Inviscid Damping Research Articles

Nonlinear enhanced dissipation and inviscid damping for the 2D Couette flow

Nonlinear enhanced dissipation and inviscid damping for the 2D Couette flow

Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

AbstractWe investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $${\mathbb {T}}\times {\mathbb {R}}$$ . The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on $${\mathbb {T}}\times {\mathbb {R}}$$ . The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

Linear inviscid damping and vortex axisymmetrization via the vector field method

In this paper, we prove the inviscid damping and vortex axisymmetrization for the linearized Euler equation around the radially symmetric, strictly monotonic vorticity distribution via the vector field method. One of the key ingredients is to establish the limiting absorption principle for the Rayleigh equation. Compared with shear flows in a channel, the main difficulty comes from the degeneration of the pipe flow at the original point and infinity.

Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

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 that are arbitrarily close to the Kolmogorov flow on the square torus mathbb {T}^2 in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on mathbb {T}^2. Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on mathbb {T}^2, and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on {mathbb T}^2.

Traveling Waves Near Couette Flow for the 2D Euler Equation

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $$H^s$$ , with $$s<3/2$$ , at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces [see Bedrossian and Masmoudi (Publ Math Inst Hautes Etudes Sci 122:195–300, 2015)], where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $$H^{\frac{3}{2}+}$$ neighborhoods of Couette flow [see Lin and Zeng (Arch Ration Mech Anal 200:1075–1097, 2011)].

Linear Inviscid Damping and Enhanced Dissipation for Monotone Shear Flows

Linear Inviscid Damping and Enhanced Dissipation for Monotone Shear Flows

Inviscid damping of an elliptical vortex subject to an external strain flow

Inviscid spatial Landau damping is studied experimentally for the case of oscillatory motion of a two-dimensional vortex about its elliptical equilibrium in the presence of an applied strain flow. The experiments are performed using electron plasmas in a Penning–Malmberg trap. They exploit the isomorphism between the two-dimensional Euler equations for an ideal fluid and the drift-Poisson equations for the plasma, where plasma density is the analog of vorticity. Perturbed elliptical vortex states are created using E×B strain flows, which are generated by applying voltages to electrodes surrounding the plasma. Measurements of spatial Landau damping (also called critical-layer damping) are in agreement with previous studies in the absence of an applied strain, where the damping is due to a resonance between the local fluid motion and the vortex oscillations. Interestingly, the damping rate does not change significantly over a wide range of applied strain rates. This can be accurately predicted from the initial vorticity profile, even though the resonant frequency is reduced substantially due to the applied strain. For higher amplitude perturbations, nonlinear trapping oscillations also exhibit behavior similar to the strain-free case. In principle, higher-order effects of the applied strain, such as separatrix crossing of peripheral vorticity and interactions with harmonics of the fundamental resonance, are expected to change the damping rate. However, this occurs only for conditions that are not realized in the experiments described here. Vortex-in-cell simulations are used to investigate the possible roles of these effects.

Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime

We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids.

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain mathbb {T}times mathbb {R}. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their L^2 norm grows as t^{1/2} and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order nu ^{-1/6} (with nu ^{-1} being proportional to the Reynolds number) on a time-scale nu ^{-1/3}, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Linear inviscid damping in Sobolev and Gevrey spaces

In a recent article (Jia, 2020) Jia established linear inviscid damping in Gevrey regularity for compactly supported Gevrey regular shear flows in a finite channel, which is of great interest in view of existing nonlinear results (Deng and Masmoudi, 2018; Masmoudi and Masmoudi, 2015; Ionescu and Jia, 2019). In this article we provide an alternative short proof of stability in Gevrey regularity for those flows which admit an approach by a Fourier-based Lyapunov functional. For these flows we show that stability in L2 by Fourier methods as in Zillinger (2017a) and Zillinger (2016) immediately upgrades to stability in Gevrey regularity. Furthermore, in the setting of a finite channel we do not need to assume compact support but only vanishing of infinite order and also establish Sobolev stability results for perturbations vanishing to finite order.

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

AbstractWe prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as t → ∞ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping. © 2021 Wiley Periodicals LLC.

On the smallness condition in linear inviscid damping: monotonicity and resonance chains

We consider the effects of mixing by smooth bilipschitz shear flows in the linearized Euler equations on . Here, we construct a model which is closely related to a small high frequency perturbation around Couette flow, which exhibits linear inviscid damping for L sufficiently small, but for which damping fails if L is large. In particular, similar to the instability results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of an eigenvalue-based argument the underlying mechanism here is shown to be based on a new cascade of resonances moving to higher and higher frequencies in y, which is distinct from the echo chain mechanism in the nonlinear problem.

Inviscid Damping and Enhanced Dissipation of the Boundary Layer for 2D Navier–Stokes Linearized Around Couette Flow in a Channel

We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $\mathbb T \times [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $\nu \to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O(\nu^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $\nu$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $\nu$. Note both such estimates are false for the vorticity in the interior. To the authors' knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(\exp(-\delta\nu^{1/3}\alpha^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.

Enhanced dissipation for the 2D couette flow in critical space

We consider the 2 D incompressible Navier-Stokes equations on with initial vorticity that is δ close in to −1(the vorticity of the Couette flow ). We prove that if where ν denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time by a mixing-enhanced dissipation effect and then converges back to Couette flow when In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space

Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. We prove that for sufficiently regular initial data of size $\epsilon \leq c_0\textbf{Re}^{-1}$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t \rightarrow \infty$. For times $t \gtrsim \textbf{Re}^{1/3}$, 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 $O(\textbf{Re}^{-1})$ to $O(c_0)$ 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 $\textbf{Re}$, 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.

On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field

We study the stability of the Couette flow $$(y,0,0)^T$$ in the presence of a uniform magnetic field $$\alpha (\sigma , 0, 1)$$ on $${{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}$$ using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit $$\mathbf{Re} ^{-1}$$, $$\mathbf{R }_m^{-1} \ll 1$$ and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations small in the Sobolev space $$H^N$$. More precisely, we show that if $$\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]$$, $$\alpha > 0$$ and $$N > 0$$ are sufficiently large, $$\sigma \in {{\mathbb {R}}}{\setminus } {\mathbb {Q}}$$ satisfies a generic Diophantine condition, and the initial perturbations $$u_{\text{ in }}$$ and $$b_{\text{ in }}$$ to the Couette flow and magnetic field, respectively, satisfy $$\Vert u_{\text{ in }}\Vert _{H^N} + \Vert b_{\text{ in }}\Vert _{H^N} = \epsilon \ll \mathbf{Re} ^{-1}$$, then the resulting solution to the 3D MHD equations is global in time and the perturbations $$u(t,x+yt,y,z)$$ and $$b(t,x+yt,y,z)$$ remain $${\mathcal {O}}(\mathbf{Re} ^{-1})$$ in $$H^{N'}$$ for some $$1 \ll N'(\sigma ) < N$$. Our proof establishes enhanced dissipation estimates describing the decay of the x-dependent modes on the timescale $$t \sim \mathbf{Re} ^{1/3}$$, as well as inviscid damping of the velocity and magnetic field with a rate that agrees with the prediction of the linear stability analysis. In the Navier–Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption $$\epsilon \ll \mathbf{Re} ^{-3/2}$$ (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017). The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier–Stokes equations linearized around Couette flow.

Barotropic instability of shear flows

AbstractWe consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

Linear inviscid damping and enhanced dissipation for the Kolmogorov flow

In this paper, we prove the linear inviscid damping and vorticity 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 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 ν23+, here ν is the viscosity coefficient.

Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.