# Steady flows of ideal incompressible fluid

Vladimir Yu. Rovenski and Vladimir A. Sharafutdinov  Sobolev Institute of
Mathematics; 4 Koptyug Avenue, Novosibirsk, 630090, Russia.
<EMAIL_ADDRESS>Department of Mathematics, University of Haifa, Mount
Carmel, 31905 Haifa, Israel<EMAIL_ADDRESS>
###### Abstract.

We call a solution to the Euler steady equations a Gavrilov flow if the
velocity vector is orthogonal to the gradient of the pressure $p$ at any
point. Any such flow is an exception since it is a solution to an
overdetermined system of differential equations. Nevertheless, Gavrilov flows
exist and are worth of studying. Theory of these flows turns out very
geometric one. In a neighborhood of a regular point, a Gavrilov flow can be
described in terms of geometry of isobaric hypersurfaces. We obtain a system
of PDEs for an axisymmetric Gavrilov flow and find consistency conditions for
the overdetermined system. Two numerical examples of such axisymmetric flows
are given: (1) with a pressure function periodic in the $z$-direction and (2)
with isobaric surfaces diffeomorphic to a torus.

Keywords: Euler equations for ideal fluid, Gavrilov flow, geodesic vector
field

Mathematics Subject Classifications (2010) 76B03, 76M99, 76A02, 53Z05

The second author was supported by RFBR, Grant 20-51-15004 (joint French –
Russian grant).

##  1\. Introduction

In dimensions 2 and 3, the Euler equations

$\displaystyle u\cdot\nabla u+\nabla p=0,$ (1.1) $\displaystyle\nabla\cdot
u=0$ (1.2)

describe steady flows of ideal incompressible fluid. The equations are also of
some mathematical interest in an arbitrary dimension. Here
$u=\big{(}u_{1}(x),\dots,u_{n}(x)\big{)}$ is a vector field on an an open set
$U\subset{\mathbb{R}}^{n}$ (the fluid velocity) and $p$ is a scalar function
on $U$ (the pressure). We consider only smooth real solutions to the Euler
equations, i.e., $u_{i}\in C^{\infty}(U)\ (i=1,\dots,n)$ and $p\in
C^{\infty}(U)$ are assumed to be real functions (the term “smooth” is used as
the synonym of “$C^{\infty}$”).

We say that a solution $(u,p)$ to (1.1)–(1.2) is a Gavrilov flow, if it
satisfies

$u\cdot\nabla p=0,$ (1.3)

i.e., the velocity is orthogonal to the pressure gradient at all points of
$U$.

Equations (1.1)–(1.3) constitute an overdetermined system of first order
differential equations: $n+2$ equations in $n+1$ unknown function. Therefore
every Gavrilov flow is an exception in a certain sense. Nevertheless, such
flows exist and deserve study. Such flows satisfy the following important
property: a pair of functions $(\tilde{u},\tilde{p})$ given by

$\tilde{u}=\varphi(p)u,\quad\nabla\tilde{p}=\varphi^{2}(p)\nabla p,$ (1.4)

where $\varphi(p)$ is an arbitrary smooth function, is again a Gavrilov flow.
This property underlies the following construction that will be called the
Gavrilov localization. Let $p_{0}\in\mathbb{R}$ be a regular value of the
function $p$ such that $M_{p_{0}}=\\{x\in U\mid p(x)=p_{0}\\}$ is a compact
hypersurface in $U$. Then we can construct a compactly supported smooth
solution to the Euler equations on the whole of ${\mathbb{R}}^{n}$ by choosing
$\varphi(p)$ as a cutoff function with support in a small neighborhood of
$p_{0}$. Indeed, the new velocity vector field $\tilde{u}$ and the gradient
$\nabla\tilde{p}$ are supported in some compact neighborhood $\tilde{U}\subset
U$ of the surface $M_{p_{0}}$, as seen from (1.4), and we can define
$\tilde{u}$ as zero in ${\mathbb{R}}^{n}\setminus U$. Thus, the new pressure
$\tilde{p}$ is constant on every connected component of $U\setminus\tilde{U}$.
Since only the gradient $\nabla p$ participates in (1.1)–(1.3), we can assume
without lost of generality that $\tilde{p}=0$ on the “exterior component” of
$U\setminus\tilde{U}$. It is now clear that $\tilde{p}$ can be extended to a
compactly supported function $\tilde{p}\in C^{\infty}({\mathbb{R}}^{n})$.

For some neighborhood $O({\mathcal{C}})$ of the circle
${\mathcal{C}}=\\{(x_{1},x_{2},0)\in{\mathbb{R}}^{3}\mid
x_{1}^{2}+x_{2}^{2}=R^{2}\\}$, A.V. Gavrilov [2] proved the existence of a
solution $u\in
C^{\infty}\big{(}O({\mathcal{C}})\setminus{\mathcal{C}};{\mathbb{R}}^{3}\big{)},\
p\in C^{\infty}\big{(}O({\mathcal{C}})\setminus{\mathcal{C}}\big{)}$ of the
Euler equations satisfying (1.3), and such that for some regular value $p_{0}$
of the function $p$, the surface $M_{p_{0}}\subset
O({\mathcal{C}})\setminus{\mathcal{C}}$ is diffeomorphic to the torus.
Gavrilov’s formulas involve an arbitrary positive constant $R$, without lost
of generality we set $R=1$. Using the localization procedure described above,
Gavrilov proved the existence of a solution $\tilde{u}\in
C^{\infty}({\mathbb{R}}^{3};{\mathbb{R}}^{3}),\ \tilde{p}\in
C^{\infty}({\mathbb{R}}^{3})$ of the Euler equations supported in a small
neighborhood of $M_{p_{0}}$. Thus, Gavrilov gave a positive answer to the old
question: Is there a smooth compactly supported solution to the Euler
equations on ${\mathbb{R}}^{3}$ that is not identically equal to zero?
Unfortunately, [2] is written in terse language and many details are omitted.

Actually the same idea is implemented in a subsequent article [1] by
Constantin – La – Vicol. The latter paper starts with the so called Grad –
Shafranov ansatz that has appeared in plasma physics. Unlike [2], the article
[1] involves a thorough analysis of nonlinear ODEs that arise while
constructing a solution.

We emphasize that in both papers [1, 2] the existence of a Gavrilov
axisymmetric smooth compactly supported flow on ${\mathbb{R}}^{3}$ is proved.
Indeed, [2] starts with the Euler equations in cylindrical coordinates for
axisymmetric solutions. To authors’ knowledge, the Grad – Shafranov ansatz is
adapted to the study of axisymmetric solutions only.

We say that two Gavrilov flows $(u,p)$ and $(\tilde{u},\tilde{p})$, defined on
the same open set $U\subset{\mathbb{R}}^{n}$, are equivalent if (1.4) holds
with a smooth non-vanishing function $\varphi(p)$. For example, $(u,p)$ and
$(-u,p)$ are equivalent Gavrilov flows. In the present article, Gavrilov flows
are considered up to the equivalence. We mostly study the structure of such a
flow in a neighborhood of the hypersurface $M_{p_{0}}\subset U$ for a regular
value $p_{0}$ of the pressure.

The article is organized as follows. In Section 2, we describe a Gavrilov flow
in terms of first and second quadratic forms of isobaric hypersurfaces
$M_{p}$. Having received a geometric description, one can forget about the
Euler equations.

In Section 3 we discuss an example of a Gavrilov flow constructed in any even
dimension. The example can be modified using the Gavrilov localization. The
following observation is widely used in [8]. If a solution $(u,p)$ to the
Euler equations is defined on ${\mathbb{R}}^{3}$ and sufficiently fast decays
at infinity, then the quadratic form $(\nu\cdot u)(\xi\cdot u)$ integrates to
zero over every affine plane $P\subset{\mathbb{R}}^{3}$, where $\nu$ is the
normal vector to $P$ and $\xi$ is an arbitrary vector parallel to $P$. In
Section 4, we show that the double integral over $P$ (for a Gavrilov flow) can
be reduced to the integral over the curve $P\cap M_{p}$. This property is
interesting in itself, but applications of this property are still unknown.

In Section 5, we obtain a system of PDEs for a Gavrilov flow in
${\mathbb{R}}^{3}$. It is an overdetermined system: 4 equations in 3 unknown
functions. The problem of deriving consistency conditions for the system is
the main one in the study of these flows. The problem remains open in the
general case.

Sections 6–9 are devoted to axisymmetric Gavrilov flows in ${\mathbb{R}}^{3}$.
Unlike [1, 2], our analysis of these flows is based on the well-known
geometric fact: the equation for geodesics admits a first integral for
surfaces of revolution, the Clairaut integral.

In Section 6, using geometric arguments, we reduce the system from the Section
5 to a much simpler system of partial differential equations for axisymmetric
Gavrilov flows: two equations in one unknown function of two variables,
including also two functions of one variable $p$.

The consistency conditions for the last system of equations are derived in
Section 7.

The existence of an axisymmetric Gavrilov flow in the open set
$O({\mathcal{C}})\setminus{\mathcal{C}}$ is proved in [2] such that the
pressure function $p(r,z)$ is smooth in a neighborhood of the point
$(r,z)=(1,0)$ that is a nondegenerate minimum point of $p$. We study such
flows in Section 8. The corresponding system of PDEs can be solved in series.
We emphasize that [1, 2] only prove the existence of axisymmetric Gavrilov
flows but do not give numerical examples. In our opinion, numerical and
geometric examples are of a great importance since they can lead to new
hypotheses.

In Section 9, we give two geometric illustrations of the axisymmetric Gavrilov
flow: one with isobaric surfaces diffeomorphic to a torus (considered in
Section 8) and another such flow with a pressure function $p(r,z)$ periodic in
$z$.

Some open questions on Gavrilov flows are posed in Section 10.

##  2\. Geometry of a Gavrilov flow

Let $(u,p)$ be a Gavrilov flow on an open set $U\subset{\mathbb{R}}^{n}$.
Integral curves of the vector field $u$ are also called particle trajectories.

######  Proposition 2.1.

The pressure $p$ is constant on every integral curve of the vector field $u$.

###### Proof.

Let $\gamma:(a,b)\rightarrow U$ be an integral curve of $u$, i.e.,
$\dot{\gamma}(t)=u(\gamma(t))$. Then $\frac{d(p(\gamma(t))}{dt}=\nabla
p(\gamma(t))\cdot\dot{\gamma}(t)=\nabla p(\gamma(t))\cdot u(\gamma(t))=0.$ The
latter equality holds in virtue of (1.3). ∎

By Proposition 2.1, the Bernoulli law, that is
$\frac{|u|^{2}}{2}+p={\rm{const}}$ along a particle trajectory, for a Gavrilov
flow splits into two conservation laws:

$p=c={\rm{const}},\quad\frac{|u|^{2}}{2}=C={\rm{const}}\quad\mbox{along a
particle trajectory}.$ (2.1)

We say that $x\in U$ is a regular point if $\nabla p(x)\neq 0$. The vector
field $u$ does not vanish at regular points as is seen from (1.1). The sets

$M_{p_{0}}=\\{x\in U\mid p(x)=p_{0}={\rm{const}}\\}$

will be called isobaric hypersurfaces. A particle trajectory starting at a
point of an isobaric hypersurface $M_{p}$ does not leave $M_{p}$ “for ever”.
In the general case, an arbitrary closed subset of $U$ can be an isobaric
hypersurface $M_{p}$. But $M_{p}$ is indeed a smooth hypersurface of
${\mathbb{R}}^{n}$ in a neighborhood of a regular point $x\in M_{p}$. We say
that $M_{p}$ is a regular isobaric hypersurface if it consists of regular
points.

Recall that a vector field $u$ on a manifold $M$ with a Riemannian metric $g$
is called a geodesic vector field if $\nabla_{\\!u}\,u=0$, where $\nabla$
stands for the covariant derivative with respect to the Levi-Civita connection
of $(M,g)$. Integral curves of a geodesic vector field are geodesics. In the
case of a Gavrilov flow, regular isobaric hypersurfaces
$M_{p}\subset{\mathbb{R}}^{n}$ are considered with the Riemannian metric
induced by the Euclidean metric of ${\mathbb{R}}^{n}$.

######  Proposition 2.2.

Given a Gavrilov flow $(u,p)$, the restriction of $u$ to every regular
isobaric hypersurface $M_{p}$ is a non-vanishing geodesic vector field.

###### Proof.

It consists of one line:

$\nabla_{\\!u}\,u=P(u\cdot\nabla u)=-P(\nabla p)=0.$ (2.2)

Nevertheless, the line needs the following comment. On the left-hand side of
(2.2), $\nabla$ stands for the covariant derivative corresponding to the
induced Riemannian metric on $M_{p}$. But two other $\nabla$’s stand for the
Euclidean gradient, the same operator as in the Euler equations (1.1)–(1.2);
and $P$ is the orthogonal projection (of vectors in ${\mathbb{R}}^{n}$) onto
the tangent hyperplane of $M_{p}$. The first equality in (2.2) is the main
relationship between intrinsic geometry of a hypersurface and geometry of the
ambient space; it goes back to Gauss and is valid in a more general setting,
see [6, Chapter VII, Proposition 3.1]. The second equality in (2.2) holds by
(1.1), and the last equality holds by (1.3). ∎

Since the vector field $u$ does not vanish at regular points, no integral
curve of $u$ living on a regular hypersurface $M_{p}$ degenerates to a point.
Thus, integral curves of $u$ constitute a geodesic foliation of a regular part
of any isobaric hypersurface.

######  Proposition 2.3.

Let a Gavrilov flow $(u,p)$ be defined on an open set
$U\subset{\mathbb{R}}^{n}$. Let us restrict the vector field $u$ onto a
regular isobaric hypersurface $M_{p}$, and let ${\rm{div}}\,u$ be the
$(n-1)$-dimensional divergence of the restriction which is understood in the
sense of intrinsic geometry of the hypersurface $M_{p}$. Then

${\rm{div}}\,u=u(\log|\nabla p\,|).$ (2.3)

On the right-hand side of (2.3), the vector field $u$ is considered as a
differentiation of the algebra $C^{\infty}(M_{p})$ of smooth functions on
$M_{p}$.

