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Study of an obstacle vortex free boundary problem. (English) Zbl 0904.35100

The problem here considered consists in looking for an open convex subset \(\Omega\) of \(C\) including the origin, and such that
(1) \(\text{Im }x>-a\), \(\nabla z\in\Omega\), with \(a\) given as a positive real constant,
(2) \(\Omega\) is symmetric w.r.t. the imaginary axis,
(3) the portion \(\Gamma\) of \(\partial\Omega\) lying in the half plane \(\text{Im }z>-a\) is a smooth Jordan arc.
Moreover, it is required that the holomorphic function \(F\in H(\Omega)\) characterized by \[ \text{Im }F(z)= \log| z|,\quad z\in\partial\Omega,\quad \text{Re }F(0)= 0,\tag{4} \] satisfies \[ \Biggl| F'(z)-{i\over z}\Biggr|^2+ 2g\text{ Im }z= \text{constant},\quad z\in\Gamma,\tag{5} \] for a given nonnegative real constant \(g\), and \[ F'(ib)= {1-b\over b},\quad b=\sup\{y\in\mathbb{R}: iy\in\Omega\}.\tag{6} \] The constant at the right-hand side of (5) is not specified.
The physical interpretation of this free boundary problem is the study of a plane incompressible flow with a vertex type singularity at the origin and in the presence of an obstacle. In the scheme above \(\Omega\) is the region occupied by the fluid, the obstacle is represented by (1). \(\Gamma\) is a streamline and the function \(\Psi(z)= \text{Im }z- \log| z|\) is the stream function. The constant \(g\) represents gravity and (5) is nothing but Bernoulli’s law on \(\Gamma\).
The unconstrained problem has been studied in previous papers. Here, the authors prove an existence and uniqueness theorem (for \(g\) small enough) extending the techniques used for the obstacle free problem. First \(\overline\Omega\) is transformed in the closure of the unit disc by means of the conformal mapping \(\Lambda(z)= -iz\exp(iF(z))\), then the problem is reformulated in a weak form as fixed point problem for a suitable operator.
Reviewer: A.Fasano (Firenze)

MSC:

35R35 Free boundary problems for PDEs
35Q35 PDEs in connection with fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
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