We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Prüss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate $O(e^{-\sqrtν\,t})$ when the viscosity coefficient $ν$ is sufficiently small as in the case of the plane Kolmogorov flow.
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…
We consider here the Navier-Stokes equations in $\mathbb{R}^{3}$ with a stationary, divergence-free external force and with an additional damping term that depends on two parameters. We first study the well-posedness of weak solutions for these equations and then, for a particular set of the damping parameters, we will obtain an upper and lower control for the energy dissipation rate $\varepsilon$ according to the Kolmogorov K41 theory. However, although the behavior of weak solutions corresponds to the K41 theory, we will show that in some specific cases the damping term introduced in the Nav…
We consider the Navier-Stokes equations on the two-dimensional unit sphere and study the linear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. We prove the linear stability of the two-jet Kolmogorov type flow for an arbitrary viscosity coefficient by showing the exponential decay of a solution to the linearized equation towards an equilibrium which grows as the viscosity coefficient tends to zero. The main result of this paper is the nonexistence of nonzero eigenvalues of the perturbation operator appea…
Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier-Stokes equations, is a mathematical analog of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier-Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose…
We consider the vorticity form of the Navier-Stokes equations on the two-dimensional unit sphere and study the nonlinear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. In particular, we assume that a perturbation contains a nondissipative part given by a linear combination of the spherical harmonics of degree one and investigate the effect of the nondissipative part on the long-time behavior of the perturbation through the convection term. We show that the nondissipative part of a weak solution to the n…
In this paper, we establish the pseudospectral bound for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [LWZ], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies $\big\| U_0-\big(k_f^{-2}\sin(k_fy),0,0\big)\big\|_{H^2}\le cν^{\frac{7}{4}}$ ($ν$ the viscosity coefficient) and $k_f\in (0,1)$, then the solution does not transition away from the Kolmogorov flow.
We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $ν$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(ν^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(ν^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide th…
We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov-Vicsek models that can be considered a non-local non-linear Fokker-Planck type equation describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin-Vicsek algorithm as mean-field limit \cite{B-C-C,D-M}, which governs the interactions of stochastic agents moving with a velocity of constant magnitude, i.e. the the corresponding velocity space for these type of Kolmogorov-Vicsek models are the unit sphere. Our analysis for $L^p$ estimates and compactness …
A defining feature of 3D hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon - anomalous dissipation - is sometimes called the `zeroth law of turbulence' as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions $S_p(\ell) :=\langle |δ_\ell u|^p\rangle$ exhibit persistent power-law scaling in the inertial range, namely $S_p(\ell) \sim |\ell|^{ζ_p}$ for exponents $ζ_p>0$ over an ever-increasing (with Reynold…
We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $\mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an "inviscid damping" mechanism that leads t…
This PhD thesis is devoted to deterministic study of the turbulence in the Navier- Stokes equations. The thesis is divided in four independent chapters.The first chapter involves a rigorous discussion about the energy's dissipation law, proposed by theory of the turbulence K41, in the deterministic setting of the homogeneous and incompressible Navier-Stokes equations, with a stationary external force (the force only depends of the spatial variable) and on the whole space R3. The energy's dissipation law, also called the Kolmogorov's dissipation law, characterizes the energy's dissipation rate …
We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on $\mathbb{T}^3$ subjected to white-in-time (colored-in-space) forcing satisfy the Kolmogorov 4/5 law (in an averaged sense and over a suitable inertial range) using only the assumption that the kinetic energy is $o(ν^{-1})$ as $ν\rightarrow 0$ (where $ν$ is the inverse Reynolds number). This plays the role of a weak anomalous dissipation. No energy balance or additional regularity is assumed (aside from that satisfied by all martingale solutions from the energy inequality). If the force is statistic…
We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
We consider the forward Kolmogorov equation corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics, including variations of the Bolker-Pacala-Dieckmann-Law model. Under the assumption of detailed balance, we provide a rigorous generalized gradient structure, incorporating the fluxes arising from the birth and death of the particles. Moreover, in the large population limit, we show convergence of the forward Kolmogorov equation to a Liouville equation, which is a transport equation associated with the mean-field limit of the underl…
