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{d}\mathrm{i}\mathrm{v}$\tau)\big|_{L^q} \le C (1+t)^{-\frac n4-\frac {(\alpha+s)q-n}{2q}}, \quad\Lambda\stackrel{$\mathrm{d}\mathrm{e}\mathrm{f}$}{=}\sqrt{-\Delta}, \end{eqnarray*} with $\frac{n}{2}-1<s<\frac{n}{p},$ $2\leq p \leq \min(4,{2n}/({n-2})),\ p\not=4
\hbox{ if }\ n=2,$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.