On the stability of single and multiple droplets for equations of thin film type

Abstract

Some general results on the energy stability/instability of droplets with zero contact angle for thin film type equations including those governed by the power laws are established. Energy instability of droplets with nonzero contact angle and configurations of droplets is also obtained. As an application, an asymptotic stability result for droplets with zero contact angle is established.

Convergence to Equilibrium for a Thin-Film Equation on a Cylindrical Surface

“…For steady states with dry regions, the entropy is not even finite. We combine Lyapunov's method with a linear bound on the growth of the entropy to produce a sequence of times along which the solution converges weakly to a steady state, using a recent argument of [12]. We pass to convergence in norm along the full solution by proving a local coercivity estimate on the energy near the minimizer (which becomes global for α ≤ 1).…”

A symmetry result for an elliptic problem arising from the 2-D thin film equation

“…A proof of this proposition in the one dimensional case, based on an entropy estimate ( [3]), could be found in [9]. It works in the two dimensional case after some minor changes.…”

“…As seen from linearization, the constant states will lose their stability in the so-called long wave situation. The classification and stability of the steady states for the 1-D thin film equation have been studied in [18]- [21], [9], [11]. The notion of energy stability has turned out to be useful in these studies.…”

“…When u x (a) = u x (b) = 0, u is called a droplet with zero contact angle. In [9] it is shown that in many situations droplets with zero contact angle are energy stable but other droplets are not energy stable.…”

Stability and instability of axisymmetric droplets in thermocapillary-driven thin films

“…Laugesen and Pugh performed linear stability analyses of strictly positive periodic steady states and found instabilities when m < n or m > n + 1 (or more generally, when W (4) (H) < 0), and states were observed to numerically evolve towards ruptured arrays of droplets [27][28][29]. Recently, Cheung and Chou performed a stability analysis of droplet states for equation (10) and found stability of isolated droplets with zero contact slope for n − 2 < m < n + 2 but instability for m > n + 3 [30].…”