###### Proof.

We will show that the equation (2.3) is equivalent to the incompressibility
equation (1.2). To this end we will rewrite the equation (1.2) in local
curvilinear coordinates adapted to the foliation of $U$ into isobaric
hypersurfaces. Fix a regular point $x_{0}\in U$ and set $p_{0}=p(x_{0})$. For
$p\in\mathbb{R}$ sufficiently close to $p_{0}$, the isobar $M_{p}$ is a
regular hypersurface near $x_{0}$. Choose local curvilinear coordinates
$(z^{1},\dots,z^{n-1})$ on the hypersurface $M_{p_{0}}$. Let

${\mathbb{R}}^{n-1}\supset\Omega\ {\stackrel{{\scriptstyle
r}}{{\longrightarrow}}}\ {\mathbb{R}}^{n},\quad r=r(z^{1}\dots,z^{n-1})$

be the parametrization of $M_{p_{0}}$ in these coordinates. Assume that
$0\in\Omega$ and $r(0)=x_{0}$. In some neighborhood of $x_{0}$, we introduce
local curvilinear coordinates $(z^{1},\dots,z^{n})$ in ${\mathbb{R}}^{n}$ as
follows. Define the vector field

$\xi=\frac{\nabla p}{|\nabla p\,|^{2}}.$ (2.4)

For $(z^{1},\dots,z^{n-1})\in{\mathbb{R}}^{n-1}$ sufficiently close to $0$,
let

$R(z^{1},\dots,z^{n-1};z^{n}),\quad(p_{0}-\varepsilon<z^{n}<p_{0}+\varepsilon)$

be the integral curve of the vector field $\xi$ starting at the point
$r(z^{1},\dots,z^{n-1})$ at the initial moment $z^{n}=p_{0}$. Thus
$R(z)\in{\mathbb{R}}^{n}$ is the solution to the Cauchy problem

$\frac{\partial R}{\partial z^{n}}(z)=\xi\big{(}R(z)\big{)},\quad
R(z^{1},\dots,z^{n-1};p_{0})=r(z^{1},\dots,z^{n-1}).$ (2.5)

Obviously, $R$ is a diffeomorphism between some neighborhood of the point
$(0,\dots,0,p_{0})$ and a neighborhood of $x_{0}$; therefore
$(z^{1},\dots,z^{n})$ constitute a local coordinate system in
${\mathbb{R}}^{n}$ near the point $x_{0}$. By our construction, $R$ satisfies
the identity $p\big{(}R(z)\big{)}=z^{n}$, which means that the coordinate
$z^{n}$ coincides with the pressure $p$. Nevertheless, we use the different
notation for the coordinate since $z^{n}$ is considered as an independent
variable while $p$ is a function on $U$. For every $z^{n}$ sufficiently close
to $p_{0}$, $(z^{1},\dots,z^{n-1})$ are local coordinates on the isobaric
hypersurface $M_{z^{n}}=\\{x\in U\mid p(x)=z^{n}\\}.$ Let

$ds_{z^{n}}^{2}=g_{\alpha\beta}dz^{\alpha}dz^{\beta},\quad
g_{\alpha\beta}=\frac{\partial R}{\partial z^{\alpha}}\cdot\frac{\partial
R}{\partial z^{\alpha}}$

be the first quadratic form of $M_{z^{n}}$. We use the following convention:
Greek indices vary from 1 to $n-1$ and the summation from 1 to $n-1$ is
assumed over a repeating Greek index; while Roman indices vary from 1 to $n$
with the corresponding summation rule. We also write the Euclidean metric of
${\mathbb{R}}^{n}$ in coordinates $(z^{1},\dots,z^{n})$ as

$ds^{2}=h_{ij}dz^{i}dz^{j},\quad h_{ij}=\frac{\partial R}{\partial
z^{i}}\cdot\frac{\partial R}{\partial z^{j}}.$

Obviously, $h_{\alpha\beta}=g_{\alpha\beta},$ and

$h_{\alpha n}(z)=\frac{\partial R}{\partial z^{\alpha}}(z)\cdot\frac{\partial
R}{\partial z^{n}}(z)=\frac{\partial R}{\partial
z^{\alpha}}(z)\cdot\frac{\nabla p}{|\nabla p\,|^{2}}\big{(}R(z)\big{)}=0.$

The last equality holds since the vector $\frac{\partial R}{\partial
z^{\alpha}}(z)$ is tangent to the hypersurface $M_{z^{n}}$ at the point $R(z)$
while the vector $\nabla p(R(z))$ is orthogonal to $M_{z^{n}}$ at the same
point. Similarly, we get $h_{nn}(z)=\big{|}\frac{\partial R}{\partial
z^{n}}(z)\big{|}^{2}=\big{|}\nabla p\big{(}R(z)\big{)}\big{|}^{-2}.$ Thus,

$(h_{ij})=\Big{(}\begin{array}[]{cc}g_{\alpha\beta}&0\\\ 0&|\nabla
p\,|^{-2}\end{array}\Big{)},\quad(h^{ij})=(h_{ij})^{-1}=\Big{(}\begin{array}[]{cc}g^{\alpha\beta}&0\\\
0&|\nabla p\,|^{2}\end{array}\Big{)}.$

Let
$\Gamma^{\alpha}_{\beta\gamma}=\frac{1}{2}g^{\alpha\delta}\big{(}\frac{\partial
g_{\beta\delta}}{\partial z^{\gamma}}+\frac{\partial
g_{\gamma\delta}}{\partial z^{\beta}}-\frac{\partial g_{\beta\gamma}}{\partial
z^{\delta}}\big{)}$ be the Christoffel symbols of $M_{z^{n}}$ in coordinates
$(z^{1},\dots,z^{n-1})$ and $G^{i}_{jk}$ be the Christoffel symbols of the
Euclidean metric in coordinates $(z^{1},\dots,z^{n})$. As follows from the
above relations, the Christoffel symbols satisfy

$\displaystyle G^{\alpha}_{\beta\gamma}$
$\displaystyle=\Gamma^{\alpha}_{\beta\gamma},\quad
G^{n}_{\beta\gamma}=-\frac{1}{2}|\nabla p\,|^{2}\,\frac{\partial
g_{\beta\gamma}}{\partial z^{n}},\quad G^{\alpha}_{\beta
n}=\frac{1}{2}g^{\alpha\delta}\,\frac{\partial g_{\beta\delta}}{\partial
z^{n}},$ (2.6) $\displaystyle G^{\alpha}_{nn}$
$\displaystyle=-\frac{1}{2}g^{\alpha\delta}\,\frac{\partial|\nabla
p\,|^{-2}}{\partial z^{\delta}},\ \ G^{n}_{\beta n}=\frac{1}{2}|\nabla
p\,|^{2}\,\frac{\partial|\nabla p\,|^{-2}}{\partial z^{\beta}},\ \
G^{n}_{nn}=\frac{1}{2}|\nabla p\,|^{2}\,\frac{\partial|\nabla
p\,|^{-2}}{\partial z^{n}}.$

The velocity $u$ can be represented in the chosen coordinates as
$u(z)=u^{\alpha}(z)\frac{\partial R(z)}{\partial z^{\alpha}}$, since it is
tangent to $M_{z^{n}}$. The velocity $u$ can be thought as a smooth vector
field either on the $n$-dimensional open set $U$ or on each isobaric
hypersurface $M_{z^{n}}$ smoothly depending on the parameter $z^{n}$. We
remain the notation $\nabla\cdot u$ for the $n$-dimensional divergence of $u$
while the $(n-1)$-dimensional divergence of $u$ on $M_{z^{n}}$ will be denoted
by ${\rm{div}}\,u$. Thus,

${\rm{div}}\,u=\nabla_{\alpha}u^{\alpha}=\frac{\partial u^{\alpha}}{\partial
z^{\alpha}}+\Gamma^{\alpha}_{\alpha\beta}u^{\beta}.$ (2.7)

By the same formula, $\nabla\cdot u=\frac{\partial u^{i}}{\partial
z^{i}}+G^{i}_{ij}u^{j},$ since $u^{n}=0$, this becomes

$\nabla\cdot u=\frac{\partial u^{\alpha}}{\partial
z^{\alpha}}+G^{i}_{i\beta}u^{\beta}.$

In particular, the incompressibility equation (1.2) is written in the chosen
coordinates as $\frac{\partial u^{\alpha}}{\partial
z^{\alpha}}+G^{i}_{i\beta}u^{\beta}=0.$ Using this equation, formula (2.7)
becomes
${\rm{div}}\,u=-(G^{i}_{i\beta}-\Gamma^{\alpha}_{\alpha\beta})u^{\beta}$. By
the formulas for Christoffel symbols,
$G^{i}_{i\beta}-\Gamma^{\alpha}_{\alpha\beta}=-\frac{\partial(\log|\nabla
p\,|)}{\partial z^{\beta}}.$ Inserting this expression into the previous
formula with ${\rm{div}}\,u$, we obtain

${\rm{div}}\,u=\frac{\partial(\log|\nabla p\,|)}{\partial
z^{\beta}}\,u^{\beta}.$

This is equivalent to (2.3). ∎

Let ${\rm{II}}$ be the second quadratic form of a regular isobaric
hypersurface $M_{p}$. Recall that ${\rm{II}}$ depends on the choice of the
unit normal vector $N$ to a hypersurface (changes its sign if $N$ is replaced
with $-N$). We choose $N$ for $M_{p}$ to be a positive multiple of $\nabla p$.

######  Proposition 2.4.

Given a Gavrilov flow $(u,p)$, the restriction of the vector field $u$ onto a
regular isobaric hypersurface $M_{p}$ satisfies

${\rm{II}}(u,u)=-|\nabla p\,|.$ (2.8)

###### Proof.

We write down (1.1) in local coordinates $(z^{1},\dots,z^{n})$ (the same as in
the previous proof) as $(u\cdot\nabla u)^{i}+(\nabla p)^{i}=0.$ Setting
$i=\alpha$ here gives the same result: integral curves of $u$ are geodesics of
$M_{p}$. Thus, we set $i=n$, and by (2.4)–(2.5), get $(\nabla p)^{n}=|\nabla
p\,|^{2}$ and

$(u\cdot\nabla u)^{n}=-|\nabla p\,|^{2}.$ (2.9)

By a well-known formula for covariant derivatives, $(u\cdot\nabla
u)^{n}=u^{i}\big{(}\frac{\partial u^{n}}{\partial
z^{j}}+G^{n}_{ij}u^{j}\big{)}.$ Since $u^{n}=0$, this becomes $(u\cdot\nabla
u)^{n}=G^{n}_{\alpha\beta}u^{\alpha}u^{\beta}.$ Using
$G^{n}_{\alpha\beta}=-\frac{1}{2}|\nabla p\,|^{2}\,\frac{\partial
g_{\alpha\beta}}{\partial z^{n}}$ from (2.6), we get

$(u\cdot\nabla u)^{n}=-\frac{1}{2}|\nabla p\,|^{2}\,\frac{\partial
g_{\alpha\beta}}{\partial z^{n}}u^{\alpha}u^{\beta}.$ (2.10)

Differentiating the equality $g_{\alpha\beta}=\frac{\partial R}{\partial
z^{\alpha}}\cdot\frac{\partial R}{\partial z^{\beta}}$ with respect to
$z^{n}$, we obtain $\frac{\partial g_{\alpha\beta}}{\partial
z^{n}}=\frac{\partial^{2}R}{\partial z^{\alpha}\partial
z^{n}}\cdot\frac{\partial R}{\partial z^{\beta}}+\frac{\partial^{2}R}{\partial
z^{\beta}\partial z^{n}}\cdot\frac{\partial R}{\partial z^{\alpha}}$. This can
be written as

$\frac{\partial g_{\alpha\beta}}{\partial z^{n}}=\frac{\partial}{\partial
z^{\alpha}}\Big{(}\frac{\partial R}{\partial z^{n}}\cdot\frac{\partial
R}{\partial z^{\beta}}\Big{)}+\frac{\partial}{\partial
z^{\beta}}\Big{(}\frac{\partial R}{\partial z^{n}}\cdot\frac{\partial
R}{\partial z^{\alpha}}\Big{)}-2\frac{\partial^{2}R}{\partial
z^{\alpha}\partial z^{\beta}}\cdot\frac{\partial R}{\partial z^{n}}.$

Both expressions in parentheses are equal to zero, and we obtain

$\frac{\partial g_{\alpha\beta}}{\partial
z^{n}}=-2\frac{\partial^{2}R}{\partial z^{\alpha}\partial
z^{\beta}}\cdot\frac{\partial R}{\partial z^{n}}.$ (2.11)

Let $N=\frac{\nabla p}{|\nabla p\,|}=|\nabla p\,|\,\frac{\partial R}{\partial
z^{n}}$ be the unit normal vector of the hypersurface $M_{z^{n}}$. Recall the
Gauss formula, e.g., [6, Chapter 7, Section 3], $\frac{\partial^{2}R}{\partial
z^{\alpha}\partial z^{\beta}}=\Gamma^{\gamma}_{\alpha\beta}\frac{\partial
R}{\partial z^{\gamma}}+b_{\alpha\beta}N$ (with the tangent and normal to $M$
components of $\frac{\partial^{2}R}{\partial z^{\alpha}\partial z^{\beta}}$),
where $b_{\alpha\beta}$ are the coefficients of II for $M_{z^{n}}$ in
coordinates $(z^{1},\dots,z^{n-1})$. Taking the scalar product of this
equality with $N$ and using the orthogonality of $N$ to $\frac{\partial
R}{\partial z^{\gamma}}$, we obtain $\frac{\partial^{2}R}{\partial
z^{\alpha}\partial z^{\beta}}\cdot N=b_{\alpha\beta}$. Since $N=|\nabla
p\,|\,\frac{\partial R}{\partial z^{n}}$, we get
$\frac{\partial^{2}R}{\partial z^{\alpha}\partial
z^{\beta}}\cdot\frac{\partial R}{\partial z^{n}}=|\nabla
p\,|^{-1}b_{\alpha\beta}$. Using this equality, formula (2.11) becomes
$\frac{\partial g_{\alpha\beta}}{\partial z^{n}}=-2|\nabla
p\,|^{-1}b_{\alpha\beta}$. Substituting this value into (2.10), we get

$(u\cdot\nabla u)^{n}=|\nabla p\,|\,b_{\alpha\beta}u^{\alpha}u^{\beta}=|\nabla
p\,|\,{\rm{II}}(u,u).$

Inserting this expression into (2.9), we arrive to (2.8). ∎

Although we use some special coordinates in the above proof, the obtained
equations (2.3) and (2.8) are independent of a coordinates choice.

All our arguments in this section are invertible, i.e., the following
statement is valid.

######  Proposition 2.5.

Let a smooth vector field $u$ and smooth real function $p$ be defined on an
open set $U\subset{\mathbb{R}}^{n}$. Choose a point $x_{0}\in U$ such that
$\nabla p(x_{0})\neq 0$ and set $p_{0}=p(x_{0})$. Assume that in some
neighborhood of $x_{0}$ the integral curves of $u$ are geodesics of level
hypersurfaces $M_{p}$ and the equations (2.3) and (2.8) are satisfied. Then
$(u,p)$ is a Gavrilov solution to the Euler equations in some neighborhood of
$x_{0}$.

