | MR 4305007 | Zbl 1479.35694
[17] M. A. Rojas-Medar, S. A. Lorca: Global strong solution of the equations for the motion of a chemical active fluid. Mat. Contemp. 8 (1995), 319-335. MR 1330043 | Zbl 0853.35096
[18] M. A. Rojas-Medar, S. A. Lorca: The equations of a viscous incompressible chemical active fluid I: Uniqueness and existence of the local solutions. Rev. Mat. Apl. 16 (1995), 57-80. MR 1382269 | Zbl 0849.35101
[19] M. A. Rojas-Medar, S. A. Lorca: The equations of a viscous incompressible chemical active fluid II: Regularity of solutions. Rev. Mat. Apl. 16 (1995), 81-95. MR 1382270 | Zbl 1126.35350
[20] F. Wu: Blowup criterion of strong solutions to the three-dimensional double-diffusive convection system. Bull. Malays. Math. Sci. Soc. (2) 43 (2020), 2673-2686. DOI 10.1007/s40840-019-00828-3 | MR 4089665 | Zbl 1443.76208
[21] F. Wu: Blowup criterion via only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces to the 3D double-diffusive convection equations. J. Math. Fluid Mech. 22 (2020), Article ID 24, 9 pages. DOI 10.1007/s00021-020-0483-9 | MR 4085356 | Zbl 1435.35313
[22] F. Wu: On continuation criteria for the double-diffusive convection system in Vishik spaces. Appl. Anal. 103 (2024), 1693-1703. DOI 10.1080/00036811.2023.2260419 | MR 4754792 | Zbl 1545.35127
Affiliations: Qiong Wu, Changjia Wang (corresponding author), School of Mathematics and Statistics, Changchun university of Science and Technology, No. 7089, Weixing Road, Chaoyang District, Changchun, 130022, P. R. China, e-mail: 2630750667@qq.com, wangchangjia@gmail.com
