论文标题
长期渐近性和规律性估计较弱的解决方案在Taylor-Couette设置中的双重变性薄膜方程
Long-time asymptotics and regularity estimates for weak solutions to a doubly degenerate thin-film equation in the Taylor-Couette setting
论文作者
论文摘要
我们研究了在流体动力学应用中产生的大时段的四阶准抛物线抛物线方程的动态行为。脱落性既是相对于未知的，也是关于其第一和第三空间衍生物的总和。建模方程式是将两种不混溶的粘性流体膜分隔在两个圆柱体之间的薄膜极限，这些膜被限制在两个圆柱体之间，这些圆柱体以小相对角速度旋转。更确切地说，占据外缸旁边层的流体被认为是牛顿的，即它具有恒定的粘度，而我们假设内缸旁边的层被剪切薄的幂律流体填充。 使用能量方法，傅立叶分析和适当的高阶抛物线方程式估计值，在初始能量低的情况下，我们证明了阳性弱解决方案的全球存在。此外，这些全局解决方案在多项式方面是稳定的，从最初接近圆的接口，以$ 1/t^{1/β} $收敛到一个$β> 0 $，因为时间趋于无限。 此外，我们还提供了第四阶的一般非线性退化抛物线方程的规律性估计。
We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate $1/t^{1/β}$ for some $β> 0$ to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.
