Abstract:

By a new energy approach involved in the high frequency and low frequency 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 $\begin{eqnarray*} \big\|\Lambda^{\alpha}(u, \Lambda^{-1}{\mathbb P}{\rm div}\tau)\big\|_{L^q} \le C (1+t)^{-\frac n4-\frac {(\alpha+s)q-n}{2q}}, \quad\Lambda\stackrel{{\rm def}}{=}\sqrt{-\Delta}, \end{eqnarray*}$ with -\frac n2 < s < \frac np, $ \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.

Key words: Oldroyd-B model, Time decay estimates, Besov space