Resolvent estimates with mild trapping

Resolvent Estimates with Mild Trapping

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.

Semiclassical Resolvent Estimates for Trapping Perturbations

We study the semiclassical estimates of the resolvent R(+ ii); 2 J R + ; 2]0;1] of a self-adjoint operator L(h) in the space of bounded operators L(H 0;s ; H 0;?s); s > 1=2. In the general case of long-range trapping "black-box" perturbations we prove that the estimate of the cut-oo resolvent k(x)R(+i0)(x)k H!H C exp(Ch ?p); (x) 2 C 1 0 (R n); p 1 implies the estimate kR(+ ii)k s;?s C 1 exp(C 1...

Resolvent Estimates of the Dirac Operator

Resolvent estimates for non-selfadjoint operators with double characteristics

We present recent progress in the understanding of the spectral and subelliptic properties of non-elliptic quadratic operators with application to the study of return to equilibrium for some systems of chains of oscillators. We then explain how these results allow to describe the spectral properties and to give sharp resolvent estimates for some classes of non-selfadjoint pseudodi erential oper...

Resolvent Estimates for Plane Couette Flow

We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three dimensional perturbations of plane Couette flow, and determining its dependence on the Reynolds number R. Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that deriving estimates for the resolvent ca...