Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 762-769.

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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;;
  • 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)


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

CLC Number: 

  • O37