Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

TL;DR

This paper investigates the bursting behavior of 3D Euler equations in cylindrical domains with large vorticity, revealing how resonances can lead to singularity-like events in fluid flow.

Contribution

It introduces a novel analysis of resonant interactions in cylindrical geometries, connecting fast oscillations to nonlinear depletion and burst phenomena in Euler flows.

Findings

Resonances deplete Euler nonlinearity.

Resonant systems exhibit homoclinic cycles.

Orbits near these cycles cause solution bursts.

Abstract

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy.

Equations

$${\partial }_t\text{mbox}\mathbf{V}+(\text{mbox}\mathbf{V}\cdot \nabla )\text{mbox}\mathbf{V}=-\nabla p,\text{mbox}\nabla \cdot \text{mbox}\mathbf{V}=0,$$

$$\text{mbox}\mathbf{V}\cdot \mathbf{N}=0\text{mbox}on\partial D,$$

$${\partial }_t\mathbf{V}+curl\mathbf{V}\times \mathbf{V}+\nabla \left(p+\frac{1}{2}|\mathbf{V}{|}^2\right)=0,$$

$$\nabla \cdot \mathbf{V}=0,$$

$${\partial }_t\mathbf{\omega }+curl(\mathbf{\omega }\times \mathbf{V})=0,$$

$$\displaystyle\boldsymbol{\omega}=curl\mathbf{V},\ \ \ \$$

$$E(t)=\frac{1}{2}{\int }_D|\mathbf{V}(t,y){|}^{2}dy.$$

$$Hel(t)={\int }_D\mathbf{V}\cdot \mathbf{\omega }dy,$$

$${\partial }_t\mathbf{\omega }+[\mathbf{\omega },\mathbf{V}]=0,$$

$${\mathbf{m}}_t+\mathbf{\omega }\times \mathbf{m}=0,\mathbf{m}=A\mathbf{\omega },$$

$${\mathbf{m}}_t+[\mathbf{\omega },\mathbf{m}]=0,$$

$$\frac{d{x}^{i_1}}{dt}=\frac{1}{2}\sum _{i_2,...,i_n}{ϵ}^{i_1{i}_2...i_n}{p}_{i_4...i_n}\left(\frac{\partial F_1}{\partial x^{i_2}}\frac{\partial F_2}{\partial x^{i_3}}-\frac{\partial F_1}{\partial x^{i_3}}\frac{\partial F_2}{\partial x^{i_2}}\right),$$

$${\dot{U}}_k+({\lambda }_m-{\lambda }_n)U_m{U}_n$$

$${\dot{U}}_m+({\lambda }_n-{\lambda }_k)U_n{U}_k$$

$${\dot{U}}_n+({\lambda }_k-{\lambda }_m)U_k{U}_m$$

$${\dot{a}}_k$$

$${\dot{a}}_m$$

$${\dot{a}}_n$$

$${\dot{a}}_{\tilde{m}}$$

$${\dot{a}}_{\tilde{k}}$$

$$||\mathbf{u}(t)|{|}_{H^s}^2/||\mathbf{u}(0)|{|}_{H^s}^2,s\ge 1$$

$$||\mathbf{u}(t)|{|}_{H^s}^2=\sum _n({\lambda }_n{)}^{2s}|u_n(t){|}^2.$$

$${\partial }_t\text{mbox}\mathbf{V}+(\text{mbox}\mathbf{V}\cdot \nabla )\text{mbox}\mathbf{V}=-\nabla p,\text{mbox}\nabla \cdot \text{mbox}\mathbf{V}=0,$$

$$\text{mbox}\mathbf{V}(t,y){|}_{t=0}=\text{mbox}\mathbf{V}(0)={\widetilde{\text{mbox}\mathbf{V}}}_0(y)+\frac{\Omega }{2}{\mathbf{e}}_3\times y$$

$$curl\text{mbox}\mathbf{V}(0,y)=curl{\widetilde{\text{mbox}\mathbf{V}}}_0(y)+\Omega {\mathbf{e}}_3,$$

$${\mathbf{V}}_s(y)=\frac{\Omega }{2}{\mathbf{e}}_3\times y,curl{\mathbf{V}}_s(y)=\Omega {\mathbf{e}}_3$$

$$\mathbf{V}(t,y)=\frac{\Omega }{2}{\mathbf{e}}_3\times y+\tilde{\mathbf{V}}(t,y),$$

$$curl\mathbf{V}(t,y)=\Omega {\mathbf{e}}_3+curl\tilde{\mathbf{V}}(t,y),$$

$${\partial }_t\tilde{\mathbf{V}}+curl\tilde{\mathbf{V}}\times \tilde{\mathbf{V}}+\Omega {\mathbf{e}}_3\times \tilde{\mathbf{V}}+curl\tilde{\mathbf{V}}\times {\mathbf{V}}_s(y)+\nabla p^{\prime }=0,\nabla \cdot \tilde{\mathbf{V}}=0,$$

$$\tilde{\mathbf{V}}(t,y){|}_{t=0}={\tilde{\mathbf{V}}}_0(y).$$

$$\text{mbox}\mathbf{C}=\{(y_1,y_2,y_3)\in \text{mbox}{\mathbf{R}}^3:0<y_3<2\pi /\alpha ,\text{mbox}{y}_1^2+y_2^2<R^2\}$$

$$\Gamma =\{(y_1,y_2,y_3)\in \text{mbox}{\mathbf{R}}^3:0<y_3<2\pi /\alpha ,\text{mbox}{y}_1^2+y_2^2=R^2\}.$$

$$\text{mbox}\mathbf{V}(y_1,y_2,y_3)=\text{mbox}\mathbf{V}(y_1,y_2,y_3+2\pi /\alpha )$$

$$\text{mbox}\mathbf{V}\cdot \mathbf{N}=\tilde{\mathbf{V}}\cdot \mathbf{N}=0\text{mbox}on\Gamma ;$$

$$\mathbf{\omega }(t,y)\cdot \nabla \mathbf{V}(t,y)={\omega }_1\frac{\partial \mathbf{V}}{\partial y_1}+{\omega }_2\frac{\partial \mathbf{V}}{\partial y_2}+{\omega }_3\frac{\partial \mathbf{V}}{\partial y_3},t\ge 0$$

$$curl{\mathbf{\Phi }}_n^{\pm }=\pm |{\lambda }_n|{\mathbf{\Phi }}_n^{\pm }$$

$${\mathbf{J}}^0=\left\{\mathbf{U}\in L^2(D):\nabla \cdot \mathbf{U}=0\text{mbox}and\mathbf{U}\cdot \mathbf{N}{|}_{\partial D}=0\text{mbox}and{\int }_o^h\mathbf{U}dz=0\right\}.$$

Full text

Key-Words: Incompressible Euler Equations, Rotating Fluids, Rigid Body Dynamics, Enstrophy Bursts

MSC: 35Q35, 76B03, 76U05

1 Introduction

The issues of blowup of smooth solutions and finite time singularities of the vorticity field for 3D incompressible Euler equations are still a major open problem. The Cauchy problem in 3D bounded axisymmetric cylindrical domains is attracting considerable attention: with bounded, smooth, non-axisymmetric 3D initial data, under the constraints of conservation of bounded energy, can the vorticity field blow up in finite time? Outstanding numerical claims for this have recently been disproven [Ke], [Hou1], [Hou2]. The classical analytical criterion of Beale-Kato-Majda [B-K-M] for non-blow up in finite time requires the time integrability of the $L^{\infty}$ norm of the vorticity. DiPerna and Lions [Li] have given examples of global weak solutions of the 3D Euler equations which are smooth (hence unique) if the initial conditions are smooth (specifically in $\mathbf{W}^{1,p}(D),\ p>1$). However, these flows are really 2-Dimensional in $x_{1},x_{2}$, 3-components flows, independent from the third coordinate $x_{3}$. Their examples [DiPe-Li] show that solutions (even smooth ones) of the 3D Euler equations cannot be estimated in $\mathbf{W}^{1,p}$ for $1<p<\infty$ on any time interval $(0,\,T)$ if the initial data are only assumed to be bounded in $\mathbf{W}^{1,p}$. Classical local existence theorems in 3D bounded or periodic domains by Kato [Ka], Bourguignon-Brézis [Bou-Br] and Yudovich [Yu1], [Yu2] require some minimal smoothness for the initial conditions (IC), e.g., in $\mathbf{H}^{s}(D),\ s>\frac{5}{2}$.

