论文标题
几何薄膜方程的收敛分析
Convergence Analysis of the Geometric Thin-Film Equation
论文作者
论文摘要
几何薄膜方程是在长波极限内的液滴扩散的数学模型，其中包括接触线奇异性的正则化。我们表明，鉴于初始rad ra的数据，问题的弱公式允许全球定义的解决方案，并且可以作为Borel可测量函数的推动力表达，其行为受一组普通微分方程（ODES）的影响。这种存在首先在有限的三角洲函数的特殊情况下证明，其中心随着时间的推移而演变 - 这些被称为“粒子溶液”。在一般情况下，我们构建了粒子溶液的收敛序列，该粒子溶液的极限产生上述形式的溶液。此外，我们证明，以这种方式构建的所有弱解决方案都是$ 1/2 $-Hölder的及时及时，并由初始条件唯一决定。
The Geometric Thin-Film equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time -- these are known as `particle solutions'. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are $1/2$-Hölder continuous in time and are uniquely determined by the initial conditions.
