临界Besov空间中密度有界的三维非齐次向列液晶流动的全局可解性

IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED
Dongxiang Chen, Xingyu Liang, Xia Ye
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引用次数: 0

摘要

本文致力于证明三维非齐次不可压缩向列液晶流的整体弱解的存在唯一性。在初始密度位于下界为正的有界函数空间的条件下,结果成立;初始速度在临界贝索夫空间B˙2内足够小,1 1 2 y3 $$ {\dot{B}}_{2,1}&amp;#x0005E;{\frac{1}{2}}\left({\mathbb{R}}&amp;#x0005E;3\right) $$,初始分子取向梯度在相同临界贝索夫空间B˙2内也足够小;1 1 2 y3 . $$ {\dot{B}}_{2,1}&amp;#x0005E;{\frac{1}{2}}\left({\mathbb{R}}&amp;#x0005E;3\right) $$。这些结果与Zhang建立的非齐次不可压缩Navier-Stokes方程的Fujita-Kato定理一致。此外,我们的工作为三维非均匀不可压缩向列液晶流的光滑解的寿命提供了一个下界。这些发现也解决了De Anna提出的独特性问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Global Solvability of 3D Inhomogeneous Nematic Liquid Crystal Flows With Only Bounded Density in Critical Besov Space

This paper is devoted to proving the global existence and uniqueness of weak solutions to the three-dimensional inhomogeneous incompressible nematic liquid crystal flows. The results hold under the conditions that the initial density lies in the bounded function space with a positive lower bound, the initial velocity is sufficiently small in the critical Besov space B ˙ 2 , 1 1 2 3 $$ {\dot{B}}_{2,1}&amp;#x0005E;{\frac{1}{2}}\left({\mathbb{R}}&amp;#x0005E;3\right) $$ , and the gradient of the initial molecular orientation is also small enough in the same critical Besov space B ˙ 2 , 1 1 2 3 $$ {\dot{B}}_{2,1}&amp;#x0005E;{\frac{1}{2}}\left({\mathbb{R}}&amp;#x0005E;3\right) $$ . These results align with the Fujita-Kato theorem for the inhomogeneous incompressible Navier-Stokes equations, as established by Zhang. Additionally, our work provides a lower bound for the lifespan of smooth solutions to the three-dimensional inhomogeneous incompressible nematic liquid crystal flows. The findings also address the uniqueness problem raised by De Anna.

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来源期刊
CiteScore
4.90
自引率
6.90%
发文量
798
审稿时长
6 months
期刊介绍: Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.
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