The classical formulation for the Euler equations is

[TABLE]

where $\partial D$ is the boundary of a bounded, connected domain $D$, $\mathbf{N}$ the normal to $\partial D$, $\mathbf{V}(t,y)=(V_{1},V_{2},V_{3})$ the velocity field, $y=(y_{1},y_{2},y_{3})$, and $p$ is the pressure.

The equivalent Lamé form [Ar-Khe]

[TABLE]

[TABLE]

implies conservation of Energy:

[TABLE]

The helicity $Hel(t)$  [Ar-Khe], [Mof], is conserved:

[TABLE]

for $D=\mathbf{R}^{3}$ and when $D$ is a periodic lattice. Helicity is also conserved for cylindrical domains, provided that $\boldsymbol{\omega}\cdot\mathbf{N}=0$ on the cylinder’s lateral boundary at $t=0$ (see [M-N-B-G]).

From the theoretical point of view, the principal difficulty in the analysis of 3D Euler equations is due to the presence of the vortex stretching term $(\boldsymbol{\omega}\cdot\nabla)\mathbf{V}$ in the vorticity equation (1.5a). The equations (1.3) and (1.5a) are equivalent to:

[TABLE]

where $[a,b]=\mbox{ curl }(\mathbf{a}\times\mathbf{b})$ is the commutator in the infinite dimensional Lie algebra of divergence-free vector fields [Ar-Khe]. This point of view has led to celebrated developments in Topological Methods in Hydrodynamics [Ar-Khe], [Mof]. The striking analogy between the Euler equations for hydrodynamics and the Euler equations for a rigid body (the latter associated to the Lie Algebra of the Lie group $SO(3,\mathbf{R}))$ had already been pointed out by Moreau [Mor1]; Moreau was the first to demonstrate conservation of Helicity (1961) [Mor2]. This has led to extensive speculations to what extent/in what cases are the solutions of the 3D Euler equations “close” to those of coupled 3D rigid body equations in some asymptotic sense. Recall that the Euler equations for a rigid body in $\mathbf{R}^{3}$ is:

[TABLE]

where $\mathbf{m}$ is the vector of angular momentum relative to the body, $\boldsymbol{\omega}$ the angular velocity in the body and $A$ the inertia operator [Ar1], [Ar-Khe].

The Russian school of Gledzer, Dolzhansky, Obukhov [G-D-O] and Vishik [Vish] has extensively investigated dynamical systems of hydrodynamic type and their applications. They have considered hydrodynamical models built upon generalized rigid body systems in $SO(n,\mathbf{R})$, following Manakhov [Man]. Inspired by turbulence physics, they have investigated “shell” dynamical systems modeling turbulence cascades; albeit such systems are flawed as they only preserve energy, not helicity. To address this, they have constructed and studied in depth $n$-dimensional dynamical systems with quadratic homogeneous nonlinearities and two quadratic first integrals $F_{1},F_{2}$. Such systems can be written using sums of Poisson brackets:

[TABLE]

where constants $p_{i_{4}...i_{n}}$ are antisymmetric in $i_{4},...,i_{n}$.

A simple version of such a quadratic hydrodynamic system was introduced by Gledzer [Gl1] in 1973. A deep open issue of the work by the Gledzer-Obukhov school is whether there exist indeed classes of I.C. for the 3D Cauchy Euler problem (1.1) for which solutions are actually asymptotically close in strong norm, on arbitrary large time intervals to solutions of such hydrodynamic systems, with conservation of both energy and helicity. Another unresolved issue is the blowup or global regularity for the “enstrophy” of such systems when their dimension $n\rightarrow\infty$.

This article reviews some current new results of a research program in the spirit of the Gledzer-Obukhov school; this program builds-up on the results of [M-N-B-G] for 3D Euler in bounded cylindrical domains. Following the original approach of [B-M-N1]-[B-M-N4] in periodic domains, [M-N-B-G] prove the non blowup of the 3D incompressible Euler equations for a class of three-dimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The initial vortex stretching is large. The approach of proving regularity is based on investigation of fast singular oscillating limits and nonlinear averaging methods in the context of almost periodic functions [Bo-Mi], [Bes], [Cor]. Harmonic analysis tools based on curl eigenfunctions and eigenvalues are crucial. One establishes the global regularity of the 3D limit resonant Euler equations without any restriction on the size of 3D initial data. The resonant Euler equations are characterized by a depleted nonlinearity. After establishing strong convergence to the limit resonant equations, one bootstraps this into the regularity on arbitrary large time intervals of the solutions of 3D Euler Equations with weakly aligned uniformly large vorticity at $t=0$. [M-N-B-G] theorems hold for generic cylindrical domains, for a set of height/radius ratios of full Lebesgue measure. For such cylinders, the 3D limit resonant Euler equations are restricted to two-wave resonances of the vorticity waves and are vested with an infinite countable number of new conservation laws. The latter are adiabatic invariants for the original 3D Euler equations.

Three-wave resonances exist for a nonempty countable set of $h/R$ ($h$ height, $R$ radius of the cylinder) and moreover accumulate in the limit of vanishingly small vertical (axial) scales. This is akin to Arnold tongues [Ar2] for the Mathieu-Hill equations and raises nontrivial issues of possible singularities/lack thereof for dynamics ruled by infinitely many resonant triads at vanishingly small axial scales. In such a context, the 3D resonant Euler equations do conserve the energy and helicity of the field.

In this review, we consider cylindrical domains with parametric resonances in $h/R$ and investigate in depth the structure and dynamics of 3D resonant Euler systems. These parametric resonances in $h/R$ are proven to be non-empty. Solutions to Euler equations with uniformly large initial vorticity are expanded along a full complete basis of elementary swirling waves ($\mathbf{T}^{2}$ in time). Each such quasiperiodic, dispersive vorticity wave is a quasiperiodic Beltrami flow; these are exact solutions of 3D Euler equations with vorticity parallel to velocity. There are no Galerkin-like truncations in the decomposition of the full 3D Euler field. The Euler equations, restricted to resonant triplets of these dispersive Beltrami waves, determine the “resonant Euler systems”. The basic “building block” of these (a priori $\infty$-dimensional) systems are proven to be $SO(3;\mathbf{C})$ and $SO(3;\mathbf{R})$ rigid body systems:

[TABLE]

These $\lambda$’s are eigenvalues of the curl operator in the cylinder, $curl\mathbf{\Phi}_{n}^{\pm}=\pm\lambda_{n}\mathbf{\Phi}_{n}^{\pm}$; the curl eigenfunctions are steady elementary Beltrami flows, and the dispersive Beltrami waves oscillate with the frequencies $\pm\frac{h}{2\pi\epsilon}\frac{n_{3}}{\lambda_{n}},\ n_{3}$ vertical wave number (vertical shear), $0<\epsilon<1$. Physicists [Ch-Ch-Ey-H] have computationally demonstrated the physical impact of the polarization of Beltrami modes $\mathbf{\Phi}^{\pm}$ on intermittency in the joint cascade of energy and helicity in turbulence.

