Optimal decay for the $n$-dimensional incompressible Oldroyd-B model without damping mechanism

By a new energy approach involved in the high frequencies and low frequencies 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 \begin{eqnarray*} \big\|\Lambda^{\alpha}(u,\Lambda^{-1}\mathbb{P}\mathrm{div}\tau)\big\|_{L^q} \le C (1+t)^{-\frac n4-\frac {(\alpha+s)q-n}{2q}}, \quad\Lambda\stackrel{\mathrm{def}}{=}\sqrt{-\Delta}, \end{eqnarray*} with $\frac n2-1<s<\frac np,$ $2\leq p \leq \min(4,{2n}/({n-2})),\ p\not=4\ \hbox{ if }\ n=2,$ 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.