Laplacian增长的多割解
Multi-Cuts Solutions of Laplacian Growth
摘要：
提出了一类新的零表面张力拉普拉斯增长的解，并证明了它包含所有其他已知解作为特殊或极限情况。这些解决方案，这是随时间变化的共形映射与分支切割内的单位圆，由一个非线性积分方程和描述油峡湾的非平行壁在Hele-Shaw细胞的粘性指进实验。运动积分的多切割拉普拉斯增长的解决方案的奇异性的Schwarz函数的发现，并推导出动力学的密度（跳跃）的削减。这些解的子类在Schwarz函数的截集上具有线性柯西密度，这是特别有趣的，因为在这种情况下，共形映射的积分方程变成线性的。这些解决方案也可以是物理上的重要性，通过代表油/空气界面，形成油峡湾与一个恒定的开口角，根据最近的实验中的Hele-shaw细胞。
英文摘要：
A new class of solutions to Laplacian growth with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut Laplacian growth solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.
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