Another “building block” for resonant Euler systems is a pair of $SO(3;\mathbf{C})$ or $SO(3;\mathbf{R})$ rigid bodies coupled via a common principal axis of inertia/mo- ment of inertia:

[TABLE]

where $\Gamma$ and $\tilde{\Gamma}$ are parameters in $\mathbf{R}$ defined in Theorem 4.10. Both resonant systems (1.11) and (1.12) conserve energy and helicity. We prove that the dynamics of these resonant systems admit equivariant families of homoclinic cycles connecting hyperbolic critical points. We demonstrate bursting dynamics: the ratio

[TABLE]

can burst arbitrarily large on arbitrarily small times, for properly chosen parametric domain resonances $h/R$. Here

[TABLE]

The case $s=1$ is the enstrophy. The “bursting” orbits are topologically close to the homoclinic cycles.

Are such dynamics for the resonant systems relevant to the full 3D Euler equations (1.1)-(1.8)? The answer lies in the following crucial “shadowing” Theorem 2.10. Given the same initial conditions, given the maximal time interval $0\leq t<T_{m}$ where the resonant orbits of the resonant Euler equations do not blow up, then the strong norm $\mathbf{H}^{s}$ of the difference between the exact Euler orbit and the resonant orbit is uniformly small on $0\leq t<T_{m}$, provided that the vorticity of the I.C. is large enough. Paradoxically, the larger the vortex streching of the I.C., the better the uniform approximation. This deep result is based on cancellation of fast oscillations in strong norms, in the context of almost periodic functions of time with values in Banach spaces (Section 4 of [M-N-B-G]). It includes uniform approximation in the spaces $\mathbf{H}^{s},\ s>5/2$. For instance, given a quasiperiodic orbit on some time torus $\mathbf{T}^{l}$ for the resonant Euler systems, the exact solutions to the Euler equations will remain $\epsilon$-close to the resonant quasiperiodic orbit on a time interval $0\leq t\leq\max T_{i}$, $1\leq i\leq l$, $T_{i}$ elementary periods, for large enough initial vorticity. If orbits of the resonant Euler systems admit bursting dynamics in the strong norms $\mathbf{H}^{s},\ s\geq 7/2$, so do some exact solutions of the full 3D Euler equations, for properly chosen parametrically resonant cylinders.

2 Vorticity waves and resonances of elementary swirling flows

We study initial value problem for the three-dimensional Euler equations with initial data characterized by uniformly large vorticity:

[TABLE]

where $y=(y_{1},y_{2},y_{3})$, $\mbox{\bf{V}}(t,y)=(V_{1},V_{2},V_{3})$ is the velocity field and $p$ is the pressure. In Eqs. (1.1) $\mathbf{e}_{3}$ denotes the vertical unit vector and $\Omega$ is a constant parameter. The field $\tilde{\mbox{\bf{V}}}_{0}(y)$ depends on three variables $y_{1}$, $y_{2}$ and $y_{3}$. Since $curl(\frac{\Omega}{2}\mathbf{e}_{3}\times y)=\Omega\mathbf{e}_{3}$, the vorticity vector at initial time $t=0$ is

[TABLE]

and the initial vorticity has a large component weakly aligned along $\mathbf{e}_{3}$, when $\Omega>>1$. These are fully three-dimensional large initial data with large initial 3D vortex stretching. We denote by $\mathbf{H}_{\sigma}^{s}$ the usual Sobolev space of solenoidal vector fields.

The base flow

[TABLE]

is called a steady swirling flow and is a steady state solution (1.1)-(1.4), as $curl(\Omega\mathbf{e}_{3}\times\mathbf{V}_{s}(y))=0$. In (2.2) and (2.3), we consider I.C. which are an arbitrary (not small) perturbation of the base swirling flow $\mathbf{V}_{s}(y)$ and introduce

[TABLE]

Eqs. (2.1) and (2.7) are studied in cylindrical domains

[TABLE]

where $\alpha$ and $R$ are positive real numbers. If $h$ is the height of the cylinder, $\alpha=2\pi/h$. Let

[TABLE]

Without loss of generality, we can assume that $R=1$. Eqs. (2.1) are considered with periodic boundary conditions in $y_{3}$

[TABLE]

and vanishing normal component of velocity on $\Gamma$

[TABLE]

where ${\bf N}$ is the normal vector to $\Gamma$. From the invariance of 3D Euler equations under the symmetry $y_{3}\rightarrow-y_{3}$, $V_{1}\rightarrow V_{1}$, $V_{2}\rightarrow V_{2}$, $V_{3}\rightarrow-V_{3}$, all results in this article extend to cylindrical domains bounded by two horizontal plates. Then the boundary conditions in the vertical direction are zero flux on the vertical boundaries (zero vertical velocity on the plates). One only needs to restrict vector fields to be even in $y_{3}$ for $V_{1}$, $V_{2}$ and odd in $y_{3}$ for $V_{3}$, and double the cylindrical domain to $-h\leq y_{3}\leq+h$.

We choose $\tilde{\mbox{\bf{V}}}_{0}(y)$ in ${\bf H}^{s}(\mathbf{C}),\ s>5/2$. In [M-N-B-G], for the case of “non-resonant cylinders”, that is, non-resonant $\alpha=2\pi/h$, we have established regularity for arbitrarily large finite times for the 3D Euler solutions for $\Omega$ large, but finite. Our solutions are not close in any sense to those of the 2D or “quasi 2D” Euler and they are characterized by fast oscillations in the $\mathbf{e}_{3}$ direction, together with a large vortex stretching term

[TABLE]

with leading component \Big{|}\Omega\frac{\partial}{\partial y_{3}}\mathbf{V}(t,y)\Big{|}\gg 1. There are no assumptions on oscillations in $y_{1}$, $y_{2}$ for our solutions (nor for the initial condition $\tilde{\mbox{\bf{V}}}_{0}(y))$.

Our approach is entirely based on sturying fast singular oscillating limits of Eqs. (1.1)-(1.5a), nonlinear averaging and cancelation of oscillations in the nonlinear interactions for the vorticity field for large $\Omega$. This has been developed in [B-M-N2], [B-M-N3], and [B-M-N4] for the cases of periodic lattice domains and the infinite space $\mathbf{R}^{3}$.

It is well known that fully three-dimensional initial conditions with uniformly large vorticity excite fast Poincaré vorticity waves [B-M-N2], [B-M-N3], [B-M-N4], [Poi]. Since individual Poincaré wave modes are related to the eigenfunctions of the curl operator, they are exact time-dependent solutions of the full nonlinear 3D Euler equations. Of course, their linear superposition does not preserve this property. Expanding solutions of (2.1)-(2.8) along such vorticity waves demonstrates potential nonlinear resonances of such waves. First recall spectral properties of the $curl$ operator in bounded, connected domains:

Proposition 2.1

([M-N-B-G]) * The curl operator admits a self-adjoint extension under the zero flux boundary conditions, with a discrete real spectrum $\lambda_{n}=\pm|\lambda_{n}|,\ |\lambda_{n}|>0$ for every $n$ and $|\lambda_{n}|\rightarrow+\infty$ as $|n|\rightarrow\infty$. The corresponding eigenfunctions $\boldsymbol{\Phi}_{n}^{\pm}$*

[TABLE]

are complete in the space

[TABLE]

Remark 2.2

In cylindrical domains, with cylindrical coordinates $(r,\theta,z)$, the eigenfunctions admit the representation:

[TABLE]

with $n_{2}=0,\pm 1,\pm 2,...$, $n_{3}=\pm 1,\pm 2,...$ and $n_{1}=0,1,2,...$. Here $n_{1}$ indexes the eigenvalues of the equivalent Sturm-Liouville problem in the radial coordinates, and $n=(n_{1},n_{2},n_{3})$. See [M-N-B-G] for technical details. From now on, we use the generic variable $z$ for any vertical (axial) coordinate $y_{3}$ or $x_{3}$. For $n_{3}=0$ (vertical averaging along the axis of the cylinder), 2-Dimensional, 3-component solenoidal fields must be expanded along a complete basis for fields derived from 2D stream functions:

[TABLE]

Here \big{(}\big{(}\mathbf{a},\,b\mathbf{e}_{3}\big{)}\big{)} denotes a 3-component vector whose horizontal projection is $\mathbf{a}$ and vertical projection is $b\mathbf{e}_{3}$.

