The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains

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.

Resolvent Estimates of the Dirac Operator

Degenerated codimension 1 crossings and resolvent estimates

In this article, we analyze the propagation of Wigner measures of a family of solutions to a system of semi-classical pseudodifferential equations presenting eigenvalues crossings on hypersurfaces. We prove the propagation along classical trajectories under a geometric condition which is satisfied for example as soon as the Hamiltonian vector fields are transverse or tangent at finite order to ...

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 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...