31 research outputs found
Regularization of point vortices for the Euler equation in dimension two
In this paper, we construct stationary classical solutions of the
incompressible Euler equation approximating singular stationary solutions of
this equation.
This procedure is carried out by constructing solutions to the following
elliptic problem [ -\ep^2 \Delta
u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad
& x\in\partial\Omega, ] where , is a bounded
domain, is a harmonic function.
We showed that if is simply-connected smooth domain, then for any
given non-degenerate critical point of Kirchhoff-Routh function
with the same strength , there is a
stationary classical solution approximating stationary points vortex
solution of incompressible Euler equations with vorticity .
Existence and asymptotic behavior of single point non-vanishing vortex
solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page
Desingularization of vortices for the Euler equation
We study the existence of stationary classical solutions of the
incompressible Euler equation in the plane that approximate singular
stationnary solutions of this equation. The construction is performed by
studying the asymptotics of equation -\eps^2 \Delta
u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet
boundary conditions and a given function. We also study the
desingularization of pairs of vortices by minimal energy nodal solutions and
the desingularization of rotating vortices.Comment: 40 page