论文标题
具有非标准生长条件的广义牛顿液的强溶液的收敛速率的上限和下限
Upper and lower bounds of convergence rates for strong solutions of the generalized Newtonian fluids with non-standard growth conditions
论文作者
论文摘要
我们考虑了不可压缩的剪切厚的幂律的运动，该动力是在$ r^3 $中带有可变的幂律指数的运动。这种非线性偏微分方程的系统是在电流动流体的数学模型中产生的。本文的目的是调查差异$ u- \ tilde {u} $的大型行为，其中$ u $是给定方程的强大解决方案，具有初始数据$ u_0 $和$ \ tilde {u} $是具有良好的初始数据$ u_0+u_0+w_0 $的相同方程的强大解决方案。初始扰动$ W_0 $不需要很小，但假定满足某些衰减条件。特别是，我们可以证明$（1+t）^{ - \fracγ{2}} \ lyseSim \ | u（t） - \ tilde {u}（t）\ | _2 \ | _2 \ sillesim（1+t） $γ\ in（2，\ frac {5} {2}）$。该证明是基于这样的观察结果，即线性热方程的溶液描述了充分的大时$ t> 0 $的电流流体溶液的渐近行为，以及具有迭代论点的广义傅立叶分裂方法。此外，还将讨论本文中使用的论点可以改善具有恒定幂律指数的牛顿流体的先前结果。
We consider the motion of an incompressible shear-thickening power-law-like non-Newtonian fluid in $R^3$ with a variable power-law index. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is the strong solution of the same equations with perturbed initial data $u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is assumed to satisfy certain decay condition. In particular, we can show that $(1+t)^{-\fracγ{2}}\lesssim \|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\fracγ{2}}$, for sufficiently large $t>0$, where $γ\in(2,\frac{5}{2})$. The proof is based on the observation that the solution of the linear heat equation describes the asymptotic behaviour of the solutions of the electrorheological fluids well for sufficiently large time $t>0$, and the generalized Fourier splitting method with an iterative argument. Furthermore, it will also be discussed that the argument used in the present paper can improve the previous results for the generalized Newtonian fluids with a constant power-law index.
