闭超曲面在磁场运动中的动力学稳定性与分支

In this project, we are devoted to study the well-posedness, nonlinear stability, Hopf bifurcation and exchange stability of the motion of supersurface in electromagnetic fields .. The motion of supersurface in electromagnetic fields is a kind of parital differential equations which decribes the relationship between the motion of supersurface and electromagnetic fields. It is composed by the mean curvature flows coupled with Maxwell equations. Due to the importance and complexity of mean curvature flows, it is still an active research subject in Mathematics and Physics until now. As we known, the nonlinear stability phenomenon exists in the motion of supersurface in electromagnetic fields, for example, the lightlike supersurface.nearby black hole. But there is no rigorous proof on it. Expecially, the bifurcation. This project will study the nonlinear stability and bifurcation for the motion of supersurface in electromagnetic fields by using dynamical methods. Therefore, our research will make people more familiar with the.motion of supersurface in electromagnetic fields by giving the proofs for the nonlinear stability and bifurcation of the motion of supersurface in.electromagnetic fields.

本项目以电磁场中超曲面运动模型为研究对象致力于研究其适定性, 非线性稳定性,分支理论以及交换稳定性。.电磁场中超曲面运动模型是研究运动的超曲面和磁场相互作用中各物理量间变化关系的偏 微分方程组。它主要由几何中的超曲面运动模型,即平均曲率流方程与电动力学中Maxwell方程组耦合而成。由于其重要性与复杂性,直至今日依然是数学物理领域中非常活跃的研究课题之一。稳定性现象存在于电磁场中超曲面运动中,例如:黑洞视界的类光曲面运动。但目前为止还没有严格的理论证明,特别对非线性不稳定性所可能引起的分支现象。本项目计划应用动力系统的方法来分析研究电磁场中超曲面运动的非线性稳定性以及分支现象。因此,我们的研究将会为人们深入理解和认识电磁场中超曲面运动的非线性稳定性提供一定的理论依据。

本项目以电磁场中超曲面运动模型为研究对象,致力于研究其适定性和非线性稳定性等动力系统现象。该模型对应物理学弦论中的带电膜运动。通过本项目的实施，得到3个重要成果：1. 建立了平坦时空中带电膜运动模型，得到其适定性和对初值的稳定性；2. 得到一维弦运动（也就是著名的Born-Infeld方程）的精确奇性衍射以及稳定性；3. 发现高维类时曲面运动的精确奇性可以是自相似球，并且这类奇性是非线性稳定的。上述三个成果分别发表在国际期刊Journal of Differential Equations, Nonlinearity 和Calculus of Variable and Partial Differential Equations上。科学意义在于给出带电膜薄运动的机制，对进一步研究黑洞附近的极值曲面运动提供一定的参考。