05318cam  22005898i 450000100090000000300050000900500170001400600190003100700150005000800410006502000270010604000350013305000260016808400570019408200180025110000390026924501710030826300090047926400610048830000340054933600260058333700280060933800270063749000740066450000440073850400410078250506130082350600500143652015840148653300950307053800360316558800470320165000390324865000150328765000160330265000190331865000120333765000240334965000320337365000930340565001460349865001430364465001690378765001560395665001160411265001720422870000370440070000370443777601610447485600450463585600480468021655465RPAM20210409154632.0m      b    000 0 cr/|||||||||||210409s2020    riu     ob    000 0 eng    a9781470462512 (online)  aLBSOR/DLCbengerdacDLCdRPAM00aTA357.5.V56bB43 2020  a35B35a76E05a76E30a76F06a76F10a35B40a76F252msc00a620.1/0642231 aBedrossian, Jacob,d1984-eauthor.10aDynamics near the subcritical transition of the 3D Couette flow I :h[electronic resource] bbelow threshold case /cJacob Bedrossian, Pierre Germain, Nader Masmoudi.  a2010 1aProvidence, RI :bAmerican Mathematical Society,c[2020]  a1 online resource (pages cm.)  atextbtxt2rdacontent  aunmediatedbn2rdamedia  avolumebnc2rdacarrier0 aMemoirs of the American Mathematical Society, x1947-6221 ; vv. 1294  a"Forthcoming, volume 266, number 1294."  aIncludes bibliographical references.00tChapter 1. IntroductiontChapter 2. Outline of the prooftChapter 3. Regularization and continuationtChapter 4. High norm estimate on $Q^2$tChapter 5. High norm estimate on $Q^3$tChapter 6. High norm estimate on $Q^1_0$tChapter 7. High norm estimate on $Q^1_\neq $tChapter 8. Coordinate system controlstChapter 9. Enhanced dissipation estimatestChapter 10. Sobolev estimatestAcknowledgmentstAppendix A. Fourier analysis conventions, elementary inequalities, and Gevrey spacestAppendix B. Definition and analysis of normstAppendix C. Multiplier and paraproduct toolstAppendix D. Elliptic estimates1 aAccess is restricted to licensed institutions  a"We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. We prove that for sufficiently regular initial data of size [epsilon] [less than or equal to] c0Re-1 for some universal c0 > 0, the solution is global, remains within O(c0) of the Couette flow in L2, and returns to the Couette flow as t [right arrow] [infinity]. For times t >/-Re1/3, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks. Our analysis contains perturbations that experience a transient growth of kinetic energy from O(Re-1) to O(c0) due to the algebraic linear instability known as the lift-up effect. Furthermore, solutions can exhibit a direct cascade of energy to small scales. The behavior is very different from the 2D Couette flow, in which stability is independent of Re, enstrophy experiences a direct cascade, and inviscid damping is dominant (resulting in a kind of inverse energy cascade). In 3D, inviscid damping will play a role on one component of the velocity, but the primary stability mechanism is the mixing-enhanced dissipation. Central to the proof is a detailed analysis of the interplay between the stabilizing effects of the mixing and enhanced dissipation and the destabilizing effects of the lift-up effect, vortex stretching, and weakly nonlinear instabilities connected to the non-normal nature of the linearization"--cProvided by publisher.  aElectronic reproduction.bProvidence, Rhode Island :cAmerican Mathematical Society.d2020  aMode of access : World Wide Web  aDescription based on print version record. 0aViscous flowxMathematical models. 0aStability. 0aShear flow. 0aInviscid flow. 0aMixing. 0aDamping (Mechanics) 0aThree-dimensional modeling. 7aPartial differential equations -- Qualitative properties of solutions -- Stability.2msc 7aFluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Hydrodynamic stability -- Parallel shear flows.2msc 7aFluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Hydrodynamic stability -- Nonlinear effects.2msc 7aFluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Turbulence [See also 37-XX, 60Gxx, 60Jxx] -- Transition to turbulence.2msc 7aFluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Turbulence [See also 37-XX, 60Gxx, 60Jxx] -- Shear flows.2msc 7aPartial differential equations -- Qualitative properties of solutions -- Asymptotic behavior of solutions.2msc 7aFluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} -- Turbulence [See also 37-XX, 60Gxx, 60Jxx] -- Turbulent transport, mixing.2msc1 aGermain, Pierre,d1979-eauthor.1 aMasmoudi, Nader,d1974-eauthor.0 iPrint version: aBedrossian, Jacob, 1984-tDynamics near the subcritical transition of the 3D Couette flow I :w(DLC)   2020032339x0065-9266z97814704421704 3Contentsuhttps://www.ams.org/memo/1294/4 3Contentsuhttps://doi.org/10.1090/memo/1294