9,995 research outputs found
Nonlinear stability of mKdV breathers
Breather solutions of the modified Korteweg-de Vries equation are shown to be
globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov
functional, at the H^2 level, which allows to describe the dynamics of small
perturbations, including oscillations induced by the periodicity of the
solution, as well as a direct control of the corresponding instability modes.
In particular, degenerate directions are controlled using low-regularity
conservation laws.Comment: 24 pp., submitte
A non-variational approach to nonlinear stability in stellar dynamics applied to the King model
In previous work by Y. Guo and G. Rein, nonlinear stability of equilibria in
stellar dynamics, i.e., of steady states of the Vlasov-Poisson system, was
accessed by variational techniques. Here we propose a different,
non-variational technique and use it to prove nonlinear stability of the King
model against a class of spherically symmetric, dynamically accessible
perturbations. This model is very important in astrophysics and was out of
reach of the previous techniques
Nonlinear stability of viscous roll waves
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic
conservation laws, we show that spectral stability implies nonlinear stability
for spatially periodic viscous roll wave solutions of the one-dimensional St.
Venant equations for shallow water flow down an inclined ramp. The main new
issues to be overcome are incomplete parabolicity and the nonconservative form
of the equations, which leads to undifferentiated quadratic source terms that
cannot be handled using the estimates of the conservative case. The first is
resolved by treating the equations in the more favorable Lagrangian
coordinates, for which one can obtain large-amplitude nonlinear damping
estimates similar to those carried out by Mascia--Zumbrun in the related shock
wave case, assuming only symmetrizability of the hyperbolic part. The second is
resolved by the observation that, similarly as in the relaxation and detonation
cases, sources occurring in nonconservative components experience greater than
expected decay, comparable to that experienced by a differentiated source.Comment: Revision includes new appendix containing full proof of nonlinear
damping estimate. Minor mathematical typos fixed throughout, and more
complete connection to Whitham averaged system added. 42 page
Isotropic steady states in galactic dynamics revised
The present paper completes our earlier results on nonlinear stability of
stationary solutions of the Vlasov-Poisson system in the stellar dynamics case.
By minimizing the energy under a mass-Casimir constraint we construct a large
class of isotropic, spherically symmetric steady states and prove their
nonlinear stability against general, i. e., not necessarily symmetric
perturbations. The class is optimal in a certain sense, in particular, it
includes all polytropes of finite mass with decreasing dependence on the
particle energy.Comment: 31 pages, LaTe
Linear and nonlinear stability of heat exchangers
The hydrodynamical problem of one-dimensional flow with a uniform heat input resulting in a change of phase is considered. Equations of mass, momentum, energy and state representing the dynamic behaviour of such a system are reduced to two coupled equations for the density p{x, t) and the inlet velocity 1/(0 on the assumption that the pressure drop applied between the inlet and the outlet is "small". A linear stability analysis is carried out which leads to the problem of computing the zeros of a complicated analytic function. A non-linear analysis is applied to the case of weak instability to find the evolution of the slowly varying amplitude of a small oscillation: in certain circumstances, a "burst " occurs, and in such cases no such small oscillation can exist. 1
On the nonlinear stability of mKdV breathers
A mathematical proof for the stability of mKdV breathers is announced. This
proof involves the existence of a nonlinear equation satisfied by all breather
profiles, and a new Lyapunov functional which controls the dynamics of small
perturbations and instability modes. In order to construct such a functional,
we work in a subspace of the energy one. However, our proof introduces new
ideas in order to attack the corresponding stability problem in the energy
space. Some remarks about the sine-Gordon case are also considered.Comment: 7 p
- …