## Abstract

An attempt is made to remove, at least partially, the non-local nature associated with the pressure term of the incompressible Euler equations. We consider the dynamics of the Jacobian matrix ** J**(

*t*) relating spatial and material coordinates in incompressible three-dimensional Euler equations. By applying the theory of matrix-valued Riccati equations to the velocity gradient tensor

**(**

*V**t*) assuming time-symmetric pressure Hessian, we derive an identity which is local in material coordinates for a fluid particle with det

**(0)≠0, as a result of invariance under time-reversal. This imposes constraints on the time evolution of the velocity gradients and generates a chain of exact infinite relationships on the Taylor coefficients of**

*V***(**

*V**t*) in time. Some of the first few are explicitly given. As a corollary, we prove that if the evolution of the velocity gradient is symmetric with regard to time

**(**

*V**t*)=

**(−**

*V**t*), it must be a constant and . The key results herein may be summarized in equations (3.10) and (3.11).

## 1. Introduction

The three-dimensional Euler equations for an incompressible fluid are known to have non-local terms stemming from the pressure gradient term:
1.1
1.2
where *D*/*Dt*=∂/∂*t*+(** u**·∇) is the material derivative. If we use vorticity

**=∇×**

*ω***we may eliminate the pressure term in its governing equations to obtain but the non-local character remains in the integral relationship between vorticity and rate of strain essentially through Biot–Savart law.**

*u*In this paper we attempt to remove, at least partially, this cumbersome nonlocal nature of the Euler equations. To this end we consider yet another representation, that is, the one in terms of the velocity gradient ** V**≡∇

**(**

*u**V*

_{ij}=∂

_{j}

*u*

_{i}). Its dynamical equations are given by 1.3 and are also affected by the pressure Hessian

**=∇∇**

*P**p*. These may be regarded as a matrix version of Riccati equations. For matrix formulations in fluid dynamics, see e.g. Yudovich (2000), Childress (2001), Yakubovich & Zenkovich (2001) and Bennett (2006). Regarding various aspects of the role of the pressure Hessian in vortex dynamics, see e.g. Ohkitani & Kishiba (1995), Gibbon

*et al.*(1999), Chae (2006) and Constantin (2008).

As an example we consider a relationship 1.4 which can be derived from the Euler equations or the vorticity equations.

An illustrative way of derivation is given as follows, e.g. Craik (1994). Introducing an auxiliary variable ** W** in the spirit of Riccati equation as
1.5
then it follows from equation (1.3) that
1.6
By writing equation (1.5) as
it is clear that each column vector of

**=[**

*W*

*w*_{1},

*w*_{2},

*w*_{3}] is material:

We deduce equation (1.4) by taking one of them, say, *w*_{1} as the vorticity vector. This derivation is based upon the observation that equation (1.3) is a Riccati equation. We will pursue this line further in what follows. In particular, we will study the dynamics of the Jacobian matrix ** J** which satisfy
and
We note some other works on Riccati equations in fluid mechanics, e.g. Drazin & Reid (1981), Constantin (2000) and Gibbon (2002).

## 2. Riccati equation

We recapitulate two basic properties of Riccati ordinary differential equations (ODE) here. Consider an ODE of the form
2.1
where *P*(*x*) is a given function of *x*. The following two basic properties are well known.

(i) By setting *y*=*u*′/*u*, we have a linear equation of the second-order
2.2
where we have denoted ′=*d*/*dx*. No methods of obtaining general solutions are known. However, if a particular solution, say, *y*_{1}(*x*) is obtained, then we may get general solutions by a quadrature.

(ii) In fact, by setting *y*=*y*_{1}(*x*)+*u*(*x*) we find
2.3
By further setting *z*=1/*u* we reduce the problem to
2.4
which is an easily solvable linear equation.

Its solution is given by 2.5 or, in the original variable, we have 2.6 Below we will apply a similar method to three-dimensional Euler equations.

## 3. Application to three-dimensional Euler equations

### (a) Heart of the matter

In §1 we have already applied the property (i) to the three-dimensional Euler equations. The aim of this paper is to apply the other property (ii) to them.

But there is a catch to that. In the case of ODEs, *P*(*x*) is an externally given function of *x*, which has nothing whatsoever to do with initial data and which is not affected by the solution. So, we may choose *u*(0) at our disposal, and generate infinitely many different solutions. However, in the case of the three-dimensional Euler equations we do not have such a wide range of freedom, because the velocity and the pressure are inherently related with each other. At first glance, it may seem impossible to apply and to make use of the property (ii).

However, it is crucial to recall that the three-dimensional Euler equations describe ideal fluids and that they form a conservative system which satisfies invariance under time-reversal. We then realize that we can still apply property (ii), because if there is one solution, then there is another solution which is running backward in time. So, we have two solutions and it is worth relating them by property (ii).

