Article
| Title: | Weak Serrin-type finite time blowup and global strong solutions for three-dimensional density-dependent heat conducting magnetohydrodynamic equations with vacuum (English) |
| Author: | Li, Huanyuan |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 68 |
| Issue: | 5 |
| Year: | 2023 |
| Pages: | 593-621 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the global existence and uniqueness of strong solution to the 3D nonhomogeneous heat conducting MHD equations under a smallness condition on the initial data. In addition, the initial vacuum is allowed. (English) |
| Keyword: | heat conducting MHD |
| Keyword: | Cauchy problem |
| Keyword: | blowup criterion |
| Keyword: | global strong solution |
| Keyword: | vacuum |
| MSC: | 35Q35 |
| MSC: | 76W05 |
| DOI: | 10.21136/AM.2022.0141-22 |
| . | |
| Date available: | 2023-10-05T15:10:27Z |
| Last updated: | 2023-10-09 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151835 |
| . | |
| Reference: | [1] Bie, Q., Wang, Q., Yao, Z.: Global well-posedness of the 3D incompressible MHD equations with variable density.Nonlinear Anal., Real World Appl. 47 (2019), 85-105. Zbl 1411.35013, MR 3873567, 10.1016/j.nonrwa.2018.10.008 |
| Reference: | [2] Chen, F., Guo, B., Zhai, X.: Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density.Kinet. Relat. Models 12 (2019), 37-58. Zbl 1410.35110, MR 3835531, 10.3934/krm.2019002 |
| Reference: | [3] Chen, F., Li, Y., Xu, H.: Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data.Discrete Contin. Dyn. Syst. 36 (2016), 2945-2967. Zbl 1332.35282, MR 3485426, 10.3934/dcds.2016.36.2945 |
| Reference: | [4] Chen, Q., Tan, Z., Wang, Y.: Strong solutions to the incompressible magnetohydrodynamic equations.Math. Methods Appl. Sci. 34 (2011), 94-107. Zbl 1254.35187, MR 2778978, 10.1002/mma.1338 |
| Reference: | [5] Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum.J. Differ. Equations 228 (2006), 377-411. Zbl 1139.35384, MR 2289539, 10.1016/j.jde.2006.05.001 |
| Reference: | [6] Davidson, P. A.: Introduction to Magnetohydrodynamics.Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2017). Zbl 1376.76001, MR 3699477, 10.1017/9781316672853 |
| Reference: | [7] Desjardins, B., Bris, C. Le: Remarks on a nonhomogeneous model of magnetohydrodynamics.Differ. Integral Equ. 11 (1998), 377-394. Zbl 1067.76097, MR 1745545 |
| Reference: | [8] Feireisl, E.: Dynamics of Viscous Compressible Fluids.Oxford Lecture Series in Mathematics and Its Applications 26. Oxford University Press, Oxford (2004). Zbl 1080.76001, MR 2040667, 10.1093/acprof:oso/9780198528388.001.0001 |
| Reference: | [9] Gerbeau, J.-F., Bris, C. Le: Existence of solution for a density-dependent magnetohydrodynamic equation.Adv. Differ. Equ. 2 (1997), 427-452. Zbl 1023.35524, MR 1441851 |
| Reference: | [10] Giga, M.-H., Giga, Y., Saal, J.: Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions.Progress in Nonlinear Differential Equations and their Applications 79. Birkhäuser, Basel (2010). Zbl 1215.35001, MR 2656972, 10.1007/978-0-8176-4651-6 |
| Reference: | [11] Grafakos, L.: Classical Fourier Analysis.Graduate Texts in Mathematics 249. Springer, New York (2008). Zbl 1220.42001, MR 2445437, 10.1007/978-0-387-09432-8 |
| Reference: | [12] He, C., Li, J., Lü, B.: Global well-posedness and exponential stability of 3D Navier-Stokes equations with density-dependent viscosity and vacuum in unbounded domains.Arch. Ration. Mech. Anal. 239 (2021), 1809-1835. Zbl 1462.35243, MR 4215202, 10.1007/s00205-020-01604-5 |