##  3\. One example of Gavrilov flow

The following example in even dimensions was suggested by Ya. Bazaykin in a
private communication. Most likely, this was known before. In the 2D case it
is called a vortex [7].

Let $(x_{1},\dots,x_{2n})$ be Cartesian coordinates in ${\mathbb{R}}^{2n}$.
Set

$u_{2j-1}(x)=-x_{2j},\ u_{2j}(x)=x_{2j-1}\ (j=1,\dots,n),\quad
p(x)=\frac{1}{2}\,|x|^{2}.$ (3.1)

It is easy to check that the equations (1.1)–(1.3) hold in
${\mathbb{R}}^{2n}$. Observe that $|u|^{2}=2p$. All points $x\neq 0$ are
regular. Isobaric hypersurfaces are spheres
$M_{p}=\\{x\in{\mathbb{R}}^{2n}:|x|^{2}=2p\\}$. Integral curves of $u$
(particle trajectories) are circles centered at the origin. Every sphere
$M_{p}$ is foliated by particle trajectories. This foliation coincides with
the Hopf fiber bundle ${\mathbb{S}}^{2n-1}\rightarrow{\mathbb{C}}P^{n-1}$ of
an odd-dimensional sphere over the complex projective space.

A Gavrilov flow on ${\mathbb{R}}^{2n+1}$ can be obtained as a direct product
of the flow (3.1) with a constant velocity flow. Namely,

$u_{2j-1}(x)=-x_{2j},\ u_{2j}(x)=x_{2j-1}\ (j=1,\dots,n),\quad
u_{2n+1}(x)=a=\mbox{const}$ (3.2)

and $p(x)=\frac{1}{2}(x_{1}^{2}+\dots+x_{n}^{2})$. Isobaric hypersurfaces are
cylinders ${\mathbb{S}}^{2n-1}\times\mathbb{R}$, and particle trajectories are
either circles (when $a=0$) or helices (when $a\neq 0$).

In aim to apply the Gavrilov localization to the flow (3.1), choose a
compactly supported smooth function $\alpha:[0,\infty)\rightarrow\mathbb{R}$
such that $\alpha(r)=0$ for $r\leq\varepsilon$ with some $\varepsilon>0$ and
define the function $\beta:[0,\infty)\rightarrow\mathbb{R}$ by
$\beta(r)=-\int_{r}^{\infty}s\alpha^{2}(s)\,ds$. Then

$\tilde{u}(x)=\alpha(|x|)u(x),\quad\tilde{p}(x)=\beta(|x|)$

is a smooth compactly supported Gavrilov flow on ${\mathbb{R}}^{2n}$
satisfying $|\tilde{u}|^{2}=\psi(\tilde{p})$ with a function $\psi$ uniquely
determined by $\alpha$. In particular, if $\alpha$ is supported in
$(r_{0}-\delta,r_{0}+\delta)$ for some $r_{0}>\delta>0$, then the velocity
$\tilde{u}$ is supported in the spherical layer
$\\{x\in\mathbb{R}^{2n}:r_{0}-\delta<|x|<r_{0}+\delta\\}$, and the pressure
$\tilde{p}$ is supported in the ball
$\\{x\in\mathbb{R}^{2n}:|x|<r_{0}+\delta\\}$ with $\tilde{p}=\mbox{const}$ in
the smaller ball $\\{x\in\mathbb{R}^{2n}:|x|\leq r_{0}-\delta\\}$. Then we can
take a linear combination of several such localized flows with disjoints
supports. In particular, a periodic Gavrilov flow can be constructed in this
way.

##  4\. Plain sections of a Gavrilov flow

Let a smooth solution $(u,p)$ of the Euler equations (1.1)–(1.2) be defined on
${\mathbb{R}}^{n}$. Assume the solution to decay sufficiently fast at infinity
together with first order derivatives (e.g., it can be a smooth compactly
supported solution). Then, see [8], the equality

$\int\nolimits_{P}\big{(}\xi\cdot u(x)\big{)}\big{(}\nu\cdot
u(x)\big{)}\,dx=0$ (4.1)

holds for every affine hyperplane $P\subset{\mathbb{R}}^{n}$ and every vector
$\xi\in{\mathbb{R}}^{n}$ parallel to $P$, where $\nu$ is the normal vector to
the hyperplane $P$ and $dx$ stands for the $(n-1)$-dimensional Lebesgue
measure on $P$. Actually there are $(n-1)$ independent equations in (4.1)
since the vector $\xi$ can take $(n-1)$ linearly independent values from the
space $\nu^{\bot}=\\{\xi\in{\mathbb{R}}^{n}\mid\nu\cdot\xi=0\\}$.

For a Gavrilov flow, the equation (4.1), combined with the Gavrilov
localization, yields an interesting statement. The following theorem can be
easily generalized to the case of an arbitrary dimension.

######  Theorem 4.1.

Let $(u,p)$ be a smooth Gavrilov flow defined on ${\mathbb{R}}^{3}$ and
sufficiently fast decaying at infinity together with first order derivatives.
Let $M_{p_{0}}$ be a regular isobaric surface and let an affine plane $P_{0}$
transversally intersect $M_{p_{0}}$. Then, for any $p$ sufficiently close to
$p_{0}$ and for any affine plane $P$ sufficiently close to $P_{0}$, we get

$\int\limits_{M_{p}\cap P}\frac{1}{|\nabla q(x)|}\big{(}\xi\cdot
u(x)\big{)}\big{(}\nu\cdot u(x)\big{)}\,ds=0,$ (4.2)

where $q\in C^{\infty}(P)$ is the restriction of the function $p$ to the plane
$P$, $\nu$ is the unit normal vector to $P$, and $\xi$ is an arbitrary vector
parallel to $P$. The integration in (4.2) is performed with respect to the arc
length $ds$ of the curve $M_{p}\cap P$.

###### Proof.

Since $M_{p_{0}}$ and $P_{0}$ intersect transversally, the same is true for
$M_{p}$ and $P$ for any $p$ close to $p_{0}$ and for any plane $P$ close to
$P_{0}$. We fix such $p$ and $P$, set $q=p\,|_{P}$ and $\gamma=M_{p}\cap P$.
Observe that $\gamma$ is a regular curve on the plane $P$ and the gradient
$\nabla q$ does not vanish in some neighborhood of $\gamma$. Therefore the
integral in (4.2) is well defined.

We parameterize the curve $\gamma$ by the arc length, $\gamma=\gamma(s)$. Then
we choose local coordinates $(s,\tau)$ in a neighborhood $U\subset P$ of
$\gamma$ in the same way as in the proof of Proposition 2.3. Namely, the
coordinates are chosen so that $x(s,0)=\gamma(s)$ and $q(x(s,\tau))=p+\tau$.
For every $s_{0}$, the coordinate line $\delta(\tau)=x(s_{0},\tau)$ starts at
$\gamma(s_{0})$ orthogonally to $\gamma$ with the initial speed
$|\dot{\delta}(0)|=\frac{1}{|\nabla q(\gamma(s_{0}))|}$. Therefore the area
form $dx$ of the plane $P$ is written in the chosen coordinates as
$dx=\big{(}\frac{1}{|\nabla q(\gamma(s))|}+o(\tau)\big{)}ds\,d\tau$.

Fix a smooth function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ such that
$\mu(r)=0$ for $|r|\geq 1$, $\mu(r)>0$ for $|r|<1$, and
$\int_{-1}^{1}\mu(r)\,dr=1$. For $\varepsilon>0$, set
$\alpha_{\varepsilon}(r)=\sqrt{\mu((r-c)/\varepsilon)}$. Using the latter
function, we define the localized Gavrilov flow $(\tilde{u},\tilde{p})$ by
$\tilde{u}=\alpha_{\varepsilon}(p)\,u,\
\nabla\tilde{p}=\alpha_{\varepsilon}^{2}(p)\nabla p$. Applying (4.1) to
$(\tilde{u},\tilde{p})$, we obtain
$\int_{P}\alpha_{\varepsilon}(q(x))\big{(}\xi\cdot u(x)\big{)}\big{(}\nu\cdot
u(x)\big{)}\,dx=0$. In the chosen coordinates, this is written as

$\int\limits_{\gamma}\int\limits_{-\varepsilon}^{\varepsilon}\mu(\tau/\varepsilon)\Big{(}\frac{1}{|\nabla
q(\gamma(s))|}+o(\tau)\Big{)}\big{(}\xi\cdot u(x(s,\tau))\big{)}\big{(}m\cdot
u(x(s,\tau))\big{)}\,d\tau\,ds=0.$ (4.3)

The integrand can be represented as

$\displaystyle\mu(\tau/\varepsilon)\Big{(}\frac{1}{|\nabla
q(\gamma(s))|}+o(\tau)\Big{)}\big{(}\xi\cdot
u(x(s,\tau))\big{)}\big{(}\nu\cdot u(x(s,\tau))\big{)}$
$\displaystyle\qquad=\mu(\tau/\varepsilon)\frac{1}{|\nabla
q(\gamma(s))|}\big{(}\xi\cdot u(\gamma(s))\big{)}\big{(}\nu\cdot
u(\gamma(s))\big{)}+o(1).$

The variables $s$ and $\tau$ are separated up to $o(\tau)$ on the right-hand
side of the latter formula. Using this representation and
$\int_{-\varepsilon}^{\varepsilon}\mu(\tau/\varepsilon)\,d\tau=\varepsilon$,
we obtain from (4.3) $\varepsilon\int\limits_{\gamma}\frac{1}{|\nabla
q(\gamma(s))|}\big{(}\xi\cdot u(\gamma(s))\big{)}\big{(}\nu\cdot
u(\gamma(s))\big{)}\,ds+o(\varepsilon)=0.$ In the limit as
$\varepsilon\rightarrow 0$, this gives (4.2). ∎

##  5\. Differential equations for a Gavrilov flow

Let a Gavrilov flow $(u,p)$ be defined on an open set of ${\mathbb{R}}^{3}$
with Cartesian coordinates $(x,y,z)$. Assume that, for $p\in(-p_{0},p_{0})$,
the regular isobaric surface $M_{p}$ is the graph

$z=f(p;x,y)\quad\big{(}(x,y)\in U\big{)}$ (5.1)

of a smooth function $f$ for some open set $U\subset{\mathbb{R}}^{2}$. The
first and the second quadratic forms of this $M_{p}$ are

$\displaystyle
I=(1+f^{\prime}_{x}{}^{2})dx^{2}+2f^{\prime}_{x}f^{\prime}_{y}\,dxdy+(1+f^{\prime}_{y}{}^{2})dy^{2},$
$\displaystyle
II=\pm\frac{1}{\sqrt{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}}\big{(}f^{\prime\prime}_{xx}\,dx^{2}+2f^{\prime\prime}_{xy}\,dxdy+f^{\prime\prime}_{yy}\,dy^{2}\big{)},$
(5.2)

where the sign depends on the choice of the unit vector normal to $M_{p}$.
Christoffel symbols of this metric are

$\displaystyle\Gamma^{x}_{xx}$
$\displaystyle=\frac{f^{\prime}_{x}f^{\prime\prime}_{xx}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}},\quad\Gamma^{x}_{xy}=\frac{f^{\prime}_{x}f^{\prime\prime}_{xy}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}},\quad\Gamma^{x}_{yy}=\frac{f^{\prime}_{x}f^{\prime\prime}_{yy}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}},$
(5.3) $\displaystyle\Gamma^{y}_{xx}$
$\displaystyle=\frac{f^{\prime}_{y}f^{\prime\prime}_{xx}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}},\quad\Gamma^{y}_{xy}=\frac{f^{\prime}_{y}f^{\prime\prime}_{xy}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}},\quad\Gamma^{y}_{yy}=\frac{f^{\prime}_{y}f^{\prime\prime}_{yy}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}.$

Let $(u^{x},u^{y})$ be geometric coordinates of the vector field $u$, i.e.,
$u=u^{x}(p;x,y)\frac{\partial}{\partial
x}+u^{y}(p;x,y)\frac{\partial}{\partial y}$, where $\frac{\partial}{\partial
x}$ and $\frac{\partial}{\partial y}$ are considered as coordinate vector
fields tangent to the surface $M_{p}$. Using (5.3), we compute covariant
derivatives

$\displaystyle\nabla_{\\!x}u^{x}$ $\displaystyle=\frac{\partial
u^{x}}{\partial
x}+\frac{f^{\prime}_{x}(f^{\prime\prime}_{xx}u^{x}+f^{\prime\prime}_{xy}u^{y})}{1+f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}},\quad\nabla_{\\!y}u^{x}=\frac{\partial
u^{x}}{\partial
y}+\frac{f^{\prime}_{x}(f^{\prime\prime}_{xy}u^{x}+f^{\prime\prime}_{yy}u^{y})}{1+f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}},$
(5.4) $\displaystyle\nabla_{\\!x}u^{y}$ $\displaystyle=\frac{\partial
u^{y}}{\partial
y}+\frac{f^{\prime}_{y}(f^{\prime\prime}_{xx}u^{x}+f^{\prime\prime}_{xy}u^{y})}{1+f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}},\quad\nabla_{\\!y}u^{y}=\frac{\partial
u^{y}}{\partial
y}+\frac{f^{\prime}_{y}(f^{\prime\prime}_{xy}u^{x}+f^{\prime\prime}_{yy}u^{y})}{1+f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}.$

In particular,

$\mbox{div}\,u=\nabla_{\\!x}u^{x}{+}\nabla_{\\!y}u^{y}=\frac{\partial
u^{x}}{\partial x}{+}\frac{\partial u^{y}}{\partial
y}+\frac{(f^{\prime}_{x}f^{\prime\prime}_{xx}{+}f^{\prime}_{y}f^{\prime\prime}_{xy})u^{x}{+}(f^{\prime}_{x}f^{\prime\prime}_{xy}{+}f^{\prime}_{y}f^{\prime\prime}_{yy})u^{y}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}.$
(5.5)

Substituting expressions (5.4) into the formula
$\nabla_{\\!u}u=(u^{x}\nabla_{\\!x}u^{x}+u^{y}\nabla_{\\!y}u^{x})\frac{\partial}{\partial
x}+(u^{x}\nabla_{\\!x}u^{y}+u^{y}\nabla_{\\!y}u^{y})\frac{\partial}{\partial
y},$ we get

