数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 762-769.

• 论文 • 上一篇    下一篇

N维不可压无阻尼Oldroyd-B模型的最优衰减

谢倩倩1,3(),翟小平2(),董柏青2,3,*()   

  1. 1 合肥学院中德应用数学研究所&数学与统计系 合肥 230601
    2 深圳大学数学与统计学院 深圳 518060
    3 安徽大学数学科学学院 合肥 230601
  • 收稿日期:2020-04-17 出版日期:2021-06-26 发布日期:2021-06-09
  • 通讯作者: 董柏青 E-mail:qianqianxieahu@163.com;pingxiaozhai@163.com;bqdong@ahu.edu.cn
  • 作者简介:谢倩倩, E-mail: qianqianxieahu@163.com|翟小平, E-mail: pingxiaozhai@163.com
  • 基金资助:
    国家自然科学基金(11601533);国家自然科学基金(11871346);广东省自然科学基金(2018A030313024);深圳市自然科学基金(JCYJ20180305125554234);深圳大学科研基金(2017056)

Optimal Decay for the N-Dimensional Incompressible Oldroyd-B Model Without Damping Mechanism

Qianqian Xie1,3(),Xiaoping Zhai2(),Boqing Dong2,3,*()   

  1. 1 Department of Mathematics and Statistics, Hefei University, Hefei 230601
    2 School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060
    3 School of Mathematical Science, Anhui University, Hefei 230601
  • Received:2020-04-17 Online:2021-06-26 Published:2021-06-09
  • Contact: Boqing Dong E-mail:qianqianxieahu@163.com;pingxiaozhai@163.com;bqdong@ahu.edu.cn
  • Supported by:
    the NSFC(11601533);the NSFC(11871346);the NSF of Guangdong Province(2018A030313024);the NSF of Shenzhen City(JCYJ20180305125554234);the Research Fund of Shenzhen University(2017056)

摘要:

对于$n(n\geq 2)$维不可压无阻尼Oldroyd-B模型,运用能量方法以及Besov空间中高低频分解技术,该文得到了关于强解的最优衰减速率.具体来说,充分利用方程的特殊结构,交换子估计以及Besov空间之间的各种插值定理,对于任意的初值$(u_0,\tau_0)\in{\dot{B}_{2,1}^{-s}}({\mathbb R}^n)$,该文得到了关于强解(参见文献[18])的最优衰减速率 其中指标满足-\frac n2 < s <\frac np,$\leq p\leq\min(4,{2n}/({n-2}))(n=2\mbox{时},p\not=4),$$p\leq q\leq\infty,$$\frac nq-\frac np-s<\alpha\leq\frac nq-1$.该文的方法也使用于其它抛物-双曲耦合的系统.

关键词: Oldroyd-B模型, 时间衰减估计, Besov空间

Abstract:

By a new energy approach involved in the high frequency and low frequency decomposition in the Besov spaces, we obtain the optimal decay for the incompressible Oldroyd-B model without damping mechanism in ${\mathbb R}^n$ ($n\ge 2$). More precisely, let $(u, \tau)$ be the global small solutions constructed in[18], we prove for any $(u_0, \tau_0)\in{\dot{B}_{2, 1}^{-s}}({\mathbb R}^n)$ that with -\frac n2 < s < \frac np, $ \leq p \leq \min(4, {2n}/({n-2})), \ p\not=4\ \hbox{ if }\ n=2, $ and $p\leq q\leq\infty$, $\frac nq-\frac np-s<\alpha \leq\frac nq-1$. The proof relies heavily on the special dissipative structure of the equations and some commutator estimates and various interpolations between Besov type spaces. The method also works for other parabolic-hyperbolic systems in which the Fourier splitting technique is invalid.

Key words: Oldroyd-B model, Time decay estimates, Besov space

中图分类号: 

  • O37