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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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