$\displaystyle\nabla_{\\!u}u$ $\displaystyle=\Big{(}u^{x}\frac{\partial
u^{x}}{\partial x}+u^{y}\frac{\partial u^{x}}{\partial
y}{+}\frac{f^{\prime}_{x}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}\big{(}f^{\prime\prime}_{xx}(u^{x})^{2}{+}2f^{\prime\prime}_{xy}u^{x}u^{y}{+}f^{\prime\prime}_{yy}(u^{y})^{2}\big{)}\Big{)}\frac{\partial}{\partial
x}$ $\displaystyle+\Big{(}u^{x}\frac{\partial u^{y}}{\partial
x}+u^{y}\,\frac{\partial u^{y}}{\partial
y}{+}\frac{f^{\prime}_{y}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}\big{(}f^{\prime\prime}_{xx}(u^{x})^{2}{+}2f^{\prime\prime}_{xy}u^{x}u^{y}{+}f^{\prime\prime}_{yy}(u^{y})^{2}\big{)}\Big{)}\frac{\partial}{\partial
y}\,.$

Being a geodesic vector field, $u$ satisfies the condition $\nabla_{\\!u}u=0$.
We arrive to the system

$\displaystyle u^{x}\,\frac{\partial u^{x}}{\partial x}+u^{y}\,\frac{\partial
u^{x}}{\partial
y}+\frac{f^{\prime}_{x}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}\big{(}f^{\prime\prime}_{xx}(u^{x})^{2}+2f^{\prime\prime}_{xy}u^{x}u^{y}+f^{\prime\prime}_{yy}(u^{y})^{2}\big{)}$
$\displaystyle=0,$ (5.6) $\displaystyle u^{x}\,\frac{\partial u^{y}}{\partial
x}+u^{y}\,\frac{\partial u^{y}}{\partial
y}+\frac{f^{\prime}_{y}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}\big{(}f^{\prime\prime}_{xx}(u^{x})^{2}+2f^{\prime\prime}_{xy}u^{x}u^{y}+f^{\prime\prime}_{yy}(u^{y})^{2}\big{)}$
$\displaystyle=0.$

Let us express $|\nabla p\,|$ in terms of the function $f$. Let $p(x,y,z)$ be
the pressure in Cartesian coordinates. We have the identity
$p(x,y,f(p;x,y))=p$. Differentiate the identity to get

$\displaystyle
p^{\prime}_{x}(x,y,f(p;x,y))+p^{\prime}_{z}(x,y,f(p;x,y))f^{\prime}_{x}(p;x,y)$
$\displaystyle=0,$ $\displaystyle
p^{\prime}_{y}(x,y,f(p;x,y))+p^{\prime}_{z}(x,y,f(p;x,y))f^{\prime}_{y}(p;x,y)$
$\displaystyle=0,$ $\displaystyle
p^{\prime}_{z}(x,y,f(p;x,y))f^{\prime}_{p}(p;x,y)$ $\displaystyle=1.$

From this we get $p^{\prime}_{x}=-\frac{f^{\prime}_{x}}{f^{\prime}_{p}},\
p^{\prime}_{y}=-\frac{f^{\prime}_{y}}{f^{\prime}_{p}},\
p^{\prime}_{z}=\frac{1}{f^{\prime}_{p}}$. The derivative $f^{\prime}_{p}$ does
not vanish on a regular isobaric surface (5.1). Thus,

$|\nabla
p\,|^{2}=p^{\prime}_{x}{}^{2}+p^{\prime}_{y}{}^{2}+p^{\prime}_{z}{}^{2}=\frac{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}{f^{\prime}_{p}{}^{2}}.$
(5.7)

Substituting expressions (5.5) and (5.7) into (2.3), we arrive to the equation

$\displaystyle\frac{\partial u^{x}}{\partial x}+\frac{\partial u^{y}}{\partial
y}+\frac{(f^{\prime}_{x}f^{\prime\prime}_{xx}+f^{\prime}_{y}f^{\prime\prime}_{xy})u^{x}+(f^{\prime}_{x}f^{\prime\prime}_{xy}+f^{\prime}_{y}f^{\prime\prime}_{yy})u^{y}}{1\\!+\\!f^{\prime}_{x}{}^{2}\\!+\\!f^{\prime}_{y}{}^{2}}$
$\displaystyle=\frac{1}{2}\cdot\frac{f^{\prime}_{p}{}^{2}}{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}\Big{[}u^{x}\frac{\partial}{\partial
x}\Big{(}\frac{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}{f^{\prime}_{p}{}^{2}}\Big{)}+u^{y}\frac{\partial}{\partial
y}\Big{(}\frac{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}{f^{\prime}_{p}{}^{2}}\Big{)}\Big{]}.$

After the obvious simplification, it becomes

$\frac{\partial u^{x}}{\partial x}+\frac{\partial u^{y}}{\partial
y}+\frac{1}{f^{\prime}_{p}}(u^{x}f^{\prime\prime}_{px}+u^{y}f^{\prime\prime}_{py})=0.$

Substituting expressions (5.2) and (5.7) into (2.8), we arrive to the equation
$f^{\prime\prime}_{xx}(u^{x})^{2}+2f^{\prime\prime}_{xy}u^{x}u^{y}+f^{\prime\prime}_{yy}(u^{y})^{2}=\frac{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}{|f^{\prime}_{p}|}$.
Using this, equations (5.6) can be written as

$u^{x}\,\frac{\partial u^{x}}{\partial x}+u^{y}\,\frac{\partial
u^{x}}{\partial y}+\frac{f^{\prime}_{x}}{|f^{\prime}_{p}|}=0,\quad
u^{x}\,\frac{\partial u^{y}}{\partial x}+u^{y}\,\frac{\partial u^{y}}{\partial
y}+\frac{f^{\prime}_{y}}{|f^{\prime}_{p}|}=0.$

We have thus proved the following

######  Proposition 5.1.

Let a Gavrilov flow $(u,p)$ be defined on an open set of ${\mathbb{R}}^{3}$.
Assume that for $p\in(-p_{0},p_{0})$ the isobaric surface $M_{p}$ is regular
and coincides with the graph of a smooth function $z=f(p;x,y)\ \big{(}(x,y)\in
U\subset{\mathbb{R}}^{2}\big{)}$. Write the restriction of the vector field
$u$ to the surface $M_{p}$ in the form $u=u^{x}(p;x,y)\frac{\partial}{\partial
x}+u^{y}(p;x,y)\frac{\partial}{\partial y}$. Then the derivative
$f^{\prime}_{p}$ does not vanish and the functions $f(p;x,y),u^{x}(p;x,y)$ and
$u^{y}(p;x,y)$ satisfy the equations

$\displaystyle\frac{\partial(f^{\prime}_{p}u^{x})}{\partial
x}+\frac{\partial(f^{\prime}_{p}u^{y})}{\partial y}=0,$ (5.8) $\displaystyle
u^{x}\,\frac{\partial u^{x}}{\partial x}+u^{y}\,\frac{\partial u^{x}}{\partial
y}+\frac{f^{\prime}_{x}}{|f^{\prime}_{p}|}=0,$ (5.9) $\displaystyle
u^{x}\,\frac{\partial u^{y}}{\partial x}+u^{y}\,\frac{\partial u^{y}}{\partial
y}+\frac{f^{\prime}_{y}}{|f^{\prime}_{p}|}=0,$ (5.10) $\displaystyle
f^{\prime\prime}_{xx}(u^{x})^{2}+2f^{\prime\prime}_{xy}u^{x}u^{y}+f^{\prime\prime}_{yy}(u^{y})^{2}=\frac{1+f^{\prime}_{x}{}^{2}+f^{\prime}_{y}{}^{2}}{|f^{\prime}_{p}|}.$
(5.11)

It is easy to check that the system (5.8)–(5.11) has the following solution:

$\displaystyle
f(x,y,p)=f(x,p)=-\sqrt{r^{2}(p)-x^{2}}\quad\big{(}-r(p)<x<r(p)\big{)},$ (5.12)
$\displaystyle
u^{x}(x,y,p)=u^{x}(x,p)=\frac{\sqrt{r^{2}(p)-x^{2}}}{\sqrt{r(p)|r^{\prime}(p)|}},\quad
u^{y}(x,y,p)=u^{y}(p)=b(p),$ (5.13)

where $r(p)$ is a smooth positive function with non-vanishing derivative, and
$b(p)$ is an arbitrary smooth function. For every $p$, the graph $M_{p}$ of
the function $(x,y)\mapsto f(x,p)$ is the half of the cylinder
${\tilde{M}}_{p}=\\{(x,y,z)\mid x^{2}+z^{2}=r^{2}(p)\\}$. Observe that
${\tilde{M}}_{p}$ is a surface of revolution around the $y$-axis. Thus,
(5.12)–(5.13) is an axisymmetric Gavrilov flow, and we do not know any
solution to the system (5.8)–(5.11) different of the (modified) solution
(5.12)–(5.13). We will study axisymmetric Gavrilov flows in the next section.

##  6\. Axisymmetric Gavrilov flows

Let $(r,z,\theta)$ be cylindrical coordinates in ${\mathbb{R}}^{3}$ related to
Cartesian coordinates $(x_{1},x_{2},x_{3})$ by $x_{1}=r\cos\theta,\
x_{2}=r\sin\theta,\ x_{3}=z.$ We study a Gavrilov flow $(u,p)$ invariant under
rotations around the $z$-axis. The flow is defined in an open set
$\tilde{U}\subset\\{(r,z,\theta):r>0\\}$ invariant under rotations around the
$z$-axis. Such a rotationally invariant set $\tilde{U}$ is uniquely determined
by the two-dimensional set
$U=\tilde{U}\cap\\{\theta=0\\}\subset\\{(r,z):r>0\\}$. For brevity we say that
an axisymmetric Gavrilov flow is defined in $U$.

A regular isobaric surface $M_{p}$ is a surface of revolution determined by
its generatrix $\Gamma_{p}=M_{p}\cap\,U$. We parameterize the curve
$\Gamma_{p}$ by the arc length $r=R(p,t)>0,\ z=Z(p,t)$,

$R^{\prime}_{t}{}^{2}+Z^{\prime}_{t}{}^{2}=1.$ (6.1)

The variables $(t,\theta)$ serve as coordinates on the isobaric surface
$M_{p}$. Since the vector field $u$ is tangent to $M_{p}$, it is uniquely
represented as $u=u^{t}(p,t)\frac{\partial}{\partial
t}+u^{\theta}(p,t)\frac{\partial}{\partial\theta}$, where
$\big{(}u^{t}(p,t),u^{\theta}(p,t)\big{)}$ are geometric coordinates of $u$.
Physical coordinates of $u$ are given by

$u=u_{r}(r,z)e_{r}+u_{z}(r,z)e_{z}+u_{\theta}(r,z)e_{\theta},$ (6.2)

where $e_{r},e_{z},e_{\theta}$ are unit coordinate vectors. Such coordinates
are related by

$u_{r}=R^{\prime}_{t}u^{t},\quad u_{z}=Z^{\prime}_{t}u^{t},\quad
u_{\theta}=Ru^{\theta}.$ (6.3)

We are going to write down differential equations for a Gavrilov flow in terms
of the functions $(R,Z,u^{t},u^{\theta})$. First, the first quadratic form of
$M_{p}$ in coordinates $(t,\theta)$ is

$I=dt^{2}+R^{2}d\theta^{2}.$ (6.4)

Christoffel symbols of the metric are
$\Gamma^{t}_{\theta\theta}=-RR^{\prime}_{t},\
\Gamma^{\theta}_{t\theta}=\frac{R^{\prime}_{t}}{R},\
\Gamma^{t}_{tt}=\Gamma^{t}_{t\theta}=\Gamma^{\theta}_{tt}=\Gamma^{\theta}_{\theta\theta}=0$.
Using these formulas, we calculate

$\nabla_{\\!t}u^{t}=\frac{\partial u^{t}}{\partial
t},\quad\nabla_{\\!\theta}u^{t}=-RR^{\prime}_{t}\,u^{\theta},\quad\nabla_{\\!t}u^{\theta}=\frac{\partial
u^{\theta}}{\partial
t}+\frac{R^{\prime}_{t}}{R}\,u^{\theta},\quad\nabla_{\\!\theta}u^{\theta}=\frac{R^{\prime}_{t}}{R}\,u^{t}.$

In particular,

$\mbox{div}\,u=\nabla_{\\!t}u^{t}+\nabla_{\\!\theta}u^{\theta}=\frac{\partial
u^{t}}{\partial t}+\frac{R^{\prime}_{t}}{R}\,u^{t}.$ (6.5)

The restriction of $u$ to $M_{p}$ is a geodesic vector field, i.e.,
$\nabla_{\\!u}u=0$. This gives the system

$u^{t}\,\frac{\partial u^{t}}{\partial
t}-RR^{\prime}_{t}(u^{\theta})^{2}=0,\quad u^{t}\,\frac{\partial
u^{\theta}}{\partial t}+\frac{2R^{\prime}_{t}}{R}\,u^{t}u^{\theta}=0.$ (6.6)

Let us express $|\nabla p\,|$ in terms of the functions $R(p,t)$ and $Z(p,t)$.
Let $p=p(r,z)$ be the pressure in cylindric coordinates (it is independent of
$\theta$). We have the identity $p\big{(}R(p,t),Z(p,t)\big{)}=p.$
Differentiating the identity with respect to $p$ and $t$, we arrive to the
linear algebraic system with unknowns
$p^{\prime}_{r}\big{(}R(p,t),Z(p,t)\big{)}$ and
$p^{\prime}_{z}\big{(}R(p,t),Z(p,t)\big{)}$

$\displaystyle
R^{\prime}_{p}(p,t)p^{\prime}_{r}\big{(}R(p,t),Z(p,t)\big{)}+Z^{\prime}_{p}(p,t)p^{\prime}_{z}\big{(}R(p,t),Z(p,t)\big{)}$
$\displaystyle=1,$ $\displaystyle
R^{\prime}_{t}(p,t)p^{\prime}_{r}\big{(}R(p,t),Z(p,t)\big{)}+Z^{\prime}_{t}(p,t)p^{\prime}_{z}\big{(}R(p,t),Z(p,t)\big{)}$
$\displaystyle=0.$

Solving the system, we have

$p^{\prime}_{r}\big{(}R(p,t),Z(p,t)\big{)}=J^{-1}(p,t)Z^{\prime}_{t}(p,t),\
p^{\prime}_{z}\big{(}R(p,t),Z(p,t)\big{)}=-J^{-1}(p,t)R^{\prime}_{t}(p,t),$

where

$J=\left|\begin{array}[]{cc}R^{\prime}_{p}&R^{\prime}_{t}\\\
Z^{\prime}_{p}&Z^{\prime}_{t}\end{array}\right|$ (6.7)

is the Jacobian of the transformation $r=R(p,t),\quad z=Z(p,t)$. The Jacobian
does not vanish. This implies with the help of (6.1)

$|\nabla p\,|=|J|^{-1}.$ (6.8)

Substituting expressions (6.5) and (6.8) into (2.3), we arrive to the equation

$\frac{\partial u^{t}}{\partial
t}+\Big{(}\frac{R^{\prime}_{t}}{R}+\frac{J^{\prime}_{t}}{J}\Big{)}u^{t}=0.$
(6.9)

The second quadratic form of the surface $M_{p}$ is expressed in coordinates
$(t,\theta)$ by
$II=-((R^{\prime}_{t}Z^{\prime\prime}_{tt}-Z^{\prime}_{t}R^{\prime\prime}_{tt})\,dt^{2}+RZ^{\prime}_{t}\,d\theta^{2}).$
The sign on the right-hand side is chosen taking our agreement into account:
the unit normal vector to the surface $M_{p}$ must coincide with $\frac{\nabla
p}{|\nabla p\,|}$. Recall that the curvature $\kappa=\kappa(p,t)$ of the plane
curve $r=R(p,t),z=Z(p,t)$ is expressed, under the condition (6.1), by

$\kappa=R^{\prime}_{t}Z^{\prime\prime}_{tt}-Z^{\prime}_{t}R^{\prime\prime}_{tt}.$
(6.10)

Using the latter equality, the previous formula takes the form

$II=-(\kappa\,dt^{2}+RZ^{\prime}_{t}\,d\theta^{2}).$ (6.11)

By (6.8) and (6.11), the equation (2.8) takes the form

$\kappa(u^{t})^{2}+RZ^{\prime}_{t}(u^{\theta})^{2}=|J|^{-1}.$ (6.12)

We obtained the system of five equations (6.1), (6.6a,b), (6.9), (6.12) in
four unknown functions $(R,Z,u^{t},u^{\theta})$. The functions $J$ and
$\kappa$ participating in the system are expressed through $(R,Z)$ by (6.7)
and (6.10) respectively. We proceed to the analysis of the system.

