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

Abstract

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\|Λ^α(u,Λ^{-1}\mathbb{P}\mathrm{div}τ)\big\|_{L^q} \le C (1+t)^{-\frac n4-\frac {(α+s)q-n}{2q}}, \quadΛ\stackrel{\mathrm{def}}{=}\sqrt{-Δ}, \end{eqnarray*} with $\frac{n}{2}-1<s<\frac{n}{p},$ $2\le p\le \min (4,2n/(n-2)),p\ne 4ifn=2,$ and $p\le q\le \infty$, $\frac{n}{q}-\frac{n}{p}-s<\alpha \le \frac{n}{q}-1$. 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.

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