非线性高阶抛物型偏微分方程
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摘要
流体薄膜在表面张力作用下的运动,比如流体薄膜覆盖固体表面的现象,是我们日常生活中经常遇到的。这种现象背后的流体动力学机制主要与表面化学作用有关。如何控制这种相互作用不仅在工业生产中有重要的地位而且近二十年来引起了数学上的许多关注,产生了很多与实际背景相联系的数学问题。本论文首先研究了粘性流体薄膜沿倾斜平面的运动,证明了描述这一运动过程的偏微分方程的解具有有限传播速度的性质；此外还证明了一类固体薄膜运动方程弱解的适定性和有限时间爆破的现象。主要成果如下：
1.考虑流体薄膜方程
这是一个退化的四阶抛物型方程,用来描述粘性流体薄膜沿倾斜平面的运动,其中函数h(x,t)表示流体的厚度；常数α>0和β>0分别依赖于gsinθ和gcosθ(g是重力加速度,θ∈(0,π／2)是倾斜角)；指数n>0表示附加在流体／固体界面上的边界条件(n=3表示无滑动切触,n∈(0,3)表示流体的部分滑动切触运动)。利用基于局部熵估计的逼近方法和局部能量方法,证明了当指数n满足4／3≤n<2时方程的强解具有有限传播速度的性质,并给出了解的支集的运动速度的上界估计。
在物理上有意义的自由边值问题所具有的一个非常重要的特性就是有限传播速度。我们的结果验证了用来描述上述薄膜运动的方程的确是物理上正确的模型。
2.考虑固体薄膜方程
这是一个非线性一致的六阶抛物型方程,用来描述外延拉紧的固体薄膜当和基底有浸润相交作用时在表面张力驱动下的运动,其中g,p,q是物理常数。我们利用Galerkin逼近方法和能量估计的方法证明了在周期边界条件下方程弱解的存在性,唯一性和正则性；利用第一特征值的方法证明了在Dirichlet边界条件下,当系数g,p,q满足一定条件时,方程弱解的L2范数具有有限时间爆破的现象,并给出了爆破时间的估计。
The motion of a liquid under the influence of surface tension, such as a fluid coating a solid surface, is a phenomenon we experience every day. The underlying dynamics of such phenomenon depend heavily on the surface chemistry. Many industrial processes rely on the ability to control these interactions. And such contact line problem has generated some very interesting and difficult mathemat-ical problems associated with the model equations. In this dissertation, the thin viscous flows over an inclined plane is considered. It is proved that this model has finite speed of propagation for the solutions. Furthermore, the wellposedness of the weak solution to the sixth-order nonlinear uniformly parabolic equation which arises in the description of thin, epitaxially strained films in the presence of wetting intersections with the substrate is obtained. And the finite-time blow-up is shown. The main achievements contained here are as follows:
1. Considering the thin liquid film equation
this is a fourth-order nonlinear degenerate parabolic equation which arises in the description of thin viscous flows over an inclined plane with slopeθ∈(0,π/2), where the function h(x, t) represents the thickness of the film,α> 0 andβ> 0 are constants proportional to g sin 9 and g cosθrespectively (g is the gravitational acceleration), and the exponent n> 0 characterizes the condition assumed on the fluid-solid interface (the case n= 3 corresponds to contact without slip, while n∈(0,3) corresponds to the motion of the fluid with partial slip). We prove that the finite speed propagation property for the strong solutions of this equa-tion holds if the exponent n satisfies 4/3≤n<2. Our approach for proving is perturbation method combined with the local energy method which relies on local energy/entropy estimates and differential inequalities. And the upper bound for the speed of the support of this solution is obtained.
One of the characteristics of the physically correct free boundary value prob-lem is the finite speed of propagation. Our results indeed confirm that the model we considered is physically correct one.