All isobaric surfaces $M_{p}$ under consideration are assumed to be regular
and connected. The equation (6.9) implies the following alternative for every
$p_{0}$: either $u^{t}(p_{0},t)\neq 0$ for all $t$ or $u^{t}(p_{0},t)\equiv
0$. The second case of the alternative is realized in the example (3.2) with
$a=0$. The converse statement is true at least partially: If
$u^{t}(p_{0},t)\equiv 0$, then $M_{p_{0}}$ coincides with the cylinder
$\\{r=\mbox{const}>0\\}$ and particle trajectories living on $M_{p_{0}}$ are
horizontal circles. Indeed, if $u^{t}(p_{0},t)\equiv 0$ then, as is seen from
(6.1) and (6.6b), $R^{\prime}_{t}(p_{0},t)\equiv 0$ and
$Z^{\prime}_{t}(p_{0},t)\equiv\pm 1$. Nevertheless, it is possible that
$u^{t}(p_{0},t)\equiv 0$ but $u^{t}(p,t)\neq 0$ for $p$ close to $p_{0}$; the
corresponding example can be constructed by a slight modification of (3.2).

To avoid degenerate cases of the previous paragraph, we additionally assume
$u^{t}(p,t)\neq 0$ for all $(p,t)$. Recall that we study flows up to the
equivalence (1.4). Therefore the latter assumption can be written without lost
of generality in the form

$u^{t}(p,t)>0\quad\mbox{for all}\quad(p,t).$ (6.13)

The equation (6.6a) simplifies under the assumption (6.13) to the following:

$\frac{\partial u^{\theta}}{\partial
t}+\frac{2R^{\prime}_{t}}{R}\,u^{\theta}=0.$ (6.14)

If $u^{\theta}(p,t_{0})=0$ for some $t_{0}$, then (6.14) implies that
$u^{\theta}(p,t)=0$ for all $t$. On the other hand, assuming that
$u^{\theta}(p,t)\neq 0$ for all $t$ and for a fixed $p$, we can rewrite (6.14)
in the form $\frac{\partial(\log u^{\theta})}{\partial t}+\frac{\partial(\log
R^{2})}{\partial t}=0$. From this we get

$u^{\theta}(p,t)=\frac{b(p)}{R^{2}(p,t)}$ (6.15)

with some function $b(p)$. This equality is true also for such $p_{0}$ that
$u^{\theta}(p_{0},t)=0$ for all $t$, just by setting $b(p_{0})=0$. The
equality (6.15) implies smoothness of the function $b$.

Substituting the expression (6.15) into (6.6a), we get
$\frac{\partial(u^{t})^{2}}{\partial
t}=2\,\frac{R^{\prime}_{t}}{R^{3}}b^{2}(p)$, that can be written in the form
$\frac{\partial(u^{t})^{2}}{\partial t}=-\frac{\partial}{\partial
t}\big{(}\frac{b^{2}(p)}{R^{2}}\big{)}$. From this we obtain

$u^{t}(p,t)=\frac{\sqrt{d(p)R^{2}(p,t)-b^{2}(p)}}{R(p,t)},$ (6.16)

where $d(p)$ is a smooth function satisfying

$d(p)R^{2}(p,t)-b^{2}(p)>0.$ (6.17)

By (6.4), $|u|^{2}=(u^{t})^{2}+R^{2}(u^{\theta})^{2}$ is valid. Substituting
values (6.15)–(6.16), we get

$|u(p,t)|^{2}=d(p).$ (6.18)

Thus, we have discovered an important phenomenon: for an axisymmetric Gavrilov
flow, all particles living on an isobaric surface $M_{p}$ move with the same
speed. In other words, constants $c$ and $C$ in the Bernoulli law (2.1) can be
expressed through each other. For instance, $|u|^{2}=3p$ in example [2], and
$|u|^{2}=mp$ with an arbitrary positive constant $m$ in example [1]. We guess
that $d(p)$ in (6.18) can be an arbitrary smooth positive function. Most
likely, the phenomenon is absent for a general (not axisymmetric) Gavrilov
flow, at least we cannot derive a relation of (6.18) type from (5.8)–(5.11).

Studying Gavrilov flows up to equivalence, we can multiply $u(p,t)$ by a non-
vanishing smooth function $\varphi(p)$. This opportunity was already used to
fix the sign of $u^{t}$ in (6.13). We still have the freedom of multiplying
$u(p,t)$ by a positive smooth function $\varphi(p)$ together with the
corresponding change of the pressure. Choosing $\varphi(p)=d(p)^{-1/2}$ and
denoting the new Gavrilov flow by $(u,p)$ again, we simplify (6.18) to the
following:

$|u(p,t)|^{2}=1.$ (6.19)

The inequality (6.17) becomes now $|b(p)|<R(p,t)$, and formula (6.16) takes
the form

$u^{t}(p,t)=\frac{\sqrt{R^{2}(p,t)-b^{2}(p)}}{R(p,t)},$ (6.20)

We continue our analysis under assumptions (6.13) and (6.19). In virtue of
(6.13), the equation (6.9) can be written in the form $\frac{\partial(\log
u^{t})}{\partial t}+\frac{\partial(\log(|J|R))}{\partial t}=0.$ This implies

$u^{t}(p,t)=\frac{\alpha^{2}(p)}{|J(p,t)|R(p,t)}$ (6.21)

with some positive smooth function $\alpha(p)$. Comparing (6.20) and (6.21),
we arrive to the equation
$\sqrt{R^{2}(p,t)-b^{2}(p)}\,|J(p,t)|=\alpha^{2}(p)$.

Finally, we simplify the equation (6.12). Substituting expressions (6.15) and
(6.20) for $u^{\theta}$ and $u^{t}$ into (6.12), we obtain $\kappa
R(R^{2}-b^{2})+b^{2}Z^{\prime}_{t}=R^{3}|J|^{-1}$. Expressing $|J|$ from
(6.21) and substituting the expression into the latter formula, we arrive to
the equation $\kappa
R(R^{2}-b^{2})+b^{2}Z^{\prime}_{t}=\frac{R^{3}\sqrt{R^{2}-b^{2}}}{\alpha^{2}}.$
We have thus proved the following

######  Theorem 6.1.

Let an axisymmetric Gavrilov flow $(u,p)$ be defined on an open set
$U\subset\\{(r,z)\mid r>0\\}$. Assume that every isobaric surface $M_{p}$ is
regular and connected. For the surface of revolution $M_{p}$, let
$r=R(p,t)>0,\ z=Z(p,t)$ be the arc length parametrization of the generatrix
$\Gamma_{p}$ of $M_{p}$. Assume also that in the representation
$u=u^{t}(p,t)\frac{\partial}{\partial
t}+u^{\theta}(p,t)\frac{\partial}{\partial\theta}$ the function $u^{t}(p,t)$
does not vanish. Then

(1) there exists a smooth positive function $d(p)$ such that $|u|^{2}=d(p)$.
Replacing $(u,p)$ with an equivalent Gavrilov flow and denoting the new flow
by $(u,p)$ again, we can assume without lost of generality that $u^{t}(p,t)>0$
and

$|u|^{2}=1.$ (6.22)

(2) under the assumption (6.22), the functions $R(p,t)$ and $Z(p,t)$ satisfy
the equations

$\displaystyle R^{\prime}_{t}{}^{2}+Z^{\prime}_{t}{}^{2}=1,$ (6.23)
$\displaystyle\sqrt{R^{2}-b^{2}(p)}\,\big{|}R^{\prime}_{p}Z^{\prime}_{t}-R^{\prime}_{t}Z^{\prime}_{p}\big{|}=\alpha^{2}(p),$
(6.24) $\displaystyle\kappa
R\big{(}R^{2}-b^{2}(p)\big{)}+b^{2}(p)Z^{\prime}_{t}=\frac{R^{3}\sqrt{R^{2}-b^{2}(p)}}{\alpha^{2}(p)}$
(6.25)

with some smooth functions $\alpha(p)>0$ and $b(p)$, where
$\kappa=\kappa(p,t)$ is the curvature of the plane curve $r=R(p,t),z=Z(p,t)$.
The functions $u^{t}(p,t)$ and $u^{\theta}(p,t)$ are expressed through
$\big{(}R(p,t),b(p)\big{)}$ by

$u^{t}=\frac{\sqrt{R^{2}-b^{2}(p)}}{R},\quad u^{\theta}=\frac{b(p)}{R^{2}}.$
(6.26)

######  Remark 6.2.

Let us make some remarks on Theorem 6.1.

1\. First, we attract reader’s attention to the hypothesis: $M_{p}$ are
connected surfaces. Otherwise functions $\alpha(p)$ and $b(p)$ can be
different on different connected components.

2\. Equations (6.23)–(6.26) are invariant under some transformations. First,
the parameter $t$ is defined up to a shift, i.e., nothing changes after the
replacement
$R(p,t)=\tilde{R}\big{(}p,t+t_{0}(p)\big{)},Z(p,t)=\tilde{Z}\big{(}p,t+t_{0}(p)\big{)}$.
Second, the equations are invariant under the changes
$Z(p,t)=\tilde{Z}(p,t)+z_{0}$ and $Z(p,t)=-\tilde{Z}(p,t),\
\kappa=-\tilde{\kappa}$, which mean a vertical shift of the origin in
${\mathbb{R}}^{3}$ and the change of the direction of the $z$-axis,
respectively.

3\. Compared to Proposition 5.1, Theorem 6.1 has an important advantage. The
unknown functions $(f,u^{x},u^{y})$ are not separated in the system
(5.8)–(5.11) and, probably, cannot be separated. On the other hand, equations
(6.23)–(6.25) involve only the functions $(R,Z)$ that determine isobaric
surfaces $M_{p}$ (the equations involve also $\alpha(p)$ and $b(p)$ that
appear as integration constants). If the system (6.23)–(6.25) was solved, the
velocity vector field $u$ would be determined by explicit formulas (6.26). Of
course, the simplification is possible thanks to the Clairaut integral for the
equation of geodesics on a surface of revolution. Although the Clairaut
integral is not mentioned in our proof of Theorem 6.1, formulas (6.26) are
actually equivalent to the Clairaut integral.

Theorem 6.1 has two other useful forms.

Case A. Assume, under hypotheses of Theorem 6.1, that the curve $\Gamma_{p}$
is the graph of a function $r=f(p,z)$. Then $f^{\prime}_{p}\neq 0$ and
(6.23)–(6.25) are equivalent to the system

$\displaystyle{f^{\prime}_{p}}^{2}=\frac{\alpha(p)(1+{f^{\prime}_{z}}^{2})}{f^{2}-b^{2}(p)},$
(6.27)
$\displaystyle\alpha(p)ff^{\prime\prime}_{zz}=b^{2}(p){f^{\prime}_{p}}^{2}-f^{3}f^{\prime}_{p}(1+{f^{\prime}_{z}}^{2})$
(6.28)

with the same functions $\alpha(p)>0$ and $b(p)$. The function $f$ must
satisfy $f(z,p)>|b(p)|$. The velocity vector $u$ is now represented as
$u=u^{z}(z,p)\frac{\partial}{\partial
z}+u^{\theta}(z,p)\frac{\partial}{\partial\theta}$, where the functions
$u^{z}$ and $u^{\theta}$ are expressed through $(f,\alpha,b)$ by
$u^{z}=\mbox{sgn}(u^{z})\,\frac{\sqrt{\alpha(p)}}{f|f^{\prime}_{p}|},\
u^{\theta}=\frac{b(p)}{f^{2}}$, where $\mbox{sgn}(u^{z})=\pm 1$ is the sign of
$u^{z}$ that is assumed do not vanish.

Case B. Assume, under hypotheses of Theorem 6.1, that the curve $\Gamma_{p}$
is the graph of a function $z=g(p,r)$. Then equations (6.23)–(6.25) are
equivalent to the system

$\displaystyle\big{(}r^{2}-b^{2}(p)\big{)}g^{\prime}_{p}{}^{2}-\alpha(p)(1+g^{\prime}_{r}{}^{2})=0,$
(6.29) $\displaystyle
r\alpha(p)g^{\prime\prime}_{rr}+b^{2}(p)g^{\prime}_{r}g^{\prime}_{p}{}^{2}+r^{3}g^{\prime}_{p}(1+g^{\prime}_{r}{}^{2})=0$
(6.30)

with the same functions $\alpha(p)>0$ and $b(p)$. The function $g(p,r)$ is
considered for $r\in\big{(}r_{1}(p),r_{2}(p)\big{)}$ with $|b(p)|<r_{1}(p)$.
The velocity vector $u$ is now represented as
$u=u^{r}(p,r)\frac{\partial}{\partial
r}+u^{\theta}(p,r)\frac{\partial}{\partial\theta}$, where the functions
$u^{r}$ and $u^{\theta}$ are expressed through $(g,\alpha,b)$ by
$u^{r}=\mbox{sgn}(u^{r})\,\frac{\sqrt{\alpha(p)}}{r|g^{\prime}_{p}|},\
u^{\theta}=\frac{b(p)}{r^{2}}$, where $\mbox{sgn}(u^{r})=\pm 1$ is the sign of
$u^{r}$ that does not vanish.

