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Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation. (English) Zbl 1154.35072

The authors prove the existence of nodal solutions for the sinh-Poisson equation with Dirichlet boundary conditions. The aim is to construct Mallier-Maslowe-type solutions for the 2D stationary Euler equations in vorticity form satisfying null-flux boundary conditions on any 2D domain with appropriate properties. They show that there exist at least two pairs of solutions which change sign exactly once, whose nodal lines intersect on the boundary. The proof of the results is based on a Lyapunov-Schmidt reduction as in [M. del Pino, M. Kowalczyk and M. Musso, Calc. Var. Partial Differ. Equ. 24, No. 1, 47–81 (2005; Zbl 1088.35067)].

MSC:

35Q35 PDEs in connection with fluid mechanics
76F35 Convective turbulence
76F50 Compressibility effects in turbulence

Citations:

Zbl 1088.35067
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