Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1287-1301.doi: 10.1007/s10473-021-0416-6

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Yiming HE1, Ruizhao ZI2   

  1. 1. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China;
    2. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
  • Received:2020-06-28 Online:2021-08-25 Published:2021-09-01
  • Contact: Ruizhao ZI
  • Supported by:
    R. Zi is partially supported by the National Natural Science Foundation of China (11871236 and 11971193), the Natural Science Foundation of Hubei Province (2018CFB665), and the Fundamental Research Funds for the Central Universities (CCNU19QN084).

Abstract: Some sufficient conditions of the energy conservation for weak solutions of incompressible viscoelastic flows are given in this paper. First, for a periodic domain in $\mathbb R^3$, and the coefficient of viscosity $\mu=0$, energy conservation is proved for $u$ and $F$ in certain Besov spaces. Furthermore, in the whole space $\mathbb R^3$, it is shown that the conditions on the velocity $u$ and the deformation tensor $F$ can be relaxed, that is, $u\in B^{\frac 13}_{3,c(\mathbb N)}$, and $F\in B^{\frac 13}_{3,\infty}$. Finally, when $\mu>0$, in a periodic domain in $\mathbb R^d$ again, a result independent of the spacial dimension is established. More precisely, it is shown that the energy is conserved for $u\in L^r(0,T;L^s(\Omega))$ for any $\frac 1r+\frac 1s\leqslant\frac 12$, with $s\geqslant4$, and $F\in L^m(0,T;L^n(\Omega))$ for any $\frac 1m+\frac 1n\leqslant\frac 12$, with $n\geqslant4$.

Key words: Incompressible viscoelastic fluids, weak solutions, energy conservation

CLC Number: 

  • 76A10