We consider a 2-dimensional thin domain with order of thickness ε which presents oscillations of amplitude also ε on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and at the bottom. We study the behavior of the Laplace operator with Neumann boundary condition and obtain its asymptotic homogenized limit as the parameter ε goes to 0. We are interested in understanding how this different oscillatory behavior at the boundary, influences the limit problem.
The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions
Fundamental global similarity solutions of the standard form u_ \g (x,t)=t^{- \a_\ g} f_ \g (y), with the rescaled variable y= x/{t^{ \b_\ g}}, \b_\ g= \frac {1-n \a_\ g}{10}, where \a_\ g>0 are real nonlinear eigenvalues ( \g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot ( |u| ^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-simila…
The Formation and Coarsening of the Concertina Pattern
手风琴是一种磁化模式在细长的薄膜元件的软材料。它是在小元件的长轴方向上的磁化反转过程中出现的普遍存在的磁畴图案。Van den Berg认为，当外场减少时，这种模式从通量闭合域中生长出来。基于实验观察和理论，我们认为，在足够长的薄膜元件，手风琴图案，而从振荡屈曲模式分叉。使用一个简化的模型，通过渐近分析和数值模拟研究，我们定量预测的平均周期的手风琴图案和定性预测其滞后。特别是，我们认为，实验观察到的粗化的手风琴图案是由于二次分叉相关的埃克豪斯不稳定性。我们还链接的手风琴图案的磁化波纹，并讨论了弱（结晶或诱导）各向异性的影响。
