基于结构网格下的AMR技术研究.ppt

Global weak solutions of an initial boundary value problem for screw pinches in plasma physics Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing Joint work with Feng Xie Jianwen Zhang Outline： 1. Governing equations 2. Introduction of weak solutions 3. Global existence 4. Idea of the proof 1. Governing Equations MHD concerns the motion of a conducting fluid (plasma) in an electromagnetic field with a very wide range of applications. The dynamic motions of the fluid and the magnetic field strongly interact each other, and thus both the hydrodynamic and electrodynamic effects have to be considered. The general governing equations of 3D MHD read: Governing equations (General case) The system is too complex to study mathematically.  Let’s consider a special but physically very interesting case: Screw Pinches which have very important applications in plasma physics. 2. Definition of weak solutions 3. Global existence  Thanks ! 谢谢 ! The rest terms in Eqs. (1)-(4) can be similarly treated ! However, we can not exclude the concentration for Eq. (5) (energy eq.), which is included in our “ weak solution” ~ our weak solution is in the generalized weak sense ! ? Remarks: Related results: For the 3D case, Ducomet Fereisl ’05 proved the existence of so-called “variational solutions under strong growth conditions on p, e, et al. However, the polytropic gas case studied here is excluded. “Variational solution ~ mass momentum Eqs. hold in the weak sense, but the energy Eq. holds in the form of weak inequality and the energy inequality holds. 1D: G.Q. Chen D.H. Wang, weak and smooth solutions …… A similar result by Jenssen Hoff for the compressible N-S equations ‘06, but they did not exclude singularity in the momentum eqns., i.e., the momentum eqs. do not hold in the classical sense of weak solutions. ii) Our g