Let us explicit elementary swirling wave flows which are exact solutions to (2.1) and (2.7):

Lemma 2.3

For every $n=(n_{1},n_{2},n_{3})$, the following quasiperiodic ($\mathbf{T}^{2}$ in time) solenoidal fields are exact solution of the full 3D nonlinear Euler equations (2.1):

[TABLE]

$n_{3}$* is the vertical wave number of $\mathbf{\Phi}_{n}$ and $\exp(\frac{\Omega}{2}\mathbf{J}t)$ the unitary group of rigid body rotations:*

[TABLE]

Remark 2.4

These fields are exact quasiperiodic, nonaxisymmetric swirling flow solutions of the 3D Euler equations. For $n_{3}\neq 0$, their second components

[TABLE]

are Beltrami flows ($curl\tilde{\mathbf{V}}\times\tilde{\mathbf{V}}\equiv 0$) exact solutions of (2.7) with $\tilde{\mathbf{V}}(t=0,y)=\mathbf{\Phi}_{n}(y)$.

$\tilde{\mathbf{V}}(t,y)$ in Eq. (2.18) are dispersive waves with frequencies $\frac{\Omega}{2}$ and $\frac{n_{3}\alpha}{|\lambda_{n}|}\Omega$, where $\alpha=\frac{2\pi}{h}$. Moreover, each $\tilde{\mathbf{V}}(t,y)$ is a traveling wave along the cylinder’s axis, since it contains the factor

[TABLE]

Note that $n_{3}$ large corresponds to small axial (vertical) scales, albeit $0\leq\alpha|n_{3}/\lambda_{n}|\leq 1$.

Proof of Lemma 2.3. Through the canonical rigid body transformation for both the field $\mathbf{V}(t,y)$ and the space coordinates $y=(y_{1},y_{2},y_{3})$:

[TABLE]

the 3D Euler equations (2.1), (2.2) transform into:

[TABLE]

For Beltrami flows such that $curl\mathbf{U}\times\mathbf{U}\equiv 0$, these Euler equations (2.20)-(2.21) in a rotating frame reduce to:

[TABLE]

which are identical to the Poincaré-Sobolev nonlocal wave equations in the cylinder [M-N-B-G], [Poi], [Sob], [Ar-Khe]:

[TABLE]

It suffices to verify that the Beltrami flows $\mathbf{\Psi}_{n}(t,x)=\mathbf{\Phi}_{n}(x)\exp\left(\pm i\frac{\alpha n_{3}}{|\lambda_{n}|}\Omega t\right)$, where $\mathbf{\Phi}_{n}^{\pm}(x)$ and $\pm|\lambda_{n}|$ are curl eigenfunctions and eigenvalues, are exact solutions to the Poincaré-Sobolev wave equation, in such a rotating frame of reference.

Remark 2.5

The frequency spectrum of the Poincaré vorticity waves (solutions to (2.22)) is exactly $\pm i\frac{\alpha n_{3}}{|\lambda_{n}|}\Omega,\ n=(n_{1},n_{2},n_{3})$ indexing the spectrum of curl. Note that $n_{3}=0$ (zero frequency of rotating waves) corresponds to 2-Dimensional, 3-Components solenoidal vector fields.

We now transform the Cauchy problem for the 3D Euler equations (2.1)-(2.2) into an infinite dimensional nonlinear dynamical system by expanding $\mathbf{V}(t,y)$ along the swirling wave flows (2.16)-(2.18):

[TABLE]

where $\mathbf{\Phi}_{n}$ denotes the curl eigenfunctions of Proposition 2.1 if $n_{3}\neq 0$, and \mathbf{\Phi}_{n}=\big{(}\big{(}curl(\phi_{n}\mathbf{\mathbf{e}_{3}}),\,\phi_{n}\mathbf{e}_{3}\big{)}\big{)} if $n_{3}=0$ (2D case, Remark 2.2).

As we focus on the case where helicity is conserved for (2.1)-(2.2), we consider the class of initial data $\tilde{\mathbf{V}}_{0}$ such that [M-N-B-G]:

[TABLE]

where $\Gamma$ is the lateral boundary of the cylinder.

The infinite dimensional dynamical system is then equivalent to the 3D Euler equations (2.1)-(2.2) in the cylinder, with $n=(n_{1},n_{2},n_{3})$ ranging over the whole spectrum of curl, e.g.:

[TABLE]

here

[TABLE]

(2D, 3-components, Remark 2.2), similarly for $m_{3}=0$ and $n_{3}=0$. The inner product $<\ ,\ >$ denotes the $L^{2}$ complex-valued inner product in $D$.

This is an infinite dimensional system of coupled equations with quadratic nonlinearities, which conserve both the energy

[TABLE]

and the helicity

[TABLE]

The quadratic nonlinearities split into resonant terms where the exponential oscillating phase factor in (2.25) reduces to unity and fast oscillating non-resonant terms ($\Omega>>1$). The resonant set $K$ is defined in terms of vertical wavenumbers $k_{3},m_{3},n_{3}$ and eigenvalues $\pm\lambda_{k}$, $\pm\lambda_{m}$, $\pm\lambda_{n}$ of $curl$:

[TABLE]

Here $k_{2},m_{2},n_{2}$ are azimuthal wavenumbers.

We shall call the “resonant Euler equations” the following $\infty$-dimensional dynamical system restricted to $(k,m,n)\in K$:

[TABLE]

here $curl\mathbf{\Phi}_{k}^{\pm}=\pm\lambda_{k}\mathbf{\Phi}_{k}^{\pm}$ if $k_{3}\neq 0$, curl\mathbf{\Phi}_{k}=\big{(}\big{(}curl(\phi_{k}\mathbf{e}_{3}),\,\mu_{k}\phi_{k}\mathbf{e}_{3}\big{)}\big{)} if $k_{3}=0$; similarly for $m_{3}=0$ and $n_{3}=0$ (2D components, Remark 2.2). If there are no terms in (2.28a) satisfying the resonance conditions, then there will be some modes for which $\frac{d\mathbf{u}_{j}}{dt}=0$.

Lemma 2.6

The resonant 3D Euler equations (2.28) conserve both energy $E(t)$ and helicity $Hel(t)$. The energy and helicity are identical to that of the full exact 3D Euler equations (2.1)-(2.2).

The set of resonances $K$ is studied in depth in [M-N-B-G]. To summarize, $K$ splits into:

• (i )

[math]-wave resonances, with $n_{3}=k_{3}=m_{3}=0$; the corresponding resonant equations are identical to the 2-Dimensional, 3-Components Euler equations, with I.C.

