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On uniqueness of weak solutions for the thin-film equation. (English) Zbl 1322.35084
The existence of non-negative weak solutions to the thin film equation (TFE) \[ \partial_t h + \mathrm{div} \left( h \nabla\Delta h \right) = 0 \quad\text{ in }\quad (0,\infty)\times\mathbb{R}^n \] is now well-known but uniqueness is a more delicate issue, partly due to the degenerate diffusion when \(h\) vanishes and the lack of a comparison principle. It is shown herein that, if the initial condition \(h_0\) is sufficiently close to the stationary solution \(h_{\mathrm{st}}: y=(y_1,\ldots,y_n) \mapsto (y_n)_+^2\), then the thin film equation has a unique solution in a suitable class which stays close to \(h_{\mathrm{st}}\) in the sense that \(\sup\{ |\nabla\sqrt{h(s,x)} - e_n|\;:\;x\in\mathcal{P}(s)\}\) is small for all times \(s\geq 0\). Here, \(e_n\) is the unit vector corresponding to the \(y_n\)-axis and \(\mathcal{P}(s)\) denotes the positivity set of \(h(s)\) at time \(s\geq 0\). A by-product of the analysis is the analyticity of the level sets of the solution.
The proof relies on the expected monotonicity with respect to \(y_n\) of solutions close to \(h_{\mathrm{st}}\) which allows the author to use the so-called von Mises transformation \[ \sqrt{h}(s,y',v(s,y',z)) = z\;, \quad y'=(y_1,\ldots,y_{n-1})\in \mathbb{R}^{n-1}\;, \quad z>0\;. \] Introducing \(u(t,x) = v(t,x) - x_n\) for \(t>0\) and \(x=(x_1,\ldots,x_n) \in H = \mathbb{R}^{n-1}\times (0,\infty)\), the function \(v\) solves a semilinear equation of the form \[ \partial_t u + Lu = f(u) \quad\text{ in }\quad (0,\infty)\times H\;, \] where \(L\) is the non-uniform parabolic operator \[ Lu(x) = \frac{1}{x_n} \Delta \left( x_n^3 \Delta u(x) \right) - 4 \Delta_{x'} u(x)\;. \] A suitable functional framework is identified in which the linear evolution equation associated to \(L\) is well-posed. Various regularity estimates and a representation formula featuring the Green’s function are then derived. A fixed point argument is then used to construct a unique solution \(u\) to the nonlinear equation for small initial data.

MSC:
35K65 Degenerate parabolic equations
35K46 Initial value problems for higher-order parabolic systems
35B20 Perturbations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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