We prove the local exponential stabilizability for the MHD system in space dimension 2, with internally distributed feedback controllers. These controllers take values in a finite dimensional space which is the unstable manifold of the elliptic part of the linearized operator. They are represented through two scalar functions supported in a subdomain. Mathematics Subject Classification 2000: 93...

The stability of travelling wave solutions of singularly perturbed, diffusive predator-prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at k 0 is simple. The proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues.

This contribution is devoted to the Oseen equations, a linearized form of the Navier-Stokes equations. We give here some results concerning the scalar Oseen operator and we prove Hardy inequalities concerning functions in Sobolev spaces with anisotropic weights that appear in the investigation of the Oseen equations. Mathematics Subject Classification (2000). 76D05, 35Q30, 26D15, 46D35.

We continue our study of the inverse scattering problem for diffuse light. In particular, we derive inversion formulas for this problem that are based on the functional singular-value decomposition of the linearized forward-scattering operator in the slab, cylindrical, and spherical geometries. Computer simulations are used to illustrate our results in model systems.

We show how to stabilize the uniform oscillations of the complex Ginzburg–Landau equation with periodic boundary conditions by means of some global delayed feedback. The proof is based on an abstract pseudo-linearization principle and a careful study of the spectrum of the linearized operator.

We present an approach for instabilities of solitons that is based on the spectrum of a fourth-order linearized operator. Unlike the standard approach which is based on the slope (Vakhitov-Kolokolov) condition, this approach provides the quantitative value of the instability rate and the qualitative nature of the instability dynamics.

We establish sharp L2-Sobolev estimates for classes of pseudodifferential operators with singular symbols [27, 20] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in R2 with simple inflection points and normal operators arising in linearized seismic imaging in the ...

Intrinsic fluctuations around the solution of the lattice Boltzmann equation are described or modeled by addition of a white Gaussian noise source. For stationary states a fluctuation-dissipation theorem relates the variance of the fluctuations to the linearized Boltzmann collision operator and the pair correlation function.

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to per...