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Desingularization of vortex rings and shallow water vortices by a semilinear elliptic problem. (English) Zbl 1294.35083
Summary: Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problem \[ \begin{cases}\begin{alignedat}{2} -\text{div} \left(\frac{\nabla u_{\varepsilon}}{b}\right) & = \frac{1}{\varepsilon^2} bf \bigl(u_{\varepsilon} - \log \frac{1}{\varepsilon} q \bigr), \quad && \text{ in } \Omega, \\ u_\varepsilon & = 0 && \text{ on } \partial\Omega, \end{alignedat}\end{cases} \] for small values of \(\varepsilon > 0\).

MSC:
35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35J61 Semilinear elliptic equations
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