##  7\. Consistency conditions

First, we will recall some basic facts from theory of first order PDEs
following [5, Part I, Section 14]. Let us consider the system of two first
order PDEs

$F(x,y,z,p,q)=0,\quad G(x,y,z,p,q)=0,$ (7.1)

where $z=z(x,y)$ is an unknown function and
$p=z^{\prime}_{x},q=z^{\prime}_{y}$. We assume $F$ and $G$ to be sufficiently
smooth functions defined for $(x,y,z)\in U$ and for all
$(p,q)\in{\mathbb{R}}^{2}$, where $U\subset{\mathbb{R}}^{3}$ is an open set.
The system (7.1) is supplied with the initial condition

$z(x_{0},y_{0})=z_{0}$ (7.2)

for a point $(x_{0},y_{0},z_{0})\in U$. The Jacobi brackets (sometimes also
called Mayer brackets) of functions $F(x,y,z,p,q)$ and $G(x,y,z,p,q)$ are
defined by

$[F,G]=(F^{\prime}_{x}+pF^{\prime}_{z})G^{\prime}_{p}-(G^{\prime}_{x}+pG^{\prime}_{z})F^{\prime}_{p}+(F^{\prime}_{y}+qF^{\prime}_{z})G^{\prime}_{q}-(G^{\prime}_{y}+qG^{\prime}_{z})F^{\prime}_{q}.$
(7.3)

The system (7.1) is said to be an involutory system if $[F,G]\equiv 0$ for
$(x,y;z,p,q)\in U\times{\mathbb{R}}^{2}$. The system (7.1) is said to be a
complete system on the open set $U\times{\mathbb{R}}^{2}$ if the equation

$[F,G]=0$ (7.4)

is an algebraic corollary of the system (7.1), i.e., if (7.4) holds for
$(x,y;z,p,q)\in U\times{\mathbb{R}}^{2}$ satisfying (7.1). The equation (7.4)
is called the consistency condition (or integrability condition) for the
system (7.1). In the case of a complete system, for an arbitrary point
$(x_{0},y_{0},z_{0})\in U$, the initial value problem (7.1)–(7.2) has a unique
solution at least in some neighborhood of the point $(x_{0},y_{0})$. Several
methods are known for the numerical solution to the IVP (7.1)–(7.2), the Mayer
method is the most popular one [5].

Let us return to axisymmetric Gavrilov flows. Under certain additional
conditions, systems (6.27)–(6.28) and (6.29)–(6.30) are equivalent. We study
the system (6.27)–(6.28). Recall that the system is considered in a
neighborhood of a regular point, where the transform
$(p,z)\mapsto\big{(}f(p,z),z\big{)}$ is one-to-one. Therefore the derivative
$f^{\prime}_{p}$ does not vanish. Note that only $b^{2}$ is involved in
(6.27)–(6.28), not the function $b$ itself. To simplify our formulas a bit, we
introduce the function $\beta(p)=b^{2}(p)\geq 0$ and rewrite the system
(6.27)–(6.28) as

$\displaystyle(f^{2}-\beta){f^{\prime}_{p}}^{2}-\alpha(1+{f^{\prime}_{z}}^{2})=0,$
(7.5) $\displaystyle\alpha
ff^{\prime\prime}_{zz}-\beta{f^{\prime}_{p}}^{2}+f^{3}f^{\prime}_{p}(1+{f^{\prime}_{z}}^{2})=0.$
(7.6)

It is assumed that the function $f$ satisfies the inequality

$f(p,z)>\sqrt{\beta(p)}.$ (7.7)

Given functions $\alpha(p)>0$ and $\beta(p)\geq 0$, (7.5)–(7.6) is an
overdetermined system of two PDEs in one unknown function $f(p,z)$. The
overdeterminess is caused by the circumstance mentioned in Introduction: a
Gavrilov flow is defined by the overdetermined system (1.1)–(1.3). We pose the
question: What conditions should be imposed on
$\big{(}\alpha(p),\beta(p)\big{)}$ for solvability of the system (7.5)–(7.6)
at least locally, i.e., in a neighborhood of a given point $(p_{0},z_{0})$?
For a fixed $p$, (7.6) can be considered as a second order ODE with an unknown
function $f_{p}(z)=f(p,z)$. Note that the variable $z$ does not explicitly
participate in (7.6). Such an equation can be reduced to a first order ODE,
see [4, Ssection 15.3]. This observation is realized by the following
statement.

######  Lemma 7.1.

Let $C^{1}$-functions $\alpha(p)>0$ and $\beta(p)\geq 0$ be defined on an
interval $(p_{1},p_{2})$, where $-\infty\leq p_{1}<p_{2}\leq\infty$. Then the
following (1) and (2) are valid.

(1) Let $f(p,z)$ be a solution to the system (7.5)–(7.6) on a rectangle

$(p,z)\in(p_{1},p_{2})\times(z_{1},z_{2}),$ (7.8)

and let the inequality (7.7) be valid on the rectangle. Then there exists a
$C^{1}$-function $\gamma(p)$ on the interval $(p_{1},p_{2})$ satisfying the
equation

$f^{2}(\varepsilon
f^{2}+\gamma)^{2}(1+f^{\prime}_{z}{}^{2})-4\alpha(f^{2}-\beta)=0$ (7.9)

and the inequalities

$0<\varepsilon f^{2}+\gamma\leq\frac{2\alpha^{1/2}\sqrt{f^{2}-\beta}}{f},$
(7.10)

where $\varepsilon=\pm 1$ is the sign of $f^{\prime}_{p}$ that does not
vanish.

(2) Conversely, let $f(p,z)>0$ and $\gamma(p)$ satisfy (7.5) and (7.9)–(7.10).
Assume additionally that, for every $p\in(p_{1},p_{2})$, the derivative
$f^{\prime}_{z}(p,z)$ is not identically equal to zero on any interval
$(z^{\prime}_{1},z^{\prime}_{2})\subset(z_{1},z_{2})$. Then $f$ solves
(7.5)–(7.6) on the rectangle (7.8).

###### Proof.

First, we find from (7.5)

$f^{\prime}_{p}=\varepsilon\,\frac{\alpha^{1/2}\sqrt{1+{f^{\prime}_{z}}^{2}}}{\sqrt{f^{2}-\beta}},$
(7.11)

where $\varepsilon=\pm 1$ is the sign of $f^{\prime}_{p}$, and then substitute
the expression into (7.6)

$f^{\prime\prime}_{zz}=\frac{\beta(1+{f^{\prime}_{z}}^{2})}{f(f^{2}-\beta)}-\varepsilon\,\frac{f^{2}(1+{f^{\prime}_{z}}^{2})^{3/2}}{\alpha^{1/2}\sqrt{f^{2}-\beta}}.$
(7.12)

Let us show that equations (7.11) and (7.12) imply

$\frac{\partial}{\partial
z}\Big{(}\frac{\sqrt{f^{2}-\beta}}{f\sqrt{1+{f^{\prime}_{z}}^{2}}}-\varepsilon\,\frac{f^{2}}{2\alpha^{1/2}}\Big{)}=0.$
(7.13)

For this, we implement the differentiation in (7.13). The result can be
written as

$\Big{(}f^{\prime\prime}_{zz}-\frac{\beta(1+{f^{\prime}_{z}}^{2})}{f(f^{2}-\beta)}+\varepsilon\,\frac{f^{2}(1+{f^{\prime}_{z}}^{2})^{3/2}}{\alpha^{1/2}\sqrt{f^{2}-\beta}}\Big{)}f^{\prime}_{z}=0.$
(7.14)

By (7.12), the left-hand side of (7.14) is identically zero. This proves
(7.13). The equation (7.13) means the existence of a function $\gamma(p)$ such
that
$\frac{\sqrt{f^{2}-\beta}}{f\sqrt{1+{f^{\prime}_{z}}^{2}}}-\varepsilon\,\frac{f^{2}}{2\alpha^{1/2}}=\frac{\gamma}{2\alpha^{1/2}}.$
This can be written in the form

$2\alpha^{1/2}\sqrt{f^{2}-\beta}=f\sqrt{1+{f^{\prime}_{z}}^{2}}(\varepsilon
f^{2}+\gamma).$ (7.15)

By (7.7), $f>0$ and $f^{2}-\beta>0$. Therefore (7.15) implies the inequalities
(7.10). Squaring the equation (7.15), we get (7.9). We have proved the first
statement of the lemma.

The second assertion of the lemma is proved by interchanging the above
arguments using the following additional remark. To pass from (7.14) to
(7.12), we need to remove the factor $f^{\prime}_{z}$ on the left-hand side of
(7.14). To do this, it suffices to assume that for every $p\in(p_{1},p_{2})$,
the derivative $f^{\prime}_{z}(p,z)$ is not identically zero on any interval
$(z^{\prime}_{1},z^{\prime}_{2})\subset(z_{1},z_{2})$. ∎

By Lemma 7.1, the system (7.5)–(7.6) can be replaced with the following:

$\displaystyle(f^{2}-\beta){f^{\prime}_{p}}^{2}-\alpha(1+{f^{\prime}_{z}}^{2})$
$\displaystyle=0,$ (7.16) $\displaystyle f^{2}(\varepsilon
f^{2}+\gamma)^{2}(1+f^{\prime}_{z}{}^{2})-4\alpha(f^{2}-\beta)$
$\displaystyle=0.$

Introducing the notations $\pi=f^{\prime}_{p},\quad\zeta=f^{\prime}_{z}$, we
write the system as

$F(p,z,f,\pi,\zeta)=0,\quad G(p,z,f,\pi,\zeta)=0,$ (7.17)

where

$\displaystyle F(p,z,f,\pi,\zeta)$
$\displaystyle=(f^{2}-\beta)\pi^{2}-\alpha(\zeta^{2}+1),$ (7.18)
$\displaystyle G(p,z,f,\pi,\zeta)$ $\displaystyle=f^{2}(\varepsilon
f^{2}+\gamma)^{2}(\zeta^{2}+1)-4\alpha(f^{2}-\beta).$

Assume that $\alpha(p)>0,\,\beta(p)\geq 0$ and $\gamma(p)$ are defined and
smooth on an interval $(p_{1},p_{2})$, where $-\infty\leq
p_{1}<p_{2}\leq\infty$. Then $F$ and $G$ are defined and smooth in
$U\times{\mathbb{R}}^{2}$, where

$U=\big{\\{}(p,z,f)\in{\mathbb{R}}^{3}|:p_{1}<p<p_{2},\ f>\\!\sqrt{\beta},\
0<\varepsilon f^{2}+\gamma<2\sqrt{\alpha}\sqrt{f^{2}-\beta}/f\big{\\}}.$
(7.19)

The functions $F$ and $G$ are actually independent of $z$ and depend on $p$
through the functions $\alpha(p),\beta(p),\gamma(p)$ only. Up to notations,
the system (7.17) is of the form (7.1).

######  Theorem 7.2.

Given $C^{1}$-functions $\alpha(p)>0,\beta(p)\geq 0$ and $\gamma(p)$ on an
interval $(p_{1},p_{2})$, define $F$ and $G$ by (7.18) and consider the system
of PDEs (7.17), where $f=f(p,z)$ is an unknown function and
$\pi=f^{\prime}_{p},\zeta=f^{\prime}_{z}$. Define an open set
$U\subset{\mathbb{R}}^{3}$ by (7.19). Then

1. The system (7.17) is complete on $U\times{\mathbb{R}}^{2}$ if the functions $\alpha,\beta,\gamma$ satisfy

$\alpha(p)=\alpha_{0}e^{3p}\quad\mbox{\rm with some constant}\ \alpha_{0}>0,$
(7.20)
$\beta^{\prime}+2\varepsilon\gamma^{\prime}+3\beta+\varepsilon\gamma=0,$
(7.21)
$\gamma\beta^{\prime}-2\beta\gamma^{\prime}+3\beta\gamma-4\varepsilon\alpha+\varepsilon\gamma^{2}=0,$
(7.22)

where either $\varepsilon=1$ or $\varepsilon=-1$ and
$\beta+\varepsilon\gamma\neq 0$. But (7.17) is not an involutory system.

2. Conversely, assume that (7.17) is a complete system on $U\times{\mathbb{R}}^{2}$, where the open set $U\subset{\mathbb{R}}^{3}$ is defined by (7.19). Assume additionally that for every $p_{0}\in(p_{1},p_{2})$ there exists $z_{0}$ such that (7.17) has a solution $f(p,z)$ in a neighborhood of $(p_{0},z_{0})$ satisfying $f^{\prime}_{z}(p_{0},z_{0})\neq 0$. Then the functions $\alpha,\beta,\gamma$ satisfy (7.20)–(7.22) with some $\varepsilon=\pm 1$.