[TABLE]

• (ii)

Two-Wave resonances, with $k_{3}m_{3}n_{3}=0$, but two of them are not null; the corresponding resonant equations (called “catalytic equations”) are proven to possess an infinite, countable set of new conservation laws [M-N-B-G].

• (iii)

Strict three-wave resonances for a subset $K^{*}\subset K$.

Definition 2.7

The set $K^{*}$ of strict 3 wave resonances is:

[TABLE]

Note that $K^{*}$ is parameterized by $h/R$, since $\alpha=\frac{2\pi}{h}$ parameterizes the eigenvalues $\lambda_{n},\lambda_{k},\lambda_{m}$ of the curl operator.

Proposition 2.8

There exist a countable, non-empty set of parameters $\frac{h}{R}$ for which $K^{*}\neq\varnothing$.

Proof. The technical details, together with a more precise statement, are postponed to the proof of Lemma 3.7. Concrete examples of resonant axisymmetric and helical waves are discussed in [Mah] ( cf. Figure 2 in the article).

Corollary 2.9

Let $\int_{0}^{h}\tilde{\mathbf{V}}_{0}(y_{1},y_{2},y_{3})\,dy_{3}=0$, i.e. zero vertical mean for the I.C. $\tilde{\mathbf{V}}_{0}(y)$ in (2.2), (2.8), (2.24d) and (2.28b). Then the resonant 3D Euler equations are invariant on $K^{*}$:

[TABLE]

(where $\tilde{\mathbf{V}}_{0}$ has spectrum restricted to $n_{3}\neq 0$).

Proof. This is an immediate corollary of the “operator splitting” Theorem 3.2 in [M-N-B-G]. $\ \ \blacksquare$

We shall call the above dynamical systems the “strictly resonant Euler system”. This is an $\infty$-dimensional Riccati system which conserves Energy and Helicity. It corresponds to nonlinear interactions depleted on $K^{*}$.

How do dynamics of the resonant Euler equations (2.28) or (2.30) approximate exact solutions of the Cauchy problem for the full Euler equations in strong norms? This is answered by the following theorem, proven in Section 4 of [M-N-B-G]:

Theorem 2.10

Consider the initial value problem

[TABLE]

for the full 3D Euler equations, with $||\tilde{\mathbf{V}}_{0}||_{\mathbf{H}_{\sigma}^{s}}\leq M_{s}^{0}$ and $curl\tilde{\mathbf{V}}_{0}\cdot\mathbf{N}=0$ on $\Gamma$.

• •

Let $\mathbf{V}(t,y)=\frac{\Omega}{2}\mathbf{e}_{3}\times y+\tilde{\mathbf{V}}(t,y)$ denote the solution to the exact Euler equations.

• •

Let $\mathbf{w}(t,x)$ denote the solution to the resonant 3D Euler equations with Initial Condition $\mathbf{w}(0,x)\equiv\mathbf{w}(0,y)=\tilde{\mathbf{V}}_{0}(y)$.

• •

Let $||\mathbf{w}(t,y)||_{\mathbf{H}_{\sigma}^{s}}\leq M_{s}(T_{M},M_{s}^{0})$ on $0\leq t\leq T_{M},\ s>7/2$.

Then, $\forall\epsilon>0,\ \exists\ \Omega^{*}(T_{M},M_{s}^{0},\epsilon)$ such that, $\forall\Omega\geq\Omega^{*}$:

[TABLE]

on $0\leq t\leq T_{M},\ \forall\beta\geq 1,\ \beta\leq s-2$. Here $||\cdot||_{H^{\beta}}$ is defined in (1.13).

The 3D Euler flow preserves the condition $curl\tilde{\mathbf{V}}_{0}\cdot\mathbf{N}=0$ on $\Gamma$, that is $curl\mathbf{V}(t,y)\cdot\mathbf{N}=0$ on $\Gamma$, for every $t\geq 0$ [M-N-B-G]. The proof of this “error-shadowing” theorem is delicate, beyond the usual Gronwall differential inequalities and involves estimates of oscillating integrals of almost periodic functions of time with values in Banach spaces. Its importance lies in that solutions of the resonant Euler equations (2.28) and/or (2.30) are uniformly close in strong norms to those of the exact Euler equations (2.1)-(2.2), on any time interval of existence of smooth solutions of the resonant system. The infinite dimensional Riccati systems (2.28) and (2.30) are not just hydrodynamic models, but exact asymptotic limit systems for $\Omega\gg 1$. This is in contrast to all previous literature on conservative 3D hydrodynamic models, such as in [G-D-O].

3 Strictly resonant Euler systems: the SO(3) case

We investigate the structure and the dynamics of the “strictly resonant Euler systems” (2.30). Recall that the set of 3-wave resonances is:

[TABLE]

From the symmetries of the curl eigenfunctions $\mathbf{\Phi}_{n}$ and eigenvalues $\lambda_{n}$ in the cylinder, the following identities hold under the transformation $n_{2}\rightarrow-n_{2}$, $n_{3}\rightarrow-n_{3}$

[TABLE]

where ∗ designates the complex conjugate (see Section 3, [M-N-B-G] for details).  The eigenfunctions  $\mathbf{\Phi}(n_{1},n_{2},n_{3})$  involve the radial functions   $J_{n_{2}}(\beta(n_{1},n_{2},\alpha n_{3})r)$ and $J_{n_{2}}^{\prime}(\beta(n_{1},n_{2},\alpha n_{3})r)$, with

[TABLE]

$\beta(n_{1},n_{2},\alpha n_{3})$ are discrete, countable roots of equation (3.30) in [M-N-B-G], obtained via an equivalent Sturm-Liouville radial problem. Since the curl eigenfunctions are even in $r\rightarrow-r,\ n_{1}\rightarrow-n_{1}$, we will extend the indices $n_{1}=1,2,...,+\infty$ to $-n_{1}=-1,-2,...$ with the above radial symmetry in mind.

Corollary 3.1

The 3-wave resonance set $K^{*}$ is invariant under the symmetries $\sigma_{j},\ j=0,1,2,3$, where

[TABLE]

Remark 3.2

For $0<i\leq 3,\ 0<j\leq 3,\ 0<l\leq 3$ $\sigma_{j}^{2}=Id,\ \sigma_{i}\sigma_{j}=-\sigma_{l}$ if $i\neq j$ and $\sigma_{i}\sigma_{j}\sigma_{l}=-Id$, for $i\neq j\neq l$. The $\sigma_{j}$ do preserve the convolution conditions in $K^{*}$.

We choose an $\alpha$ for which the set $K^{*}$ is not empty. We further take the hypothesis of a single triple wave resonance $(k,m,n)$, modulo the symmetries $\sigma_{j}$:

Hypothesis 3.3

$K^{*}$* is such that there exists a single triple wave number resonance $(n,k,m)$, modulo the symmetries $\sigma_{j},\ j=1,2,3$ and $\sigma_{j}(k)\neq k,\ \sigma_{j}(m)\neq m,\ \sigma_{j}(n)\neq n$ for $j=2$ and $j=3$.*

Under the above hypothesis, one can demonstrate that the strictly resonant Euler system splits into three uncoupled systems in $\mathbf{C}^{3}$:

Theorem 3.4

Under hypothesis 3.3, the resonant Euler system reduces to three uncoupled rigid body systems in $\mathbf{C}^{3}$:

[TABLE]

where $C_{kmn}=i<\mathbf{\Phi}_{k}\times\mathbf{\Phi}_{m},\mathbf{\Phi}_{n}^{*}>,\ C_{kmn}$ real and the other two uncoupled systems obtained with the symmetries $\sigma_{2}(k,m,n)$ and $\sigma_{3}(k,m,n)$. The energy and the helicity of each subsystem are conserved:

[TABLE]

Proof. It follows from $U_{-k}=U_{k}^{*},\ \lambda(-k)=\lambda(+k)$, similarly for $m$ and $n$; and in a very essential way from the antisymmetry of $<\mathbf{\Phi}_{k}\times\mathbf{\Phi}_{m},\mathbf{\Phi}_{n}^{*}>$, together with $curl\mathbf{\Phi}_{k}=\lambda_{k}\mathbf{\Phi}_{k}$. That $C_{kmn}$ is real follows from the eigenfunctions explicited in Section 3 of [M-N-B-G].$\ \ \blacksquare$

Remark 3.5

This deep structure, i.e. $SO(3;\mathbf{C})$ rigid body systems in $\mathbf{C}^{3}$ is a direct consequence of the Lamé form of the full 3D Euler equations, cf. Eqs. (1.3) and (2.7), and the nonlinearity $curl\mathbf{V}\times\mathbf{V}$.

