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| In the absence of surface tension, severe difficulties arise when trying to solve the stationary free surface equations for a Navier-Stokes flow, in particular when the free surface is supposed to be a compact, connected hypersurface in $\mathbf{R}^{3}$.
Let $\rho$ be a function defining a graph above the unit sphere $S_{0}$ (or any reasonable, smooth, sphere-like surface $\Sigma_{0}$), so that the domain $\Omega_{\rho}$ occupied by the fluid is just the interior of a smooth, compact and connected surface. In this talk, I will focus on a local approach, assuming that there exists a parameter (e.g. a force term, an angular velocity...) such that, for a given parameter value, there exists a particular solution $\rho_{0} \equiv 0$ to the free surface problem. The question of interest is then the existence of a solution in a neighbourhood of $\rho_{0}$.
The main object will be the linearized free surface operator $\mathcal{L}$, and the main question is that of the invertibility of $\mathcal{L}$. Unfortunately, the method introduced in a previous work (Abergel, Bailly) fails in this context, as the vectorfield $u$ has to vanish at some points (Poincar\'{e}-Bendixon's theorem). In the light of results by Holcman and Kupka, I will give some sufficient conditions ensuring that the linearized operator $\mathcal{L}$ is invertible and hence, that the free surface problem under scrutiny has solutions. |
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