论文标题
高阶双重变性抛物线方程的长期行为和稳定性
Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order
论文作者
论文摘要
我们研究了第四阶的二式抛物线抛物性问题的解决方案的长期行为。方程模型例如，非牛顿薄膜流动在平坦的不渗透底部和零接触角上的动态行为。我们认为剪切率依赖性流体是由构型幂律或Ellis-Law描述的流体粘度。在所有三种情况下，正阳性常数（即阳性平膜）是唯一的阳性稳态解决方案。此外，我们可以详细介绍解决方案相对于$ h^1（ω）$ - 规范的长期行为。在剪切厚的幂律流体的情况下，人们观察到最初接近稳态的溶液在有限的时间内会融合到平衡。在剪切薄的幂律案例中，我们发现稳态在多项式上是稳定的，因为随着时间的流逝，最初接近稳态的溶液在某些$β> 0 $中以$ 1/t^{1/β} $的速率收敛到平衡。最后，在Ellis-Fluid的情况下，稳态在$ H^1（ω）$中呈指数稳定。
We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the $H^1(Ω)$-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate $1/t^{1/β}$ for some $β> 0$. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in $H^1(Ω)$.