The system (3.3) is equivariant with respect to the symmetry operators

[TABLE]

provided $\chi_{1}=\chi_{2}+\chi_{3}$. It admits other integrals known as the Manley-Rowe relations (see, for instance [We-Wil]). It differs from the usual 3-wave resonance systems investigated in the literature, such as in [Zak-Man1], [Zak-Man2], [Gu-Ma] in that

• (1)

helicity is conserved,

• (2)

dynamics of these resonant systems rigorously “shadow” those of the exact 3D Euler equations, see Theorem 2.10.

Real forms of the system (3.3) are found in Gledzer et al. [G-D-O], corresponding to the exact invariant manifold $U_{k}\in i\mathbf{R},\ U_{m}\in\mathbf{R},\ U_{n}\in\mathbf{R}$, albeit without any rigorous asymptotic justification. The $\mathbf{C}^{3}$ systems (3.3) with helicity conservation laws are not discussed in [G-D-O].

The only nontrivial Manley-Rowe conservation laws for the resonant system (3.3), rigid body $SO(3;\mathbf{C})$, which are independent from energy and helicity, are:

[TABLE]

where $U_{j}=r_{j}\exp(i\theta_{j}),\ j=k,m,n,$ and

[TABLE]

The resonant system (3.3) is well known to possess hyperbolic equilibria and heteroclinic/homoclinic orbits on the energy surface. We are interested in rigorously proving arbitrary large bursts of enstrophy and higher norms on arbitrarily small time intervals, for properly chosen $h/R$. To simplify the presentation, we establish the results for the simpler invariant manifold $U_{k}\in i\mathbf{R}$, and $U_{m},U_{n}\in\mathbf{R}$.

Rescale time as:

[TABLE]

Start from the system

[TABLE]

Assume that $U_{k}\in i\mathbf{R}$ and that $U_{m},U_{n}\in\mathbf{R}$: set $p=iU_{k}$, $q=U_{m}$ and $r=U_{n}$, as well as ${\lambda}_{k}={\lambda}$, ${\lambda}_{m}=\mu$ and ${\lambda}_{n}=\nu$: then

[TABLE]

This system admits two first integrals:

[TABLE]

System (3.5) is exactly the $SO(3,\mathbf{R})$ rigid body dynamics Euler equations, with inertia momenta $I_{j}=\frac{1}{|\lambda_{j}|},\ j=k,m,n$ [Ar1].

Lemma 3.6

([Ar1], [G-D-O])* With the ordering $\lambda_{k}>\lambda_{m}>\lambda_{n}$, i.e. $\lambda>\mu>\nu$, the equilibria $(0,\pm 1,0)$ are hyperbolic saddles on the unit energy sphere, and the equilibria $(\pm 1,0,0),\ (0,0,\pm 1)$ are centers. There exist equivariant families of heteroclinic connections between $(0,+1,0)$ and $(0,-1,0)$. Each pair of such connections correspond to equivariant homoclinic cycles at $(0,1,0)$ and $(0,-1,0)$.*

We investigate bursting dynamics along orbits with large periods, with initial conditions close to the hyperbolic point $(0,E(0),0)$ on the energy sphere $E$. We choose resonant triads such that $\lambda_{k}>0,\ \lambda_{n}<0,\lambda_{k}\sim|\lambda_{n}|,\ |\lambda_{m}|\ll\lambda_{k},$ equivalently:

[TABLE]

Lemma 3.7

There exist $h/R$ with $K^{*}\neq\varnothing$, such that

[TABLE]

Remark 3.8

Together with the polarity $\pm$ of the curl eigenvalues, these are 3-wave resonances where two of the eigenvalues are much larger in moduli than the third one. In the limit $|k|,\ |m|,\ |n|\gg 1,\lambda_{k}\sim\pm|k|,\ \lambda_{m}\sim\pm|m|,\ \lambda_{n}\sim\pm|n|,$ the eigenfunctions $\mathbf{\Phi}$ have leading asymptotic terms which involve cosines and sines periodic in $r$, cf. Section 3 [M-N-B-G]. In the strictly resonant equations (2.30), the summation over the quadratic terms becomes an asymptotic convolution in $n_{1}=k_{1}+n_{1}$. The resonant three waves in Lemma 3.7 are equivalent to Fourier triads $k+m=n,$ with $|k|\sim|n|$ and $|m|\ll|k|,\ |n|$, in periodic lattices. In the physics of spectral theory of turbulence [Fri], [Les], these are exactly the triads responsible from transfer of energy between large scales and small scales. These are the triads which have hampered mathematical efforts at proving the global regularity of the Cauchy problem for 3D Navier-Stokes equations in periodic lattices [Fe].

Proof of Lemma 3.7 ([M-N-B-G]) The transcendental dispersion law for 3-waves in $K^{*}$ for cylindrical domains, is a polynomial of degree four in $\vartheta_{3}=1/h^{2}$:

[TABLE]

with $n_{2}=k_{2}+m_{2}$ and $n_{3}=k_{3}+m_{3}$.

Then with $h_{k}=\frac{\beta^{2}(k_{1},k_{2},\alpha k_{3})}{k_{3}^{2}},\ h_{m}=\frac{\beta^{2}(m_{1},m_{2},\alpha m_{3})}{m_{3}^{2}},\ h_{n}=\frac{\beta^{2}(n_{1},n_{2},\alpha n_{3})}{n_{3}^{2}}$, cf. the radial Sturm-Liouville problem in Section 3, [M-N-B-G], the coefficients of $\tilde{P}(\vartheta_{3})$ are given by:

[TABLE]

Similar formulas for the periodic lattice domain were first derived in [B-M-N2], [B-M-N3], [B-M-N4]. In cylindrical domains the resonance condition for $K^{*}$ is identical to

[TABLE]

with $\vartheta_{3}=\frac{1}{h^{2}},\ h_{k}=\beta^{2}(k)/k_{3}^{2},\ h_{m}=\beta^{2}(m)/m_{3}^{2},\ h_{n}=\beta^{2}(n)/n_{3}^{2}$; Eq. (3.8) is the equivalent rational form.

