Diffusion and mixing in fluid flow via the resolvent estimate

Abstract

In this paper, we first present a Gearhardt-Prüss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales.Gearhart-Prüss theorem gives the semigroup bound via the resolvent estimate.Theorem 1.2 (Gearhart-Prüss theorem). Let H be a closed operator with a dense domain D(H) generating a strongly continuous semigroup e tH . Assume that (z −H) −1 is uniformly bounded for Re z ≥ ω. Then there exists M > 0 so that e tH satisfies P (M, ω).Let us refer to [7] for more introductions.Recently, Helffer and Sjöstrand presented a quantitative version of Gearhart-Prüss theorem and gave some interesting applications to complex Airy operator, complex harmonic oscillator and Fokker-Planck operator [10]. Motivated by their work, we first present a Gearhart-Prüss type theorem with a sharp bound for m-accretive operators. A closed operator H in a Hilbert

On Pseudospectral Bound for Non-selfadjoint Operators and Its Application to Stability of Kolmogorov Flows

“…Recently Wei [27] obtained a refined version of the Gearhart-Prüss theorem for semigroup in terms of the pseudospectral bound. If one applies this general result of [27] then the semigroup estimate (3.150) is a direct consequence of the pseudospecral bound sup…”

Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

“…The estimate of the semigroup is a consequence of the pseudospectral bound obtained in the previous subsection and the Gearhard-Prüss type theorem by Wei [25], stated in Theorem A1. Let us recall that the semigroup considered here is {e −tA(ξ) } t≥0 in X ξ/|ξ|,|ξ| , where A(ξ) is defined by (14).…”

“…The operator A(rω) = irA(ω) + L(ω) becomes m-accretive, which enables us to obtain the semigroup bound directly from the pseudospectral bound, the resolvent estimate with resolvent parameters only along the imaginary axis, in virtue of the Gearhart-Püss type theorem by Wei [25] (see also [26]) (see Theorem A1). The pseudospectral bound was discussed by Gallagher, Gallay, and Nier [27], who studied the harmonic oscillator with some class of large skew-symmetric perturbations, which was also discussed, for example, by Li, Wei, and Zhang [28] and and Ibrahim, Maekawa, and Masmoudi [29] in order to study the semigroup estimate for the linearization around the stationary flows for the Navier-Stokes equations such as the Burgers vortex and the Kolmogorov flow.…”

Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

“…For the accretive operator H , the second author in [35] derives the following sharp semigroup bound from the pseudospectra ‰.H /: ke tH k Ä e t ‰.H /C =2 : Based on the pseudospectra bounds (2.1), in Proposition 4.4 and Proposition 4.5, we will show that…”

“…To obtain the semigroup bound from the pseudospectral bound, we use the following Gearhart-Prüss-type lemma with sharp bound from [35]. See also [17] for a slightly weaker version.…”