2. Considering the thin solid film equation
this is a sixth-order nonlinear uniformly parabolic equation which arises in the de-scription of thin, epitaxially strained films in the presence of wetting intersections with the substrate with constants g,p and q depending on some characteristic of the physical domain. Under periodic boundary conditions we prove the existence, uniqueness and regularity of the weak solution by means of Galerkin method and energy method. Also, we show the finite-time blow-up under the Dirichlet bound-ary conditions using the method of the first eigenvalue and obtain the estimates for the blow-up time.
1.考虑流体薄膜方程
这是一个退化的四阶抛物型方程,用来描述粘性流体薄膜沿倾斜平面的运动,其中函数h(x,t)表示流体的厚度；常数α>0和β>0分别依赖于gsinθ和gcosθ(g是重力加速度,θ∈(0,π／2)是倾斜角)；指数n>0表示附加在流体／固体界面上的边界条件(n=3表示无滑动切触,n∈(0,3)表示流体的部分滑动切触运动)。利用基于局部熵估计的逼近方法和局部能量方法,证明了当指数n满足4／3≤n<2时方程的强解具有有限传播速度的性质,并给出了解的支集的运动速度的上界估计。
在物理上有意义的自由边值问题所具有的一个非常重要的特性就是有限传播速度。我们的结果验证了用来描述上述薄膜运动的方程的确是物理上正确的模型。
2.考虑固体薄膜方程
这是一个非线性一致的六阶抛物型方程,用来描述外延拉紧的固体薄膜当和基底有浸润相交作用时在表面张力驱动下的运动,其中g,p,q是物理常数。我们利用Galerkin逼近方法和能量估计的方法证明了在周期边界条件下方程弱解的存在性,唯一性和正则性；利用第一特征值的方法证明了在Dirichlet边界条件下,当系数g,p,q满足一定条件时,方程弱解的L2范数具有有限时间爆破的现象,并给出了爆破时间的估计。
The motion of a liquid under the influence of surface tension, such as a fluid coating a solid surface, is a phenomenon we experience every day. The underlying dynamics of such phenomenon depend heavily on the surface chemistry. Many industrial processes rely on the ability to control these interactions. And such contact line problem has generated some very interesting and difficult mathemat-ical problems associated with the model equations. In this dissertation, the thin viscous flows over an inclined plane is considered. It is proved that this model has finite speed of propagation for the solutions. Furthermore, the wellposedness of the weak solution to the sixth-order nonlinear uniformly parabolic equation which arises in the description of thin, epitaxially strained films in the presence of wetting intersections with the substrate is obtained. And the finite-time blow-up is shown. The main achievements contained here are as follows:
1. Considering the thin liquid film equation
this is a fourth-order nonlinear degenerate parabolic equation which arises in the description of thin viscous flows over an inclined plane with slopeθ∈(0,π/2), where the function h(x, t) represents the thickness of the film,α> 0 andβ> 0 are constants proportional to g sin 9 and g cosθrespectively (g is the gravitational acceleration), and the exponent n> 0 characterizes the condition assumed on the fluid-solid interface (the case n= 3 corresponds to contact without slip, while n∈(0,3) corresponds to the motion of the fluid with partial slip). We prove that the finite speed propagation property for the strong solutions of this equa-tion holds if the exponent n satisfies 4/3≤n<2. Our approach for proving is perturbation method combined with the local energy method which relies on local energy/entropy estimates and differential inequalities. And the upper bound for the speed of the support of this solution is obtained.
One of the characteristics of the physically correct free boundary value prob-lem is the finite speed of propagation. Our results indeed confirm that the model we considered is physically correct one.
2. Considering the thin solid film equation
this is a sixth-order nonlinear uniformly parabolic equation which arises in the de-scription of thin, epitaxially strained films in the presence of wetting intersections with the substrate with constants g,p and q depending on some characteristic of the physical domain. Under periodic boundary conditions we prove the existence, uniqueness and regularity of the weak solution by means of Galerkin method and energy method. Also, we show the finite-time blow-up under the Dirichlet bound-ary conditions using the method of the first eigenvalue and obtain the estimates for the blow-up time.
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