| Title:
|
Global strong solutions of a 2-D new magnetohydrodynamic system (English) |
| Author:
|
Liu, Ruikuan |
| Author:
|
Yang, Jiayan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
65 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
105-120 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg's estimates for the stationary Stokes equation and Solonnikov's theorem on $L^p$-$L^q$-estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution is obtained. (English) |
| Keyword:
|
global strong solution |
| Keyword:
|
magnetohydrodynamics |
| Keyword:
|
Stokes equation |
| Keyword:
|
$L^p$-$L^q$-estimates |
| MSC:
|
35B65 |
| MSC:
|
35D35 |
| MSC:
|
35Q35 |
| MSC:
|
35Q61 |
| MSC:
|
76W05 |
| idZBL:
|
07177874 |
| idMR:
|
MR4064592 |
| DOI:
|
10.21136/AM.2020.0208-19 |
| . |
| Date available:
|
2020-02-20T09:47:40Z |
| Last updated:
|
2022-03-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147997 |
| . |
| Reference:
|
[1] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II.Commun. Pure Appl. Math. 17 (1964), 35-92. Zbl 0123.28706, MR 0162050, 10.1002/cpa.3160170104 |
| Reference:
|
[2] Amrouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension.Czech. Math. J. 44 (1994), 109-140. Zbl 0823.35140, MR 1257940, 10.21136/CMJ.1994.128452 |
| Reference:
|
[3] Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion.Adv. Math. 226 (2011), 1803-1822. Zbl 1213.35159, MR 2737801, 10.1016/j.aim.2010.08.017 |
| Reference:
|
[4] Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability.International Series of Monographs on Physics, Clarendon Press, Oxford (1961). Zbl 0142.44103, MR 0128226 |
| Reference:
|
[5] Duvaut, G., Lions, J. L.: Inéquations en thermoélasticité et magnétohydrodynamique.Arch. Ration. Mech. Anal. 46 (1972), 241-279 French. Zbl 0264.73027, MR 0346289, 10.1007/BF00250512 |
| Reference:
|
[6] Geissert, M., Hess, M., Hieber, M., Schwarz, C., Stavrakidis, K.: Maximal $L^p$-$L^q$-estimates for the Stokes equation: a short proof of Solonnikov's theorem.J. Math. Fluid Mech. 12 (2010), 47-60. Zbl 1261.35120, MR 2602914, 10.1007/s00021-008-0275-0 |
| Reference:
|
[7] 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:
|
[8] 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:
|
[9] Jiu, Q., Niu, D., Wu, J., Xu, X., Yu, H.: The 2D magnetohydrodynamic equations with magnetic diffusion.Nonlinearity 28 (2015), 3935-3955. Zbl 1328.76076, MR 3424899, 10.1088/0951-7715/28/11/3935 |
| Reference:
|
[10] Liu, R., Wang, Q.: $S^1$ attractor bifurcation analysis for an electrically conducting fluid flow between two rotating cylinders.Physica D 392 (2019), 17-33. MR 3928313, 10.1016/j.physd.2019.03.001 |
| Reference:
|
[11] Liu, R., Yang, J.: Magneto-hydrodynamical model for plasma.Z. Angew. Math. Phys. 68 (2017), Article number 114, 15 pages. Zbl 1386.76191, MR 3704274, 10.1007/s00033-017-0861-1 |
| Reference:
|
[12] Ma, T.: Theory and Method of Partial Differential Equation.Science Press, Beijing (2011), Chinese. |
| Reference:
|
[13] Ma, T., Wang, S.: Phase Transition Dynamics.Springer, New York (2014). Zbl 1285.82004, MR 3154868, 10.1007/978-1-4614-8963-4 |
| Reference:
|
[14] Masmoudi, N.: Global well posedness for the Maxwell-Navier-Stokes system in 2D.J. Math. Pures Appl. (9) 93 (2010), 559-571. Zbl 1192.35133, MR 2651981, 10.1016/j.matpur.2009.08.007 |
| Reference:
|
[15] Regmi, D.: Global weak solutions for the two-dimensional magnetohydrodynamic equations with partial dissipation and diffusion.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 144 (2016), 157-164. Zbl 1348.35196, MR 3534099, 10.1016/j.na.2016.07.002 |
| Reference:
|
[16] Ren, X., Wu, J., Xiang, Z., Zhang, Z.: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion.J. Funct. Anal. 267 (2014), 503-541. Zbl 1295.35104, MR 3210038, 10.1016/j.jfa.2014.04.020 |
| Reference:
|
[17] Sengul, T., Wang, S.: Pattern formation and dynamic transition for magnetohydrodynamic convection.Commun. Pure Appl. Anal. 13 (2014), 2609-2639. Zbl 1305.76130, MR 3248406, 10.3934/cpaa.2014.13.2609 |
| Reference:
|
[18] Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations.Commun. Pure Appl. Math. 36 (1983), 635-664. Zbl 0524.76099, MR 0716200, 10.1002/cpa.3160360506 |
| Reference:
|
[19] Solonnikov, V. A.: Estimates for solutions of nonstationary Navier-Stokes equations.J. Sov. Math. 8 (1977), 467-529. Zbl 0404.35081, 10.1007/BF01084616 |
| Reference:
|
[20] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis.American Mathematical Society, Providence (2001). Zbl 0981.35001, MR 1846644, 10.1090/chel/343 |
| Reference:
|
[21] Vialov, V.: On the regularity of weak solutions to the MHD system near the boundary.J. Math. Fluid Mech. 16 (2014), 745-769. Zbl 1308.35228, MR 3267547, 10.1007/s00021-014-0184-3 |
| Reference:
|
[22] Wang, Q.: Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders.Discrete Contin. Dyn. Syst., Ser. B. 19 (2014), 543-563. Zbl 1292.76078, MR 3170198, 10.3934/dcdsb.2014.19.543 |
| Reference:
|
[23] Wang, T.: A regularity criterion of strong solutions to the 2D compressible magnetohydrodynamic equations.Nonlinear Anal., Real World Appl. 31 (2016), 100-118. Zbl 1338.35088, MR 3490834, 10.1016/j.nonrwa.2016.01.011 |
| Reference:
|
[24] Wu, J.: Generalized MHD equations.J. Differ. Equations 195 (2003), 284-312. Zbl 1057.35040, MR 2016814, 10.1016/j.jde.2003.07.007 |
| Reference:
|
[25] Yamazaki, K.: Global regularity of $N$-dimensional generalized MHD system with anisotropic dissipation and diffusion.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 122 (2015), 176-191. Zbl 1319.35201, MR 3348070, 10.1016/j.na.2015.04.006 |
| Reference:
|
[26] 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 |
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