29 research outputs found
Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity
In this paper, a compressible viscous-dispersive Euler system in one space
dimension in the context of quantum hydrodynamics is considered. The purpose of
this study is twofold. First, it is shown that the system is locally
well-posed. For that purpose, the existence of classical solutions which are
perturbation of constant states is established. Second, it is proved that in
the particular case of subsonic equilibrium states, sufficiently small
perturbations decay globally in time. In order to prove this stability
property, the linearized system around the subsonic state is examined. Using an
appropriately constructed compensating matrix symbol in the Fourier space, it
is proved that solutions to the linear system decay globally in time,
underlying a dissipative mechanism of regularity gain type. These linear decay
estimates, together with the local existence result, imply the global existence
and the decay of perturbations to constant subsonic equilibrium states as
solutions to the full nonlinear system.Comment: 42 page
Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations
This paper establishes the spectral stability of monotone traveling front
solutions for reaction-diffusion equations where the reaction function is of
Nagumo (or bistable) type and with diffusivities which are density dependent
and degenerate at zero (one of the equilibrium points of the reaction).
Spectral stability is understood as the property that the spectrum of the
linearized operator around the wave, acting on an exponentially weighted space,
is contained in the complex half plane with non-positive real part. Three
different types of monotone waves are studied: (i) stationary
diffusion-degenerate fronts, connecting the two stable equilibria of the
reaction; (ii) traveling diffusion-degenerate fronts connecting zero with the
unstable equilibrium; and, (iii) non-degenerate fronts. In the first two cases,
the degeneracy is responsible of the loss of hyperbolicity of the asymptotic
coefficient matrices of the spectral problem at one of the end points,
precluding the application of standard techniques to locate the essential
spectrum. This difficulty is overcome with a suitable partition of the
spectrum, a generalized convergence of operators technique, the analysis of
singular (or Weyl) sequences and the use of energy estimates. The monotonicity
of the fronts, as well as detailed descriptions of the decay structure of
eigenfunctions on a case by case basis, are key ingredients to show that all
traveling fronts under consideration are spectrally stable in a suitably chosen
exponentially weighted energy space.Comment: 53 pages, 8 figures, 1 tabl
On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations
In this contribution, we summarize recent results [8, 9] on the stability analysis of periodicwavetrains for the sine-Gordon and general nonlinearKlein-Gordon equations. Stability is considered both from the point of view of spectral analysis of the linearized problem and from the point of view of the formal modulation theory of Whitham [12]. The connection between these two approaches is made through a modulational instability index [9], which arises from a detailed analysis of the Floquet spectrum of the linearized perturbation equation around the wave near the origin. We analyze waves of both subluminal and superluminal propagation velocities, as well as waves of both librational and rotational types. Our general results imply in particular that for the sine-Gordon case only subluminal rotationalwaves are spectrally stable. Our proof of this fact corrects a frequently cited one given by Scott [11]