###### Proof.

To agree (7.1) and (7.17), we need to change the variables in (7.1) as
follows: $x:=p,\ y:=z,\ z:=f,\ p:=\pi$ and $q:=\zeta$. Then the formula (7.3)
takes the form

$[F,G]=(F^{\prime}_{p}+\pi F^{\prime}_{f})G^{\prime}_{\pi}-(G^{\prime}_{p}+\pi
G^{\prime}_{f})F^{\prime}_{\pi}+(F^{\prime}_{z}+\zeta
F^{\prime}_{f})G^{\prime}_{\zeta}-(G^{\prime}_{z}+\zeta
G^{\prime}_{f})F^{\prime}_{\zeta}.$ (7.23)

We find the derivatives by differentiating (7.18):

$\begin{array}[]{ll}F^{\prime}_{p}=-\beta^{\prime}\pi^{2}{-}\alpha^{\prime}(\zeta^{2}+1),&G^{\prime}_{p}=2\gamma^{\prime}f^{2}(\varepsilon
f^{2}{+}\gamma)(\zeta^{2}{+}1)-4\alpha^{\prime}(f^{2}{-}\beta)+4\alpha\beta^{\prime},\\\\[5.0pt]
F^{\prime}_{z}=0,&G^{\prime}_{z}=0,\\\\[5.0pt]
F^{\prime}_{f}=2f\pi^{2},&G^{\prime}_{f}=2f(\varepsilon
f^{2}{+}\gamma)^{2}(\zeta^{2}{+}1){+}4\varepsilon f^{3}(\varepsilon
f^{2}{+}\gamma)(\zeta^{2}{+}1){-}8\alpha f,\\\\[5.0pt]
F^{\prime}_{\pi}=2(f^{2}-\beta)\pi,&G^{\prime}_{\pi}=0,\\\\[5.0pt]
F^{\prime}_{\zeta}=-2\alpha\zeta,&G^{\prime}_{\zeta}=2f^{2}(\varepsilon
f^{2}+\gamma)^{2}\zeta.\end{array}$ (7.24)

Since $G^{\prime}_{\pi}=F^{\prime}_{z}=G^{\prime}_{z}=0$, the formula (7.23)
simplifies to the following:

$[F,G]=-G^{\prime}_{p}F^{\prime}_{\pi}-\pi
G^{\prime}_{f}F^{\prime}_{\pi}+\zeta F^{\prime}_{f}G^{\prime}_{\zeta}-\zeta
G^{\prime}_{f}F^{\prime}_{\zeta}.$ (7.25)

Substituting values (7.24) into (7.25), we obtain

$\displaystyle\frac{1}{4}[F,G]=-\pi(f^{2}-\beta)\big{(}\gamma^{\prime}f^{2}(\varepsilon
f^{2}+\gamma)(\zeta^{2}+1)-2\alpha^{\prime}(f^{2}-\beta)+2\alpha\beta^{\prime}\big{)}$
(7.26) $\displaystyle-\pi^{2}f(f^{2}-\beta)\big{(}(\varepsilon
f^{2}+\gamma)^{2}(\zeta^{2}+1)+2\varepsilon f^{2}(\varepsilon
f^{2}+\gamma)(\zeta^{2}+1)-4\alpha\big{)}$
$\displaystyle+\pi^{2}\zeta^{2}f^{3}(\varepsilon
f^{2}+\gamma)^{2}+\zeta^{2}\alpha f\big{(}(\varepsilon
f^{2}+\gamma)^{2}(\zeta^{2}+1)+2\varepsilon f^{2}(\varepsilon
f^{2}+\gamma)(\zeta^{2}+1)-4\alpha\big{)}.$

The right-hand side of (7.26) is a 7th degree polynomial in $f$ and the
coefficient at $f^{7}$ is $-\pi^{2}(2\zeta^{2}+3)\neq 0$. Thus, (7.17) is not
an involutory system. Now, we prove that (7.17) is a complete system. For this
we derive from (7.17)–(7.18)

$\zeta^{2}=\frac{4\alpha(f^{2}-\beta)}{f^{2}(\varepsilon f^{2}+\gamma)^{2}}-1$
(7.27)

and
$\pi=\varepsilon\,\frac{\alpha^{1/2}\sqrt{\zeta^{2}+1}}{\sqrt{f^{2}-\beta}}$,
where $\varepsilon=\pm 1$ is the sign of $\pi$. We find from two last
equalities

$\pi=\frac{2\varepsilon\alpha}{f(\varepsilon f^{2}+\gamma)}.$ (7.28)

Substituting expressions (7.27)–(7.28) into (7.26), we obtain

$\displaystyle\frac{1}{16\alpha^{2}}[F,G]$
$\displaystyle=-\varepsilon\frac{2(f^{2}-\beta)}{f(\varepsilon
f^{2}+\gamma)}\big{(}\frac{\gamma^{\prime}(f^{2}-\beta)}{\varepsilon
f^{2}+\gamma}-\frac{\alpha^{\prime}}{2\alpha}(f^{2}-\beta)+\frac{\beta^{\prime}}{2}\big{)}$
(7.29) $\displaystyle-\frac{4\alpha(f^{2}-\beta)}{f(\varepsilon
f^{2}+\gamma)^{2}}\big{(}\frac{f^{2}-\beta}{f^{2}}+2\varepsilon\,\frac{f^{2}-\beta}{\varepsilon
f^{2}+\gamma}-1\big{)}+f\big{(}\frac{4\alpha(f^{2}-\beta)}{f^{2}(\varepsilon
f^{2}+\gamma)^{2}}-1\big{)}$
$\displaystyle+f\big{(}\frac{4\alpha(f^{2}-\beta)}{f^{2}(\varepsilon
f^{2}+\gamma)^{2}}-1\big{)}\big{(}\frac{f^{2}-\beta}{f^{2}}+2\varepsilon\,\frac{f^{2}-\beta}{\varepsilon
f^{2}+\gamma}-1\big{)}.$

We are interested in the case when $[F,G]=0$. Equating the right-hand side of
(7.29) to zero and multiplying the resulting equality by $f^{3}(\varepsilon
f^{2}+\gamma)^{3}$, we arrive to the equation

$\displaystyle-2\varepsilon f^{2}(f^{2}-\beta)(\varepsilon
f^{2}+\gamma)\big{(}\gamma^{\prime}(f^{2}-\beta)-\frac{\alpha^{\prime}}{2\alpha}(f^{2}-\beta)(\varepsilon
f^{2}+\gamma)+\frac{\beta^{\prime}}{2}(\varepsilon f^{2}+\gamma)\big{)}$
(7.30) $\displaystyle-4\alpha(f^{2}-\beta)\big{(}(f^{2}-\beta)(\varepsilon
f^{2}+\gamma)+2\varepsilon f^{2}(f^{2}-\beta)-f^{2}(\varepsilon
f^{2}+\gamma)\big{)}$ $\displaystyle+4\alpha f^{2}(f^{2}-\beta)(\varepsilon
f^{2}+\gamma)-f^{4}(\varepsilon f^{2}+\gamma)^{3}$
$\displaystyle+\big{(}4\alpha(f^{2}{-}\beta)-f^{2}(\varepsilon
f^{2}{+}\gamma)^{2}\big{)}\big{(}(f^{2}{-}\beta)(\varepsilon
f^{2}{+}\gamma)+2\varepsilon f^{2}(f^{2}{-}\beta)-f^{2}(\varepsilon
f^{2}{+}\gamma)\big{)}=0.$

The left-hand side of the equation (7.30) is a polynomial of 10th degree in
$f$. It is almost unbelievable, but the degree of the polynomial can be
decreased to 4. Namely, the equation (7.30) is equivalent to the following
one:

$\displaystyle-f^{2}(f^{2}-\beta)(\varepsilon f^{2}+\gamma)\big{[}$
$\displaystyle\big{(}3-\frac{\alpha^{\prime}}{\alpha}\big{)}f^{4}+\big{(}2\varepsilon\gamma^{\prime}-\varepsilon\frac{\alpha^{\prime}}{\alpha}\gamma+\frac{\alpha^{\prime}}{\alpha}\beta+\beta^{\prime}+4\varepsilon\gamma\big{)}f^{2}$
(7.31)
$\displaystyle+\big{(}-2\varepsilon\beta\gamma^{\prime}+\varepsilon\frac{\alpha^{\prime}}{\alpha}\beta\gamma+\varepsilon\gamma\beta^{\prime}-4\alpha+\gamma^{2}\big{)}\big{]}=0.$

Indeed, a simple (though rather cumbersome) calculation verifies that the
polynomials on the left-hand sides of (7.30) and (7.31) are equal. By (7.19),
the factor $f^{2}(f^{2}-\beta)(\varepsilon f^{2}+\gamma)$ does not vanish on
$U$. Therefore the equation (7.31) is equivalent to the following:

$\big{(}3-\frac{\alpha^{\prime}}{\alpha}\big{)}f^{4}+\big{(}2\varepsilon\gamma^{\prime}-\varepsilon\frac{\alpha^{\prime}}{\alpha}\gamma+\frac{\alpha^{\prime}}{\alpha}\beta+\beta^{\prime}+4\varepsilon\gamma\big{)}f^{2}+\big{(}-2\varepsilon\beta\gamma^{\prime}+\varepsilon\frac{\alpha^{\prime}}{\alpha}\beta\gamma+\varepsilon\gamma\beta^{\prime}-4\alpha+\gamma^{2}\big{)}=0.$
(7.32)

Equating coefficients of the polynomial on the left-hand side of (7.32) to
zero, we arrive to the system of ODEs which is equivalent to (7.20)–(7.22):

$\displaystyle\alpha^{\prime}-3\alpha=0,$ (7.33)
$\displaystyle\beta^{\prime}+2\varepsilon\gamma^{\prime}+3\beta+\varepsilon\gamma=0,$
$\displaystyle\varepsilon\gamma\beta^{\prime}-2\varepsilon\beta\gamma^{\prime}+3\varepsilon\beta\gamma-4\alpha+\gamma^{2}=0.$

Of course, (7.33) implies (7.32) and the determinant is
$\beta+\varepsilon\gamma$. This proves the first assertion of Theorem 7.2. To
prove that (7.32) implies (7.33), we need for each $p_{0}\in(p_{1},p_{2})$ to
have at least three distinct $z_{1},z_{2},z_{3}$ such that the values
$f^{2}(p_{0},z_{1}),\,f^{2}(p_{0},z_{2}),\,f^{2}(p_{0},z_{3})$ are pairwise
different and (7.32) holds at $(p_{0},z_{1}),(p_{0},z_{2})$ and
$(p_{0},z_{3})$. The existence of such $z_{1},z_{2},z_{3}$ is guaranteed by
the hypothesis of the second assertion of Theorem 7.2: For each
$p_{0}\in(p_{1},p_{2})$, there exists $z_{0}$ such that the system (7.17) has
a solution $f(p,z)$ in a neighborhood of $(p_{0},z_{0})$ satisfying
$f^{\prime}_{z}(p_{0},z_{0})\neq 0$. ∎

Remark. Roughly speaking, Theorem 7.2 means that the relations (7.20)–(7.22)
constitute the consistency condition for the system (7.17). Nevertheless, we
emphasize that two statements of Theorem 7.2 are not exactly converse to each
other. For example, the axisymmetric Gavrilov flow (3.2) (isobaric surfaces
are cylinders and particle trajectories are either circles or spiral lines)
corresponds to the solution $f(p,z)=2p^{1/2}$ to the system (7.16) with
$(p_{1},p_{2})=(0,\infty)$ and

$\alpha(p)=4a_{0},\ \ \beta(p)=4(1-a_{0})p,\ \ \gamma(p)=4(a_{0}-p)\ \
(a_{0}=\mbox{const},\,0<a_{0}\leq 1).$ (7.34)

The functions (7.34) do not satisfy the consistency conditions (7.20)–(7.22).
For this solution, $f^{\prime}_{z}\equiv 0$ holds, and the second statement of
Theorem 7.2 does not apply.

An analog of Lemma 7.1 is valid for the system (6.29)–(6.30) with minor
changes (the variables $(p,z)$ are replaced by $(p,r)$, the function $f$ is
replaced by $g$, etc.). The corresponding system of two first order PDEs looks
as follows:

$\displaystyle(r^{2}-\beta)g^{\prime}_{p}{}^{2}-\alpha(1+g^{\prime}_{r}{}^{2})$
$\displaystyle=0,$ (7.35) $\displaystyle r^{2}(\gamma-\tau
r^{2})^{2}(1+g^{\prime}_{r}{}^{2})-4\alpha(r^{2}-\beta)g^{\prime}_{r}{}^{2}$
$\displaystyle=0,$

where $\alpha(p)>0,\beta(p)\geq 0$ and $\gamma(p)$ are the same functions as
in (7.16) and $\tau=\pm 1$ is the sign of $g^{\prime}_{p}$ that does not
vanish. An analog of Theorem 7.2 is valid for the system (7.35) with the same
consistency conditions (7.20)–(7.22).

##  8\. Structure of an axisymmetric Gavrilov flow in a neighborhood of a
minimum point of the pressure

In [2], the existence of a pair $(\tilde{u},\tilde{p})$ is proved such that
(a) $(1,0)$ is a non-degenerate minimum point of the function
$\tilde{p}=\psi(r,z)\in C^{\infty}(U)$ and $\psi(1,0)=0$, where
$U\subset\\{(r,z)\mid r>0\\}$ is a neighborhood of $(1,0)$; (b)
$(\tilde{u},\tilde{p})$ is an axisymmetric Gavrilov flow in
$U\setminus\\{(1,0)\\}$; (c) the split Bernoulli law for the flow is of the
simplest form

$|\tilde{u}|^{2}=3\tilde{p}.$ (8.1)

The flow $(\tilde{u},\tilde{p})$ does not satisfy the normalization condition
(6.22). To apply our equations, we must replace $(\tilde{u},\tilde{p})$ with
an equivalent Gavrilov flow $(u,p)$ such that $|u|^{2}=1$. By the definition
(1.4) of equivalent Gavrilov flows, $u=\varphi(\tilde{p})\tilde{u},\ \nabla
p=\varphi^{2}(\tilde{p})\nabla\tilde{p}$ with some non-vanishing function
$\varphi(\tilde{p})$. As is seen from (8.1),
$\varphi^{2}(\tilde{p})=1/|\tilde{u}|^{2}=1/3\,\tilde{p}$. Thus

$p(r,z)=\frac{1}{3}\ln\psi(r,z)+C\quad(C=\mbox{const}),$ (8.2)

Hence, the pressure function gets a logarithmic singularity at the point
$(1,0)$ after normalization. Comparing (7.20) and (8.2), we conclude that

$\alpha=c\,\psi\quad(c=\mbox{const}>0).$ (8.3)

Since $p\to-\infty$ as $(r,z)\to(1,0)$, it is natural to choose initial
conditions for the system (7.21)–(7.22) at $p=-\infty$. The conditions are

$\beta(-\infty)=\lim\limits_{p\to-\infty}\beta(p)=1/3,\quad\gamma(-\infty)=\lim\limits_{p\to-\infty}\gamma(p)=-1.$
(8.4)

Indeed, for $p$ close to $-\infty$, $\Gamma_{p}$ is a “small” closed curve
around $(1,0)$. The tangent line to $\Gamma_{p}$ is vertical at some point
$(r_{1},z_{1})=\big{(}r_{1}(p),z_{1}(p)\big{)}\in\Gamma_{p}$. In a
neighborhood of $(r_{1},z_{1})$, the curve $\Gamma_{p}$ is the graph of a
function $r=f(p,z)$ solving the system (7.16) and satisfying
$f^{\prime}_{z}(p,z_{1})=0$. Setting $z=z_{1}$ in the second equation of
(7.16), we get

$f^{2}(p,z_{1})\big{(}\varepsilon
f^{2}(p,z_{1})+\gamma(p)\big{)}^{2}-4\alpha(p)\big{(}f^{2}(p,z_{1})-\beta(p)\big{)}=0.$
(8.5)

In view of our assumption $f^{2}(p,z)-\beta(p)>0$, see (7.7), the factor
$\big{(}f^{2}(p,z_{1})-\beta(p)\big{)}$ remains bounded when $p\to-\infty$.
Also $\alpha(p)=\alpha_{0}e^{3p}\to 0$ as $p\to-\infty$. Thus, the second term
on the left-hand side of (8.5) runs to 0 as $p\to-\infty$. The same is true
for the first term. Taking into account that $f^{2}(p,z_{1})\to 1$ as
$p\to-\infty$, we obtain $\gamma(-\infty)=-\varepsilon$.

