Lecture|Jun 13, 2017/16:30|Room 224 NanHai Building
Speaker: Pro. DONG Bo-qing, Shenzhen University
ABOUT DONG BO-QING
Prof. Dong is a doctoral tutor. He received his master's degree and doctoral degrees in June 2004 in Huazhong University of Science and Technology and in June 2007 in Nankai University respectively. In July 2007 he worked in Anhui University. Now he is a commentator of Mathematical Review in USA. In recent years, he has been working on the study of fluid dynamics equations. He has made some contributions in describing the uniqueness of existence, regularity, attenuation, stability and dynamical properties of solutions of nonlinear partial differential equations with complex flows. Prof. Dong has published more than 40 papers which have been recorded by SCI in important international journals such as Nonlinearity, J.Differential Equations, Discrete Contin. Dyn.Syst, J. Math. Phys. The research work was referenced more than 100 times by domestic and foreign counterparts in mainstream journals. He has presided over a number of National Natural Science Fund projects.
The magnetohydrodynamic (MHD) equations with only magnetic diffusion play a significant role in the study of magnetic reconnection and magnetic turbulence. In certain physical regimes and under suitable scaling the magnetic diffusion becomes partial (given by part of the Laplacian operator). Such equations are of great mathematical interest as well. There have been considerable recent developments on the fundamental issue of whether classical solutions of these partially dissipated equations remain smooth for all time. This problem remains open for the 2D MHD equations with the standard Laplacian magnetic diffusion. This talk focuses on a system of the 2D MHD equations with the kinematic dissipation given by the fractional operator (?Δ)_α and the magnetic diffusion by partial Laplacian. We are able to show that this system with any α > 0 always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion.
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