Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation

Abstract

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.Since the boundary is a function of time t, we need a condition determining its evolution which is (1.1c). Viewing (1.1a) as a nonlinear continuity equation, we read off the horizontal fluid velocity as v = ∂ 3 z h. By compatibility, this velocity has to match the speed of the contact lines at the free boundaries. A simple calculation shows that (1.1c) implies conservation of mass (volume) M :=We mention that problem (1.1) can be derived by a lubrication approximation from Darcy's flow in the Hele-Shaw cell [18,21,22]. Equation (1.1a) is a specific case of a larger class of thin-film equations,with a nonlinear mobility h n and where n ∈ (0, 3). Again, (1.2) can be derived from the Navier-Stokes system with an in general nonlinear slip condition at the liquid-solid interface by an asymptotic expansion [8,13,29]. Source-type self-similar solutions.It is instructive to study the case where the initial condition comes from a Dirac mass M δ 0 at time t = −1, where M is the mass of the droplet. Additionally we assume that the solution to (1.1) is self-similar. We note that (1.1a) remains invariant under the two-parameter scaling transformation if we additionally assume conservation of mass, i.e., H * Z * = 1. Hence our source-type self-similar solutions attain the structureThis ansatz automatically yields films for which the mass is constant in time t. We insert ansatz (1.3) in (1.1a) and obtain after some basic manipulations (cf. Appendix A)which is known as the Smyth-Hill profile [34]. Convergence to source-type self-similar solutions.It is known that the long-time dynamics of (1.1) for rather general initial data is governed by the sourcetype self-similar solution (1.4). A first observation supporting this insight was obtained by Bernis [3,4] through the study of the Cauchy problem for the thin-film equation in the 1 + 1-dimensional case. Bernis was able to show that the speed of propagation of the interface is finite and the spreading rate asymptotically matches the one of the source-type solution. For the case of mobility exponent n = 1 (the Darcy flow in the Hele-Shaw cell), a stronger result by Carrillo and Toscani is available [12]: Using analogies to the second-order counterpart of (1.1a), the porous medium equatio...

Null-controllability of perturbed porous medium gas flow

“…Embeddings for weighted Sobolev spaces. The following two useful lemmas are taken from the work of Gnann [18]. For the sake of completeness, we provide short proofs in Appendix B.…”

Pressure-dipole solutions of the thin-film equation

“…In earlier sections, we have analytically and numerically constructed classes of self-similar solutions of (1.3). In general, such solutions of parabolic PDEs are of great importance when they can be shown to be stable attracting states that capture the long-time dynamics of solutions [37] starting from broad classes of initial data. A linear stability analysis of the self-similar solutions (see for example [13,14,60]) can yield valuable insight into the solution dynamics near these states.…”

Global existence for a thin film equation with subcritical mass

“…One of the most important properties for (12) is finite speed of propagation of the support of solutions. This problem can also be regarded as a free-boundary problem and we refer to Giacomelli, Gnann, Knüpfer and Otto [17,18,19,33], and Mellet [30] for in-depth studies of this problem. The existence of weak solutions for this degenerate parabolic fourth order free boundary problem was proved when n = 1 [33].…”

“…Before defining weak solutions, we need to review literatures on possible singularities that may appear to the higher derivatives of weak solutions. For the thin film equation without the long-wave unstable term, Giacomelli, Knüpfer, and Otto [18], John [23], and Gnann [19] showed that for n = 1 solutions are smooth up to the free boundary. However, for the thin film equation with the long-wave unstable term, the control of solutions at the free boundary is subtle and we do not know if solutions have singularities or not.…”