Blow-up behavior for a degenerate elliptic $$\sinh $$ -Poisson equation with variable intensities
Tóm tắt
In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure
$$\mathcal P$$
defined on the interval
$$[-1,1]$$
. Under Dirichlet boundary conditions we establish the exclusion of boundary blow-up points, we show that the concentration mass does not have residual
$$L^1$$
-terms (“residual vanishing”) and we determine the location of blow-up points in terms of Kirchhoff’s Hamiltonian. We allow
$$\mathcal P$$
to be a general Borel measure, which could be “degenerate” in the sense that
$$\mathcal P(\{\alpha _-^*\})=0=\mathcal P(\{\alpha _+^*\})$$
, where
$$\alpha _-^*=\min \mathrm {supp}\mathcal P$$
and
$$\alpha _+^*=\max \mathrm {supp}\mathcal P$$
. Our main results are new for the standard sinh-Poisson equation as well.
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