具有分数阶非线性项的表面生长模型在Besov空间中解的局部存在性和唯一性*

doi:10.7523/j.ucas.2024.052

中国科学院大学学报

具有分数阶非线性项的表面生长模型在Besov空间中解的局部存在性和唯一性^*

王庆恺, 吴刚

中国科学院大学数学科学学院,北京 100049

收稿日期:2024-04-01 修回日期:2024-05-17 发布日期:2024-06-11
通讯作者: ^†E-mail：wugang2011@ucas.edu.cn
基金资助:
*国家自然科学基金（11771423）资助

Local existence and uniqueness in Besov spaces for the solution of a surface growth model with fractional power nonlinear term

WANG Qingkai, WU Gang

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

Received:2024-04-01 Revised:2024-05-17 Published:2024-06-11

摘要/Abstract

摘要： 在本文中,我们研究初值为${{h}_{0}}$的如下一维四阶非线性方程的柯西问题：${{\partial }_{t}}h+\partial _{x}^{4}h+\partial _{x}^{2}\left( {{\left| {{\partial }_{x}}h \right|}^{\alpha }} \right)=0$,其中$\alpha $是大于等于5的实数。利用对相应线性方程和非线性项的精细估计,Littlewood-Paley理论,双模方法,压缩映射原理,我们在非齐次Besov空间中得到了局部适定性结果。

关键词: 表面生长模型, 柯西问题, 适定性, Besov空间, Littlewood-Paley理论

Abstract: In this paper we study the Cauchy problem for the 4-th order nonlinear equation ${{\partial }_{t}}h+\partial _{x}^{4}h+\partial _{x}^{2}\left( {{\left| {{\partial }_{x}}h \right|}^{\alpha }} \right)=0$ in one dimension for the initial data ${{h}_{0}}$ where $\alpha \ge 5$ and $\alpha \in \mathbb{R}$. Making use of some subtle estimates of the corresponding linear equation and the nonlinear term, Littlewood-Paley theory, two-norm method and contraction mapping principle, we get the local well-posedness result in nonhomogeneous Besov spaces.

Key words: surface growth model, Cauchy problem, well-posedness, Besov spaces, Littlewood-Paley theory

中图分类号:

O175.29

王庆恺, 吴刚. 具有分数阶非线性项的表面生长模型在Besov空间中解的局部存在性和唯一性^*[J]. 中国科学院大学学报, DOI: 10.7523/j.ucas.2024.052.

WANG Qingkai, WU Gang. Local existence and uniqueness in Besov spaces for the solution of a surface growth model with fractional power nonlinear term[J]. Journal of University of Chinese Academy of Sciences, DOI: 10.7523/j.ucas.2024.052.

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具有分数阶非线性项的表面生长模型在Besov空间中解的局部存在性和唯一性^*

Local existence and uniqueness in Besov spaces for the solution of a surface growth model with fractional power nonlinear term

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