| Reference: | [13] He, C., Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations.J. Differ. Equations 213 (2005), 235-254. Zbl 1072.35154, MR 2142366, 10.1016/j.jde.2004.07.002 |
| Reference: | [14] Huang, X., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system.J. Differ. Equations 254 (2013), 511-527. Zbl 1253.35121, MR 2990041, 10.1016/j.jde.2012.08.029 |
| Reference: | [15] Kim, H.: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations.SIAM J. Math. Anal. 37 (2006), 1417-1434. Zbl 1141.35432, MR 2215270, 10.1137/S0036141004442197 |
| Reference: | [16] Kozono, H., Yamazaki, M.: Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains.Nonlinear Anal., Theory Methods Appl. 38 (1999), 959-970. Zbl 0934.35123, MR 1716426, 10.1016/S0362-546X(98)00145-X |
| Reference: | [17] Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models.Oxford Lecture Series in Mathematics and Its Applications 3. Oxford University Press, Oxford (1996). Zbl 0866.76002, MR 1422251 |
| Reference: | [18] Sohr, H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach.Birkhäuser Advanced Texts. Birkhäuser, Basel (2001). Zbl 0983.35004, MR 1928881, 10.1007/978-3-0348-8255-2 |
| Reference: | [19] Wang, Y.: Weak Serrin-type blowup criterion for three-dimensional nonhomogeneous viscous incompressible heat conducting flows.Physica D 402 (2020), Article ID 132203, 8 pages. Zbl 1453.76039, MR 4046368, 10.1016/j.physd.2019.132203 |
| Reference: | [20] Wang, W., Yu, H., Zhang, P.: Global strong solutions for 3D viscous incompressible heat conducting Navier-Stokes flows with the general external force.Math. Methods Appl. Sci. 41 (2018), 4589-4601. Zbl 1397.35231, MR 3828345, 10.1002/mma.4915 |
| Reference: | [21] Wu, H.: Strong solutions to the incompressible magnetohydrodynamic equations with vacuum.Comput. Math. Appl. 61 (2011), 2742-2753. Zbl 1221.76253, MR 2795397, 10.1016/j.camwa.2011.03.033 |
| Reference: | [22] Zhong, X.: Global strong solution for 3D viscous incompressible heat conducting Navier-Stokes flows with non-negative density.J. Differ. Equations 263 (2017), 4978-4996. Zbl 1377.35227, MR 3680944, 10.1016/j.jde.2017.06.004 |
| Reference: | [23] Zhong, X.: Global strong solutions for 3D viscous incompressible heat conducting magnetohydrodynamic flows with non-negative density.J. Math. Anal. Appl. 446 (2017), 707-729. Zbl 1352.35133, MR 3554752, 10.1016/j.jmaa.2016.09.012 |
| Reference: | [24] Zhong, X.: Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum.Calc. Var. Partial Differ. Equ. 60 (2021), Article ID 64, 24 pages. Zbl 1461.76571, MR 4239818, 10.1007/s00526-021-01957-z |
| Reference: | [25] Zhong, X.: Global existence and large time behavior of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum.Anal. Appl., Singap. 20 (2022), 193-219. Zbl 1490.76251, MR 4386926, 10.1142/S0219530521500056 |
| Reference: | [26] Zhong, X.: Global well-posedness and exponential decay of 2D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum.J. Geom. Anal. 32 (2022), Article ID 19, 26 pages. Zbl 1480.76101, MR 4349463, 10.1007/s12220-021-00754-6 |
| Reference: | [27] Zhou, L.: Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting magnetohydrodynamic flows with vacuum.Electron. J. Qual. Theory Differ. Equ. 2019 (2019), Article ID 81, 16 pages. Zbl 1449.35358, MR 4028913, 10.14232/ejqtde.2019.1.81 |
| Reference: | [28] Zhu, M., Ou, M.: Global strong solutions to the 3D incompressible heat-conducting magnetohydrodynamic flows.Math. Phys. Anal. Geom. 22 (2019), Article ID 8, 17 pages. Zbl 1416.35224, MR 3918713, 10.1007/s11040-019-9306-8 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
