精品文档---下载后可任意编辑两类四阶非线性抛物方程解的适定性和渐近性中期报告摘要本篇中期报告主要介绍了两类四阶非线性抛物方程解的适定性和渐近性的讨论进展。第一类方程是高维 Allen-Cahn 方程，已经得到了广泛的关注，因为它可以用于描述许多物理现象，如相变、流体动力学等。第二类方程是液晶流中的弯曲能模型，它在生物和材料科学中也有很多应用。在本篇报告中，我们首先介绍了四阶抛物方程的基本理论和相关的适定性结果。然后我们讨论了高维 Allen-Cahn 方程的适定性和渐近性问题，包括全局存在性、长时间行为、漂移不变量估量等。接着我们介绍了液晶流中的弯曲能模型的适定性和渐近性，包括解的存在性、唯一性和长时间行为。最后，我们介绍了当前讨论中的一些挑战和未来的方向。关键词：抛物方程、适定性、渐近性、高维 Allen-Cahn 方程、液晶流AbstractThis intermediate report mainly introduces the research progress of the well-posedness and asymptotic behavior for two types of fourth-order nonlinear parabolic equations. The first type of equation is the high-dimensional Allen-Cahn equation, which has received widespread attention because it can be used to describe many physical phenomena, such as phase transitions, fluid dynamics, etc. The second type of equation is the bending energy model in liquid crystal flow, which also has many applications in biology and materials science.In this report, we first introduce the basic theory of fourth-order parabolic equations and related well-posedness results. Then, we discuss the well-posedness and asymptotic behavior of the high-dimensional Allen-Cahn equation, including global existence, long-time behavior, drift invariant estimate, etc. Next, we introduce the well-posedness and asymptotic behavior of the bending energy model in liquid crystal flow, including the existence, uniqueness, and long-time behavior of solutions. Finally, we introduce some challenges and future directions in the current research.精品文档---下载后可任意编辑Keywords: parabolic equation, well-posedness, asymptotic behavior, high-dimensional Allen-Cahn equation, liquid crystal flow