From the asymptotic formula (3.44) in [M-N-B-G], for large $\beta$:

[TABLE]

where $\psi=0$ if $\lim\frac{m_{2}}{m_{3}}\frac{h}{2\pi}=0$ (e.g. $h$ fixed, $m_{2}/m_{3}\rightarrow 0$) and $\psi=\pm\frac{\pi}{2}$ if $\lim\frac{m_{2}}{m_{3}}\frac{h}{2\pi}=\pm\infty$ (e.g. $\frac{m_{2}}{m_{3}}$ fixed, $h\rightarrow\infty$). The proof is completed by taking leading terms $\tilde{P}_{0}+\vartheta_{3}\tilde{P}_{1}$ in (3.8), $\vartheta_{3}=\frac{1}{h^{2}}\ll 1$, and $m_{2}=0,\ k_{2}=\mathcal{O}(1),\ n_{2}=\mathcal{O}(1).\ \ \ \blacksquare$

We now state a theorem for bursting of the $\mathbf{H}^{3}$ norm in arbitrarily small times, for initial data close to the hyperbolic point $(0,E(0),0)$:

Theorem 3.9

(Bursting dynamics in $\mathbf{H}^{3}$). Let $\lambda>\mu>\nu,\ \lambda\nu<0,\ |\mu|\ll\lambda$ and $\lambda\sim|\nu|$. Let $W(t)=\lambda^{6}p(t)^{2}+\mu^{6}q(t)^{2}+\nu^{6}r(t)^{2}$ the $\mathbf{H}^{3}$-norm squared of an orbit of (3.5). Choose initial data such that: $W(0)=\lambda^{6}p(0)^{2}+\mu^{6}q(0)^{2}$ with $\lambda^{6}p(0)^{2}\sim\frac{1}{2}W(0)$ and $\mu^{6}q(0)^{2}\sim\frac{1}{2}W(0)$. Then there exists $t^{*}>0$, such that

[TABLE]

where $t^{*}\leq\frac{6}{\sqrt{W(0)}}\mu^{2}Ln(\lambda/|\mu|)(\lambda/|\mu|)^{-1}$.

Remark 3.10

Under the conditions of Lemma 3.7, $\left(\frac{\lambda}{\mu}\right)^{6}\gg 1$, whereas $\mu^{2}(Ln(\lambda/|\mu|))(\lambda/|\mu|)^{-1}\ll 1$. Therefore, over a small time interval of length $O(\mu^{2}(Ln(\lambda/|\mu|))(\lambda/|\mu|)^{-1})\ll 1$, the ratio $||\mathbf{U}(t)||_{H^{3}}/||\mathbf{U}(0)||_{H^{3}}$ grows up to a maximal value $O\left((\lambda/|\mu|)^{3}\right)\gg 1$. Since the orbit is periodic, the $\mathbf{H}^{3}$ semi-norm eventually relaxes to its initial state after some time (this being a manifestation of the time-reversibility of the Euler flow on the energy sphere). The “shadowing” theorem 2.10 with $s>7/2$ ensures that the full, original 3D Euler dynamics, with the same initial conditions, will undergo the same type of burst. Notice that, with the definition (1.13) of $\|\cdot\|_{H^{s}}$, one has

[TABLE]

Hence the solid rotation part of the original 3D Euler solution does not contribute to the ratio $||\mathbf{V}(t)||_{H^{3}}/||\mathbf{V}(0)||_{H^{3}}$.

Theorem 3.11

(Bursting dynamics of the enstrophy). Under the same conditions for the 3-wave resonance, let $\Xi(t)=\lambda^{2}p(t)^{2}+\mu^{2}q(t)^{2}+\nu^{2}r(t)^{2}$ the enstrophy. Choose initial data such that $\Xi(0)=\lambda^{2}p(0)^{2}+\mu^{2}q(0)^{2}+\nu^{2}r(0)^{2}$ with $\lambda^{2}p(0)^{2}\sim\frac{1}{2}\Xi(0),\ \mu^{2}q(0)^{2}\sim\frac{1}{2}\Xi(0)$. Then there exists $t^{**}>0$, such that

[TABLE]

where $t^{**}\leq\frac{1}{\sqrt{2}}\frac{1}{\sqrt{\Xi(0)}}Ln\left(\lambda/|\mu|\right)\left(\lambda/|\mu|\right)^{-1}.$

Remark 3.12

It is interesting to compare this mechanism for bursts with earlier results in the same direction obtained by DiPerna and Lions. Indeed, for each $p\in(1,\infty)$, each $\delta\in(0,1)$ and each $t>0$, Di Perna and Lions [DiPe-Li] constructed examples of 2D-3 components solutions to Euler equations such that

[TABLE]

Their examples essentially correspond to shear flows of the form

[TABLE]

where $u\in W^{1,p}_{x_{2}}$ while $w\in W^{1,p}_{x_{2}}$. Obviously

[TABLE]

Thus, all components in $curl\mathbf{V}(t,x_{1},x_{2})$ belong to $L^{p}_{loc}$, except for the term

[TABLE]

For each $t>0$, this term belongs to $L^{p}$ for all choices of the functions $u\in W^{1,p}_{x_{2}}$ and $w\in W^{1,p}_{x_{1},x_{2}}$ if and only if $p=\infty$. Whenever $p<\infty$, DiPerna and Lions construct their examples as some smooth approximation of the situation above in the strong $W^{1,p}$ topology.

In other words, the DiPerna-Lions construction works only in cases where the initial vorticity does not belong to an algebra — specifically to $L^{p}$, which is not an algebra unless $p=\infty$.

The type of burst obtained in our construction above is different: in that case, the original vorticity belongs to the Sobolev space $H^{2}$, which is an algebra in space dimension $3$. Similar phenomena are observed in all Sobolev spaces $H^{\beta}$ with $\beta\geq 2$ — which are also algebras in space dimension 3.

In other words, our results complement those of DiPerna-Lions on bursts in higher order Sobolev spaces, however at the expense of using more intricate dynamics.

We proceed to the proofs of Theorem 3.9 and 3.11. We are interested in the evolution of

[TABLE]

Compute

[TABLE]

then

[TABLE]

Using the first integrals above, one has

[TABLE]

where $Van$ is the Vandermonde matrix

[TABLE]

For ${\lambda}\not=\mu\not=\nu\not={\lambda}$, this matrix is invertible and

[TABLE]

Hence

[TABLE]

so that

[TABLE]

Later on, we shall use the notations

[TABLE]

Therefore, we find that $\Xi$ satisfies the second order ODE

[TABLE]

which can be put in the form

[TABLE]

where $P_{{\lambda},\mu,\nu}$ is the cubic

[TABLE]

and

[TABLE]

In the sequel, we assume that the initial data for $(p,q,r)$ is such that

[TABLE]

Let us compute

[TABLE]

We shall also assume that

[TABLE]

Then $K_{{\lambda},\mu,\nu}>0$ — in fact $K_{{\lambda},\mu,\nu}\sim 2$, and $\Xi$ is a periodic function of $t$ such that

[TABLE]

with half-period

[TABLE]

We are interested in the growth of the (squared) $\mathbf{H}^{3}$ norm

[TABLE]

Expressing $p^{2}$, $q^{2}$ and $r^{2}$ in terms of $E$, $H$ and $\Xi$, it is found that

[TABLE]

Hence, when $\Xi=x_{+}({\lambda},\mu,\nu)$, then

[TABLE]

Let us compute

[TABLE]

We shall pick the initial data such that

[TABLE]

Hence, when $\Xi$ reaches $x_{+}({\lambda},\mu,\nu)$, one has

[TABLE]

Hence $W$ jumps from $W(0)$ to a quantity $\sim\tfrac{1}{4}\frac{{\lambda}^{6}}{\mu^{6}}W(0)$ in an interval of time that does not exceed one period of the $\Xi$ motion, i.e. $2T_{{\lambda},\mu,\nu}$. Let us estimate this interval of time. We recall the asymptotic equivalent for the period of an elliptic integral in the modulus 1 limit.

Lemma 3.13

Assume that $x_{-}<x_{0}<x_{+}$. Then

[TABLE]

uniformly in $x_{-}$, $x_{0}$, and $x_{+}$ as $\frac{x_{+}-x_{0}}{x_{+}-x_{-}}\to 1$.

