Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation
Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation
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薄膜方程的适定性和自相似渐近性
DOI：
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发表时间：
2015
影响因子：
2
通讯作者：
M. V. Gnann
中科院分区：
文献类型：
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作者：
M. V. Gnann
研究了一类具有线性迁移率的薄膜方程在完全润湿条件下的紧支撑解。Carrillo和Toscani已经解决了这个问题，证明了源型自相似轮廓是具有紧密支持的初始数据的熵解的全局吸引子。在这里，我们研究了相应的经典自由边界问题的源型自相似解的小扰动，并在极小假设下建立了加权$L^2-空间中的整体存在唯一性理论。此外，我们还得到了解、自由边界和质心的演化的渐近性。随着空间平移在我们的参照系中向外扩展，收敛速度比Carrillo和Toscani获得的速度更快。
We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted $L^2$-spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.