论文标题
配备Oseen-Frank能量的Ericksen-Leslie系统的耗散解决方案
Dissipative solution to the Ericksen-Leslie system equipped with the Oseen-Frank energy
论文作者
论文摘要
我们分析了在三个空间维度上配备Oseen-Frank能量的Ericksen-Leslie系统。引入了耗散解决方案的新概念。最近，作者将标准解决方案的概念引入了考虑的系统，并显示了这些广义解决方案的全球存在以及弱的唯一性。在本文中，我们表明对测量值的期望是耗散解决方案。耗散解决方案的概念本身依赖于不平等而不是平等性，但由功能而不是参数化度量来描述。这些解决方案在全球存在，并实现弱的独特性。此外，我们将相对能量不等式推广到满足不同非均匀差异边界条件的解决方案，并结合了暂时恒定的电磁场的影响。依靠这种普遍的能量不平等，我们研究了长期行为，并表明所有解决方案都将大量时间限制融合到某个稳定状态。
We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state.
