Abstract. In this paper, we prove the decay estimates of the velocity and H 1 scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960.

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

In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of ν 2 3 + , here ν is the viscosity coefficient.Here P 2 denotes the orthogonal projection of L 2 (T 2 δ ) to the subspace W 2 = span{sin y, cos y}.Remark 1.7. The stability threshold ν γ for γ > 2 3 may be not optimal. We can only achieve the stability threshold ν 3 4 if we do not use the enhanced dissipation with an extra decay factor of the velocity.Remark 1.8. The case of δ = 1 is a challenging problem. In this case, we need to consider the linearized Navier-Stokes equations around the dipole states such as e −νt (− sin y, sin x), e −νt (− cos y, cos x).

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.

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This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10]. As a corollary, we also prove the global regularity under the assumption that |ru θ (r, z, t)| ≤ | ln r| −3/2 , ∀ 0 < r ≤ δ 0 ∈ (0, 1/2).

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