Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

Zhang

2020

Comm Pure Appl Math

Self Cite

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v 0 satisfies kv 0 .y; 0; 0/k H 2 c 0 Re 1 for some c 0 > 0 independent of Re, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen et al. in 1993. Moreover, we prove that the long-time dynamics of the solution behaves as: for t . Re, the solution will experience a transient growth from O.Re 1 / to O.c 0 / due to the 3D lift-up effect; for t & Re 1=3 , the solution will rapidly converge to a streak solution due to the mixing-enhanced dissipation effect.

Section: G T/mentioning

confidence: 87%

“…2 for nonmonotone flows such as Kolmogorov flow and Poiseuille flow. Following the semigroup method in [22], we can prove the threshold 5 3 for the 3D Couette flow. To improve this threshold, new ideas are needed.…”

Section: G T/mentioning

confidence: 99%

Transition Threshold for the 3D Couette Flow in Sobolev Space

Zhang

2020

Comm Pure Appl Math

Self Cite

“…Let us also mention some important progress on the enhanced dissipation of the linearized Navier-Stokes equations around shear flows such as Couette flow and Kolmogorov flow [1,3,9,11,14,16,17].…”

Section: Introductionmentioning

confidence: 99%

Diffusion and mixing in fluid flow via the resolvent estimate

2019

Sci. China Math.

Self Cite

107

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

“…• Two-dimensional Couette flow in a finite channel: If X is taken as a Sobolev space, then D 1 2 gives stability ( [6,13,14,18]). • Other shear flows: See [16,20,21,30,31,43].…”

Section: Introductionmentioning

confidence: 99%

Stability threshold of two-dimensional Couette flow in Sobolev spaces

Masmoudi

Ann. Inst. H. Poincaré Anal. Non Linéaire

2022

We study the stability threshold of the two-dimensional Couette flow in Sobolev spaces at high Reynolds number Re. We prove that if the initial vorticity in satisfies k in . 1/k H Ä "Re 1=3 , then the solution of the two-dimensional Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time tRe 1=3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t ! C1.