Applications of Mathematics, Vol. 67, No. 2, pp. 199-208, 2022
Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids
Peixin Zhang, Mingxuan Zhu
Received April 20, 2020. Published online May 20, 2021.
Abstract: We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\varepsilon}$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac32$ and any $0<\varepsilon<1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.
Keywords: non-resistive magneto-micropolar fluid; local existence
References: [1] G. Ahmadi, M. Shahinpoor: Universal stability of magneto-micropolar fluid motions. Int. J. Engin. Sci. 12 (1974), 657-663. DOI 10.1016/0020-7225(74)90042-1 | MR 0443550 | Zbl 0284.76009
[2] D. Blömker, C. Nolde, J. C. Robinson: Rigorous numerical verification of uniqueness and smoothness in a surface growth model. J. Math. Anal. Appl. 429 (2015), 311-325. DOI 10.1016/j.jmaa.2015.04.025 | MR 3339076 | Zbl 1315.65092
[3] J.-Y. Chemin, D. S. McCormick, J. C. Robinson, J. L. Rodrigo: Local existence for the non-resistive MHD equations in Besov spaces. Adv. Math. 286 (2016), 1-31. DOI 10.1016/j.aim.2015.09.004 | MR 3415680 | Zbl 1333.35183
[4] M. Chen: Global well-posedness of the 2D incompressible micropolar fluid flows with partial viscosity and angular viscosity. Acta Math. Sci., Ser. B, Engl. Ed. 33 (2013), 929-935. DOI 10.1016/S0252-9602(13)60051-X | MR 3072129 | Zbl 1299.35043
[5] M. Chen, X. Xu, J. Zhang: The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect. Z. Angew. Math. Phys. 65 (2014), 687-710. DOI 10.1007/s00033-013-0345-x | MR 3238510 | Zbl 1300.35091
[6] S. C. Cowin: Polar fluids. Phys. Fluids 11 (1968), 1919-1927. DOI 10.1063/1.1692219 | Zbl 0179.56002
[7] M. E. Erdoğan: Polar effects in the apparent viscosity of suspension. Rheol. Acta 9 (1970), 434-438. DOI 10.1007/BF01975413
[8] C. L. Fefferman, D. S. McCormick, J. C. Robinson, J. L. Rodrigo: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267 (2014), 1035-1056. DOI 10.1016/j.jfa.2014.03.021 | MR 3217057 | Zbl 1296.35142
[9] C. L. Fefferman, D. S. McCormick, J. C. Robinson, J. L. Rodrigo: Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch. Ration. Mech. Anal. 223 (2017), 677-691. DOI 10.1007/s00205-016-1042-7 | MR 3590662 | Zbl 1359.35150
[10] Q. Jiu, D. Niu: Mathematical results related to a two-dimensional magneto-hydrodynamic equations. Acta Math. Sci., Ser. B, Engl. Ed. 26 (2006), 744-756. DOI 10.1016/S0252-9602(06)60101-X | MR 2265204 | Zbl 1188.35148
[11] T. Kato, G. Ponce: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41 (1988), 891-907. DOI 10.1002/cpa.3160410704 | MR 0951744 | Zbl 0671.35066
[12] C. E. Kenig, G. Ponce, L. Vega: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4 (1991), 323-347. DOI 10.1090/S0894-0347-1991-1086966-0 | MR 1086966 | Zbl 0737.35102
[13] G. Łukaszewicz: Micropolar Fluids: Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (1999). DOI 10.1007/978-1-4612-0641-5 | MR 1711268 | Zbl 0923.76003
[14] E. E. Ortega-Torres, M. A. Rojas-Medar: Magneto-micropolar fluid motion: Global existence of strong solutions. Abstr. Appl. Anal. 4 (1999), 109-125. DOI 10.1155/S1085337599000287 | MR 1810322 | Zbl 0976.35055
[15] M. A. Rojas-Medar: Magneto-micropolar fluid motion: Existence and uniqueness of strong solution. Math. Nachr. 188 (1997), 301-319. DOI 10.1002/mana.19971880116 | MR 1484679 | Zbl 0893.76006
[16] M. A. Rojas-Medar: Magneto-micropolar fluid motion: On the convergence rate of the spectral Galerkin approximations. Z. Angew. Math. Mech. 77 (1997), 723-732. DOI 10.1002/zamm.19970771003 | MR 1479160 | Zbl 0894.76093
[17] M. A. Rojas-Medar, J. L. Boldrini: Magneto-micropolar fluid motion: Existence of weak solutions. Rev. Mat. Complut. 11 (1998), 443-460. DOI 10.5209/rev_REMA.1998.v11.n2.17276 | MR 1666509 | Zbl 0918.35114
[18] J. Yuan: Existence theorem and blow-up criterion of strong solutions to the magneto-micropolar fluid equations. Math. Methods Appl. Sci. 31 (2008), 1113-1130. DOI 10.1002/mma.967 | MR 2419091 | Zbl 1137.76071
[19] B. Yuan, X. Li: Regularity of weak solutions to the 3D magneto-micropolar equations in Besov spaces. Acta Appl. Math. 163 (2019), 207-223. DOI 10.1007/s10440-018-0220-z | MR 4008703 | Zbl 1428.35409
[20] Z. Zhang: A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces. Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Article ID 104, 6 pages. DOI 10.14232/ejqtde.2016.1.104 | MR 3578430 | Zbl 1399.35307
Affiliations: Peixin Zhang, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P. R. China, e-mail: zhpx@hqu.edu.cn; Mingxuan Zhu (corresponding author), School of Mathematical Sciences, Qufu Normal University, Qufu 273100, P. R. China, e-mail: mxzhu@qfnu.edu.cn
