arXiv 1905.026042019-05-07math.AP

无阻尼机制的$n$维不可压Oldrophil-B模型的最优衰减

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

Xiaoping Zhai

备注：: 12page

摘要：

通过一种新的能量方法，在Besov空间中对高、低频分量进行分解，得到了无阻尼机制的Oldrophil-B模型在$\mathbb{R}^n$（$n\ge 2$）中的最优衰减.更准确地说，让$(u,τ)$是在[18]中构造的全局小解，我们证明了对于任何$(u_0,τ_0)\in{\dot{B}_{2,1}^{-s}}(\mathbb{R}^n)$，\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*}与$\frac n2-1<s<\frac np, $ $2\leq p \leq \min(4,{2n}/({n-2})),\ p\not=4\ \hbox{ if }\ n=2,$和$p\leq q\leq\infty$，$\frac nq-\frac np-s<α\leq\frac nq-1$。证明在很大程度上依赖于特殊的耗散结构的方程和一些交换子估计和各种插值之间的Besov型空间。该方法也适用于其他抛物-双曲方程组，其中傅立叶分裂技术是无效的。

英文摘要：

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 $(u,τ)$ be the global small solutions constructed in [18], we prove for any $(u_0,τ_0)\in{\dot{B}_{2,1}^{-s}}(\mathbb{R}^n)$ 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 n2-1<s<\frac np, $ $2\leq p \leq \min(4,{2n}/({n-2})),\ p\not=4\ \hbox{ if }\ n=2,$ and $p\leq q\leq\infty$, $\frac nq-\frac np-s<α\leq\frac nq-1$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文