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We investigate optimal decay estimates of solutions to the 3D Cauchy problem of the rate type viscoelastic fluids. The main novelty of this article involves three aspects: first, we show that the second-order and third-order spatial derivative of the global solution converges to zero atL2{L}^{2}-rate(1+t)−74{\left(1+t)}^{-\tfrac{7}{4}}and(1+t)−94{\left(1+t)}^{-\tfrac{9}{4}}, respectively, which improve previous related decay results. Second, we prove that the decay rate of themmth order spatial derivative (wherem∈[0,3]m\in {[}0,3]) of the extra stress tensor of the field inL2{L}^{2}is(1+t)−54−m2{\left(1+t)}^{-\tfrac{5}{4}-\tfrac{m}{2}}, which is faster than that of the velocity. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates, which coincide with the upper rates. Therefore, our results are precisely optimal in this sense.

Optimal large time behavior of the 3D rate type viscoelastic fluids