Here

[TABLE]

Next

[TABLE]

so that

[TABLE]

Hence

[TABLE]

Conclusion: collecting (3.26), (3.27) and (3.29), we see that the squared $\mathbf{H}^{3}$ norm $W$ varies from $W(0)$ to a quantity $\sim\rho^{6}W(0)$ in an interval of time $\lesssim\frac{12}{\sqrt{W(0)}}\frac{\mu^{2}\ln\rho}{\rho}$. (Here $\rho={\lambda}/\mu$).

We now proceed to obtain similar bursting estimates for the enstrophy. We return to (3.21) and (3.22). Pick the initial data so that

[TABLE]

Then

[TABLE]

while

[TABLE]

Hence, in the limit as $\rho=\lambda/|\mu|\rightarrow+\infty$, one has

[TABLE]

And $\Xi$ varies from

[TABLE]

on an interval of time of length $T_{\lambda,\mu,\nu}$. $\ \ \blacksquare$

4 Strictly resonant Euler systems: the case of 3-waves resonances on

small-scales

4.1 Infinite dimensional uncoupled $SO(3)$ systems

In this section, we consider the 3-wave resonant set $K^{*}$ when

[TABLE]

i.e. 3-wave resonances on small scales; here $|k|^{2}=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$, where $(k_{1},k_{2},k_{3})$ index the curl eigenvalues, and similarly for $|m|^{2},\ |n|^{2}$. Recall that $k_{2}+m_{2}=n_{2},\ k_{3}+m_{3}=n_{3}$ (exact convolutions), but that the summation on $k_{1},\ m_{1}$ on the right hand side of Eqs. (2.30) is not a convolution. However, for $|k|^{2},\ |m|^{2},\ |n|^{2}\geq\frac{1}{\eta^{2}},$ the summation in $k_{1},\ m_{1}$ becomes an asymptotic convolution. First:

Proposition 4.1

The set $K^{*}$ restricted to $|k|^{2},|m|^{2},|n|^{2}\geq\frac{1}{\eta^{2}}$, $\forall\eta,0<\eta\ll 1$ is not empty: there exist at least one $h/R$ with resonant three waves satisfying the above small scales condition.

Proof. We follow the algebra of the exact transcendental dispersion law (3.8) derived in the proof of Lemma 3.7. Note that $\tilde{P}(\vartheta_{3})<0$ for $\vartheta_{3}=\frac{1}{h^{2}}$ large enough. We can choose $h_{m}=\frac{\beta^{2}(m_{1},m_{2},\alpha m_{3})}{m_{3}^{2}}=0,$ say in the specific limit $\frac{h}{2\pi m_{3}}\rightarrow 0$, and $\beta(m_{1},m_{2},\alpha m_{3})\sim m_{1}\pi+m_{2}\frac{\pi}{2}+\frac{\pi}{4}$. Then $\tilde{P}_{0}=h_{k}^{2}h_{n}^{2}>0$ and $\tilde{P}(\vartheta_{3})$ must possess at least one (transcendental) root $\vartheta_{3}=\frac{1}{h^{2}}.\ \ \ \blacksquare$

In the above context, the radial components of the curl eigenfunctions involve cosines and sines in $\frac{\beta r}{R}$ (cf. Section 3, [M-N-B-G]) and the summation in $k_{1},\ m_{1}$ on the right hand side of the resonant Euler equations (2.30) becomes an asymptotic convolution. The rigorous asymptotic convolution estimates are highly technical and detailed in [Fro-M-N]. The 3-wave resonant systems for $|k|^{2},\ |m|^{2},\ |n|^{2}\geq\frac{1}{\eta^{2}}$ are equivalent to those of an equivalent periodic lattice $[0,2\pi]\times[0,2\pi]\times[0,2\pi h],\ \vartheta_{3}=\frac{1}{h^{2}}$; the resonant three wave relation becomes:

[TABLE]

The algebraic geometry of these rational 3-wave resonance equations has been investigated in depth in [B-M-N3] and [B-M-N4]. Here $\vartheta_{1},\ \vartheta_{2},\ \vartheta_{3}$ are periodic lattice parameters; in the small-scales cylindrical case, $\vartheta_{1}=\vartheta_{2}=1$ (after rescaling of $n_{2}$, $k_{2}$, $m_{2}$),  $\vartheta_{3}=1/h^{2},\ h$ height. Based on the algebraic geometry of “resonance curves” in [B-M-N3], [B-M-N4], we investigate the resonant 3D Euler equations (2.30) in the equivalent periodic lattices.

First, triplets $(k,m,n)$ solution of (4.1) are invariant under the reflection symmetries $\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}$ defined in Corollary 3.1 and Remark 3.2: $\sigma_{0}=Id,\,\sigma_{j}(k)=(\epsilon_{i,j}k_{i}),\ 1\leq i\leq 3,\ \epsilon_{i,j}=+1$ if $i\neq j$, $\epsilon_{i,j}=-1$ if $i=j$, $1\leq j\leq 3$. Second the set $K^{*}$ in (4.1) is invariant under the homothetic transformations:

[TABLE]

The resonant triplets lie on projective lines in the wavenumber space, with equivariance under $\sigma_{j},\ 0\leq j\leq 3$ and $\gamma$-rescaling. For every given equivariant family of such projective lines, the resonant curve is the graph of $\frac{\vartheta_{3}}{\vartheta_{1}}$ versus $\frac{\vartheta_{2}}{\vartheta_{1}}$, for parametric domain resonances in $\vartheta_{1},\ \vartheta_{2},\ \vartheta_{3}$.

Lemma 4.2

(p.17, [B-M-N4]). For every equivariant $(k,m,n)$, the resonant curve in the quadrant $\vartheta_{1}>0,\vartheta_{2}>0,\vartheta_{3}>0$ is the graph of a smooth function $\vartheta_{3}/\vartheta_{1}\equiv F(\vartheta_{2}/\vartheta_{1})$ intersected with the quadrant.

Theorem 4.3

(p.19, [B-M-N4]). A resonant curve in the quadrant $\vartheta_{3}/\vartheta_{1}>0,$ $\vartheta_{2}/\vartheta_{1}>0$ is called irreducible if:

[TABLE]

An irreducible resonant curve is uniquely characterized by six non-negative algebraic invariants $\mathcal{P}_{1},\ \mathcal{P}_{2},\ \mathcal{R}_{1},\ \mathcal{R}_{2},\ \mathcal{S}_{1},\ \mathcal{S}_{2}$, such that

[TABLE]

and permutations thereof.

Lemma 4.4

(p. 25, [B-M-N4]). For resonant triplets $(k,m,n)$ associated to a given irreducible resonant curve, that is verifying Eq. (4.3), consider the convolution equation $n=k+m$. Let $\sigma_{i}(n)\neq n,\ \forall i,\ 1\leq i\leq 3$. Then there are no more that two solutions $(k,m)$ and $(m,k)$, for a given $n$, provided the six non-degeneracy conditions (3.39)-(3.44) in [B-M-N4] for the algebraic invariants of the irreducible curve are verified.

For more details on the technical non-degeneracy conditions, see the Appendix. An exhaustive algebraic geometric investigation of all solutions to $n=k+m$ on irreducible resonant curves is found in [B-M-N4]. The essence of the above lemma lies in that given such an irreducible, “non-degenerate” triplet $(k,m,n)$ on $K^{*}$, all other triplets on the same irreducible resonant curves are exhaustively given by the equivariant projective lines:

[TABLE]

and permutations of