More specifically, consider an equation for the Laplacian of the pressure
Normally, we use it to find the pressure, assuming that the velocity on the right-hand side is known. Now we take a look at it in a reverse direction as
and notice that it is invariant under . The relationship between the velocity and the pressure is *not* 1 to 1 but 2 to 1. This recognition makes it possible to apply property (ii) of the Riccati theory to three-dimensional Euler equations.

### (b) Exact relationship

If we replace ** V** in equation (1.3) as we have
Setting

**=**

*Z*

*U*^{−1}we find This inhomogeneous equation can be solved as follows; see Whyburn (1934), Reid (1946, 1963, 1972) and Williams (1989).

We first consider homogeneous equations
Their solutions are known to have the form
where ** J** and satisfy
3.1
and
3.2
Here, the initial conditions are
As are other variables, the Jacobian matrix

**(**

*J***,**

*a**t*) is a function of material coordinates

**and time**

*a**t*. We suppress

**for simplicity in many places in the present paper.**

*a*Second, by the method of variations of constants we may absorb the inhomogeneous term by setting
or
Thus we get
where *C*_{1} is a constant and
3.3
Fixing *C*_{1} by the initial condition, we obtain
This connects a particular solution ** V**(

*t*) with another one where we have assumed In fact, besides the original one

**(**

*V**t*), we have only one more solution which is a time-reversal of the original, that is,

We finally obtain under the assumption that ** P**(

*t*)=

**(−**

*P**t*) for a fluid particle 3.4 as the general relationship relating the Jacobian matrix and the velocity gradient. In what follows we discuss the consequences of the relationship.

### (c) Special case

For the time being, we shall assume that the time evolution of the velocity gradients is symmetric with regard to time, that is,
3.5
holds for a fluid particle.1 Now let us consider the dynamics of . If we define ** L**(

*t*)≡

*J*^{−1}(

*t*), it satisfies

Under the assumption (3.5), can be written as

Under the same assumption it is clear that both the vorticity ** ω** and the rate-of-strain

**is invariant under**

*S**t*→−

*t*as is seen in where

**is the vorticity tensor or equivalently**

*Ω**ω*

_{i}=−

*ϵ*

_{ijk}

*Ω*

_{jk}.

Since the left-hand side of Cauchy invariant ** ω**=

*Jω*_{0}is even in

*t*, we have a consistency condition It should be noted that in general

**(**

*J**t*) is not necessarily an even function of time.

By plugging equations (3.4) and (3.5) into equation (3.1), we find the following equation for the Jacobian matrices
3.6
or, equivalently
3.7
for *t*≥0. If we set
we find by equation (3.6)
Hence the matrix in equation (3.6) is always non-singular. This means if ** J**(

*t*) blows up, some of the elements of

**must do so. (In fact, this does not happen under equation (3.5) as we see below.) To summarize, under the conditions**

*A***(**

*V**t*)=

**(−**

*V**t*), , and (

**(**

*J**t*)−

**(−**

*J**t*))

*ω*_{0}=0, we obtain equation (3.6).

### (d) Solution of *J*(*t*) in the special case

*J*

In fact, it turns out that we may solve equation (3.6) exactly. By assuming that ** J**(

*t*) has a Taylor expansion in

*t*, we find by successive differentiation and we find by induction. We thus find a simple solution 3.8 This result can be checked by a direct substitution into equation (3.6).2 It has a local expression in that it depends only on quantities associated with

**. It is of interest to compare with it the general case, where non-locality enters at the second-order in time series**

*a*Under the assumption (3.5), we deduce
that is, the velocity gradient is a constant. Plugging equation (3.8) into
this is possible, only if we have ** V**(0)

**(0)=0, or which means that there is no vortex-stretching at**

*ω**t*=0.

### (e) General case

Now we release the condition of equation (3.5) and see what we get in the more general case. In this case, we have by equation (3.4)
3.9
By defining a matrix we find by direct computations
3.10
The identity (3.9) may then be recast in a neat form
3.11
For the special case ** V**(

*t*)=

**(−**

*V**t*) we obtain a closed equation for

**(**

*Φ**t*) (, in this case) 3.12 It is readily confirmed that it has a solution of the form as it should.

It should be noted that the set of equations (3.10) and (3.11) do not form a closed system; the second term on the left-hand side of equation (3.11) is similar to the right-hand side of equation (3.10) but differs by a negative sign in ** V**(−

*t*). Nevertheless, the identity (3.9) tells something about the time evolution of the Euler equations. It generates a chain of infinite exact relationships on the Taylor coefficients of

**(**

*V**t*).

Given that we find by equations (3.1) and (3.2) and

The left-hand side of equation (3.9) reads
The expressions for *J*^{−1} and read
where
and
where
respectively.

