Remarks on L 2 $L^{2}$ decay of solutions for the third-grade non-Newtonian fluid flows in R 3 $\mathbb{R}^{3}$

Abstract

Abstract This paper is concerned with the improved L 2 $L^{2}$ decay for solutions of a class of the third-grade non-Newtonian fluid flows in R 3 $\mathbb {R}^{3}$ . By developing the classic Fourier splitting methods, we prove the non-uniform decay of solutions when u 0 ∈ L 2 ( R 3 ) $u_{0}\in L^{2}(\mathbb {R}^{3})$ and improve algebraic decay rates of solutions as ( 1 + t ) − 3 2 ( 1 r − 1 2 ) − 1 2 $(1+t)^{{-\frac{3}{2}}({\frac{1}{r}}-{\frac{1}{2}})-\frac{1}{2} }$ when the initial data satisfy some moment condition. The results extend the previous result by Zhao (Nonlinear Anal., Real World Appl. 15:229-238, 2014).

Keywords

• non-Newtonian fluid flows
• improved L 2 $L^{2}$ decay