62,599 research outputs found
Unstable and Stable Galaxy Models
To determine the stability and instability of a given steady galaxy
configuration is one of the fundamental problems in the Vlasov theory for
galaxy dynamics. In this article, we study the stability of isotropic spherical
symmetric galaxy models , for which the distribution function
depends on the particle energy only. In the first part of the article, we
derive the first sufficient criterion for linear instability of
is linearly unstable if the second-order operator has a
negative direction, where is the projection onto the function
space being the angular momentum [see the explicit formula
(\ref{A0-radial})]. In the second part of the article, we prove that for the
important King model, the corresponding is positive definite. Such a
positivity leads to the nonlinear stability of the King model under all
spherically symmetric perturbations.Comment: to appear in Comm. Math. Phy
Compressible, inviscid Rayleigh-Taylor instability
We consider the Rayleigh-Taylor problem for two compressible, immiscible,
inviscid, barotropic fluids evolving with a free interface in the presence of a
uniform gravitational field. After constructing Rayleigh-Taylor steady-state
solutions with a denser fluid lying above the free interface with the second
fluid, we turn to an analysis of the equations obtained from linearizing around
such a steady state. By a natural variational approach, we construct normal
mode solutions that grow exponentially in time with rate like e^{t
\sqrt{\abs{\xi}}}, where is the spatial frequency of the normal mode. A
Fourier synthesis of these normal mode solutions allows us to construct
solutions that grow arbitrarily quickly in the Sobolev space , which leads
to an ill-posedness result for the linearized problem. Using these pathological
solutions, we then demonstrate ill-posedness for the original non-linear
problem in an appropriate sense. More precisely, we use a contradiction
argument to show that the non-linear problem does not admit reasonable
estimates of solutions for small time in terms of the initial data.Comment: 31 pages; v2: updated grant informatio
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