The bracketed part on the right-hand side of equation (3.9) reads

By taking its inverse, we finally obtain from equation (3.9) 3.13 where .

We give a few low-order constraints as examples, omitting the details of the straightforward derivations. At *n*=1 we find that both sides are equal to −4** V**(0)

^{2}consistently. At

*n*=2 we find or 3.14 This implies if

*V*_{0}and

*V*_{1}are known we can tell what

*V*_{2}is purely locally (in material coordinates). However, in order to obtain

*V*_{1}we need to solve equation (1.3) which is non-local in nature. It is clear that

*V*_{2}=0 whenever

*V*_{1}=0, which is necessary for the special solution (3.8) to hold.

At *n*=3 we find after some algebra
3.15
It is clear that equation (3.15) holds when we have equation (3.14) showing that the analysis is consistent.

At *n*=4 we find after tedious algebra,
3.16
At this order we find a local formula for *V*_{4} once (*V*_{0},*V*_{1},*V*_{2},*V*_{3}) are known. Again, if *V*_{1}=*V*_{2}=*V*_{3}=0, we automatically have *V*_{4}=0, which is consistent with the special solution ** V**(

*t*)=

**(0).**

*V*As anticipated from the left-hand side of equation (3.9), it imposes constraints on the even-order Taylor coefficients of ** V**(

*t*). In general,

*V*_{2n}may be expressed locally in terms of

*V*_{0},

*V*_{1},

*V*_{2},…,

*V*_{2n−1}. While it seems difficult to deduce the general character (e.g. the regularity issues) of the time evolution of the three-dimensional Euler equations from these constraints, they may be used as a solid check of computations of temporal Taylor coefficients for the three-dimensional Euler equations. Moreover, the identity (3.9) may be used in building a model for the three-dimensional Euler equations.

It should be noted that the time derivatives are taken in the sense of Lagrangian sense. It should also be stressed that equations (3.14) and (3.16) are obtained *without* solving potential problems associated with the pressure. In the usual Taylor expansion method, we must solve potential problems to have *V*_{n} expressed in terms of *V*_{j},(0≤*j*≤*n*−1). In this sense the application of Ricatti theory is successful, at least partially, in alleviating the difficulty associated with non-locality of the three-dimensional Euler equations.

## 4. Summary

An exact identity for the Jacobian matrix obtained by applying the theory of matrix-valued Riccati equations to the velocity gradient in the three-dimensional Euler equations. The essence of the derivation of equation (3.4) is that we have removed the pressure Hessian by the property (ii) of the Riccati equation and by the 2:1 correspondence between the pressure and the velocity.

In general, the governing equations for the Jacobian matrix in the case of may be derived as
It should be noted that in spite of the non-local character of the dynamics of the Jacobian matrix, the equations for ** Φ**(

*t*) take a neat form as in equations (3.10) and (3.11).

A few comments may be in order. The Jacobian ** J** controls regularity of the Euler equations in terms Cauchy invariant. It may be of interest to consider a BKM-like criterion in terms of (Beale

*et al.*1984).

By using time-ordered exponentials (e.g. Johnson & Lapidus 2002), we may express the Jacobian in terms of the velocity gradient
where *T* () denotes a (reverse) time-ordered operator. We also have similar formulae for and However, it seems difficult to deduce equation (3.9) by solely working with these expressions.

It is hoped that these results will be used in Taylor series analysis of the Euler equations or in developing models for them.

## Acknowledgements

Note added in proof. After completion of this paper, A. Gilbert has kindly pointed out to the author that in order to obtain equation (3.4) we need the assumption just preceding equation (3.4). He also suggested that a simple ABC flow may be a good test bed for searching for suitable particle paths.

Part of the contents of this work was presented at Issac Newton Institute Workshop ‘Euler at Newton: The Inviscid Equations,’ on 31 October 2008. I have benefited from comments made by the participants. The motivation has been augmented by attending a Workshop ‘The three-dimensional Euler and two-dimensional surface quasi-geostrophic equations’ (30 March–10 April 2009) at American Institute of Mathematics. I have also benefited from discussions with A. Gilbert, C. Meneveau, C. Rosales and T. Sakajo. I thank Y. Li for pointing out a flaw in the preprint of this work.

This work has been partially supported by EPSRC EP/F009267/1. The author has been supported by a Royal Society Wolfson Research Merit Award.

## Footnotes

↵1 For example, in the Taylor–Green vortex, the total entrophy satisfies

*Q*(*t*)=*Q*(−*t*); e.g. Brachet*et al.*(1983). This does not mean that it has the property (3.5) point-wise.↵2 It is possible that actually

(0)=0 is the only possibility. We do not discuss this here.*V*- Received September 11, 2009.
- Accepted November 3, 2009.

- © 2009 The Royal Society