Since $\varepsilon=\pm 1$ is the sign of $f^{\prime}_{p}$ in (8.5), by a
similar analysis of the system (7.35), we demonstrate that $\varepsilon=1$ in
our setting. This proves the second equality in (8.4). The first equality in
(8.4) is proved similarly.

The “Cauchy problem” (7.21)–(7.22), (8.4) (with $\varepsilon=1$) is easily
solved in series
$\beta=\frac{1}{3}+\sum\nolimits_{k=1}^{\infty}\beta_{k}\alpha^{k}$ and
$\gamma=\frac{1}{3}+\sum\nolimits_{k=1}^{\infty}\gamma_{k}\alpha^{k}.$
Equations (7.21)–(7.22) imply some recurrent relations that allow us to
compute all coefficients. In particular,

$\displaystyle\beta=\frac{1}{3}-\frac{7}{6}\alpha{+}\frac{13}{72}\alpha^{2}{-}\frac{133}{1728}\alpha^{3}{+}\frac{575}{13824}\alpha^{4}{-}\frac{2077}{82944}\alpha^{5}{+}\frac{259}{16128}\alpha^{6}+\dots,$
(8.6)
$\displaystyle\gamma=-1+\alpha-\frac{1}{8}\alpha^{2}+\frac{7}{144}\alpha^{3}-\frac{115}{4608}\alpha^{4}+\frac{67}{4608}\alpha^{5}-\frac{7}{768}\alpha^{6}+\dots.$
(8.7)

The system (7.16) can be equivalently written in terms of the function
$\psi(r,z)$. Indeed, as is seen from (8.2), a solution $f(p,z)$ to the system
(7.16) is related to $\psi$ by $\frac{1}{3}\ln\psi(f(p,z),z)+C=p.$ Starting
with this equation, we derive from (7.16) the system

$\displaystyle 2c\psi^{\prime}_{r}-3r(r^{2}+\gamma)$ $\displaystyle=0,$ (8.8)
$\displaystyle
c^{2}\psi^{\prime}_{z}{}^{2}-9c(r^{2}-\beta)\psi+\frac{9}{4}r^{2}(r^{2}+\gamma)^{2}$
$\displaystyle=0,$

where $c$ is the constant from (8.3).

By the change $\psi=\frac{1}{c}\tilde{\psi}$ of the unknown function, (8.8) is
transformed to the same system with $c=1$. Therefore we can assume $c=1$
without lost of generality, i.e.,

$\displaystyle 2\psi^{\prime}_{r}-3r(r^{2}+\gamma)$ $\displaystyle=0,$ (8.9)
$\displaystyle\psi^{\prime}_{z}{}^{2}-9(r^{2}-\beta)\psi+\frac{9}{4}r^{2}(r^{2}+\gamma)^{2}$
$\displaystyle=0.$

The equality (8.3) becomes $\alpha=\psi$. The function $\psi(r,z)$ is defined
and smooth in a neighborhood of the point $(1,0)$ and satisfies $\psi(1,0)=0$.
Let us show that $\psi(r,z)$ is an even function of $z$. Indeed, the second
equation of the system (8.9) gives
$\psi^{\prime}_{z}{}^{2}(1,0)+\frac{9}{4}r^{2}(1+\gamma(-\infty))^{2}=0.$
Since $\gamma(-\infty)=-1$ by (8.4), we obtain

$\psi^{\prime}_{z}(1,0)=0.$ (8.10)

Then we differentiate the first equation of the system (8.9) with respect to
$z$

$2\psi^{\prime\prime}_{rz}-3r\gamma^{\prime}_{z}=0.$ (8.11)

Since $\gamma$ depends on $p$ only,
$\gamma^{\prime}_{z}=\gamma^{\prime}p^{\prime}_{z},$ where
$\gamma^{\prime}=d\gamma/dp$. Together with (8.2), this gives
$\gamma^{\prime}_{z}=\frac{\gamma^{\prime}\psi^{\prime}_{z}}{3\psi}$.
Substituting this expression into (8.11) and setting $z=0$, we arrive to the
linear first order ODE for the function $\psi^{\prime}_{z}(r,0)$:

$2\frac{d\psi^{\prime}_{z}(r,0)}{dr}-3\frac{r\gamma^{\prime}(p(r,0))}{3\psi(r,0)}\,\psi^{\prime}_{z}(r,0)=0.$
(8.12)

The coefficient of the equation $\frac{r\gamma^{\prime}(p(r,0))}{3\psi(r,0)}$
is a bounded smooth function in a neighborhood of $r=1$ as is seen from (8.7).
Together with the initial condition (8.10), the equation (8.12) implies
$\psi^{\prime}_{z}(r,0)=0$. The system (8.9) is invariant under the change
$z\to-z$. Therefore the equality $\psi^{\prime}_{z}(r,0)=0$ implies that
$\psi(r,z)$ is an even function of $z$.

The system (8.9) allows us to compute term-by-term all Taylor coefficients of
the function $\psi(r,z)$ at the point $(1,0)$. In particular,

$\displaystyle\psi(r,z)$
$\displaystyle=\frac{3}{2}(r-1)^{2}+\frac{3}{2}z^{2}+\frac{9}{4}(r-1)^{3}+\frac{9}{4}(r-1)z^{2}+\frac{57}{32}(r-1)^{4}$
(8.13)
$\displaystyle+\frac{45}{16}(r-1)^{2}z^{2}+\frac{33}{32}z^{4}+\frac{9}{8}(r-1)^{5}+\frac{9}{4}(r-1)^{3}z^{2}+\frac{9}{4}(r-1)z^{4}+\dots$

Second and 3d order terms on the right-hand side of (8.13) are easily derived
from (8.9), and we used Maple for computing 4th and 5th order terms.

Recall that physical coordinates $(u_{r},u_{z},u_{\theta})$ of the velocity
vector field $u$ are defined by (6.2). The function $u_{\theta}$ is called the
swirl. As follows from (6.3) and (6.26),

$(u_{r}^{2}+u_{z}^{2})(r,z)=1-\frac{\beta(p)}{r^{2}},\quad
u_{\theta}(r,z)=\frac{\sqrt{\beta(p)}}{r}.$ (8.14)

Together with (8.6), these formulas allow us to compute the swirl and two-
dimensional vector field $u_{r}e_{r}+u_{z}e_{z}$ tangent to the curves
$\Gamma_{p}$.

##  9\. Two examples of axisymmetric Gavrilov flows

Here, we discuss the numerical method for constructing axisymmetric Gavrilov
flows on the base of the system (7.16). For the initial condition
$f(p_{0},z_{0})=f_{0}$, we can assume without lost of generality that
$p_{0}=z_{0}=0$ since the system (7.16) is invariant under the change $p\to
p+\mbox{const}$ and $z\to z+\mbox{const}$. Thus, the initial condition for $f$
is

$f(0,0)=f_{0}.$ (9.1)

First, we will find the functions $\alpha(p),\beta(p),\gamma(p)$. The function
$\alpha$ is given explicitly by (7.20) with an arbitrary constant
$\alpha_{0}=\alpha(0)>0$. The functions $\beta(p)\geq 0$ and $\gamma(p)$ solve
the system (7.21)–(7.22) supplied with the initial conditions

$\beta(0)=\beta_{0},\quad\gamma(0)=\gamma_{0}.$ (9.2)

In particular, $\beta_{0}\geq 0$. For being able to write the system
(7.21)–(7.22) in the form $\beta^{\prime}=B(\beta,\gamma),\
\gamma^{\prime}=\Gamma(\beta,\gamma)$, we have to impose the restrictions

$\beta\geq 0,\quad\beta+\varepsilon\gamma\neq 0.$ (9.3)

In particular, $\beta_{0}$ and $\gamma_{0}$ must satisfy these inequalities.
Let $(p_{1},p_{2})$ be the maximal interval such that the solution to the
Cauchy problem (7.21)–(7.22), (9.2) exists on $(p_{1},p_{2})$ and satisfies
(9.3). Here,
$p_{1}=p_{1}(\alpha_{0},\beta_{0},\gamma_{0})<0<p_{2}=p_{2}(\alpha_{0},\beta_{0},\gamma_{0})$.
By Theorem 7.2, the system (7.16) is complete on $U\times{\mathbb{R}}^{2}$,
where $U\subset{\mathbb{R}}^{3}$ is defined by (7.19). General theory [4,
Chapter 1, Section 14] guarantees the existence and uniqueness of a solution
to the IVP (7.16), (9.1) for any $(0,0,f_{0})\in U$ at least in some
neighborhood of $(p_{0},z_{0})=(0,0)$.

Thus, a solution $f(p,z)$ to the IVP (7.16), (9.1) exists in some neighborhood
of $(0,0)$ and is uniquely determined by 5 constants
$(\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$ chosen so that

$\displaystyle\alpha_{0}>0,\quad\beta_{0}\geq
0,\quad\beta_{0}+\varepsilon\gamma_{0}\neq 0,\quad
f_{0}>0,\quad\varepsilon=\pm 1,$ (9.4) $\displaystyle
f_{0}>\beta_{0}^{1/2},\quad 0<\varepsilon
f_{0}^{2}+\gamma_{0}<2\alpha_{0}^{1/2}f_{0}^{-1}\sqrt{f_{0}^{2}-\beta_{0}}.$

The inequalities on the second line of (9.4) come from (7.19). Let us denote
this unique solution by
$f(p,z;\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$.

In general, $f(p,z;\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$ is a
local solution, i.e., is defined in some neighborhood
$U(\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)\subset{\mathbb{R}}^{2}$
of the point $(0,0)$. Nevertheless, for some values of the parameters, it can
happen that $(p^{\prime}_{1},p^{\prime}_{2})\times\mathbb{R}\subset
U(\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$ for some $-\infty\leq
p^{\prime}_{1}<0<p^{\prime}_{2}\leq\infty$, in such a case we speak on a
global solution defined on $(p^{\prime}_{1},p^{\prime}_{2})\times\mathbb{R}$.
Global solutions are of particular interest. Unfortunately, so far we have
neither necessary nor sufficient conditions on the parameters
$(\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$ for the existence of a
global solution.

Everything said above in this section is valid for the numerical method based
on the system (7.35). Global solutions most often appear due to periodicity
using the following

######  Lemma 9.1.

Let a solution $f(p,z)$ of the system (7.16) be defined on a rectangle
$(p_{1},p_{2})\times(z_{1},z_{2})$ and satisfy (7.7). Assume the existence of
$(p_{0},z_{0})\in(p_{1},p_{2})\times(z_{1},z_{2})$ such that
$f^{\prime}_{z}(p_{0},z_{0})=0$ and $f^{\prime\prime}_{zz}(p_{0},z_{0})\neq
0$. Then, at least in some neighborhood of $p_{0}$, the solution is symmetric
with respect to $z_{0}$, i.e.,

$f(p,z)=f(p,-z+2z_{0})\quad\mbox{\rm for}\ p\in(p_{0}-\delta,p_{0}+\delta)\
\mbox{\rm with some}\ \delta>0.$ (9.5)

Hence, $f$ can be extended to a solution defined in
$(p_{0}-\delta,p_{0}+\delta)\times(z_{0}-\Delta,z_{0}+\Delta)$ such that
$(z_{1},z_{2})\subset(z_{0}-\Delta,z_{0}+\Delta)$.

###### Proof.

By the implicit function theorem, there exists a smooth function $z=\zeta(p)$
defined for $p\in(p_{0}-\delta,p_{0}+\delta)$ with some $\delta>0$ such that
$\zeta(p_{0})=z_{0}$ and

$f^{\prime}_{z}(p,\zeta(p))=0.$ (9.6)

We are going to prove that $\zeta(p)$ is actually a constant function. To this
end we differentiate (9.6):
$f^{\prime\prime}_{pz}(p,\zeta(p))+f^{\prime\prime}_{zz}(p,\zeta(p))\,\zeta^{\prime}(p)=0$.
By choosing a smaller $\delta$, we can assume that
$f^{\prime\prime}_{zz}(p,\zeta(p))\neq 0$. Thus, to prove the equality
$\zeta^{\prime}=0$, we have to demonstrate that
$f^{\prime\prime}_{pz}(p,\zeta(p))=0$. To this end we differentiate the first
equation of the system (7.16) with respect to $z$ (recall that $\alpha$ and
$\beta$ are independent of $z$)
$(f^{2}-\beta)f^{\prime}_{p}f^{\prime\prime}_{pz}+f(f^{\prime}_{p})^{2}f^{\prime}_{z}-\alpha
f^{\prime}_{z}f^{\prime\prime}_{zz}=0.$ Setting $z=\zeta(p)$ here and using
(9.6), we obtain
$(f^{2}-\beta)f^{\prime}_{p}f^{\prime\prime}_{pz}|_{z=\zeta(p)}=0$. Since
$f^{\prime}_{p}\neq 0$ and $f^{2}-\beta>0$, this implies
$f^{\prime\prime}_{pz}(p,\zeta(p))=0$. We have thus proved

$f^{\prime}_{z}(p,z_{0})=0\quad\mbox{for}\ p\in(p_{0}-\delta,p_{0}+\delta).$
(9.7)

The system (7.16) is invariant under the change $z\to 2z_{0}-z$. Thus (9.7)
implies (9.5). ∎

Now, under hypotheses of Lemma 9.1, assume the existence of a second point
$(p^{\prime}_{0},z^{\prime}_{0})\in(p_{1},p_{2})\times(z_{1},z_{2})\
(z_{0}\neq z^{\prime}_{0})$ such that
$f^{\prime}_{z}(p^{\prime}_{0},z^{\prime}_{0})=0$ and
$f^{\prime\prime}_{zz}(p^{\prime}_{0},z^{\prime}_{0})\neq 0$. An analog of
(9.5) holds for $z^{\prime}_{0}$ with some $\delta^{\prime}>0$. Assume
additionally that

$({\tilde{p}}_{1},{\tilde{p}}_{2})=(p_{0}-\delta,p_{0}+\delta)\cap(p^{\prime}_{0}-\delta^{\prime},p^{\prime}_{0}+\delta^{\prime})\neq\emptyset.$

Using symmetries with respect to $z_{0}$ and $z^{\prime}_{0}$, we extend
$f(p,z)$ to a global $z$-periodic solution defined on
$({\tilde{p}}_{1},{\tilde{p}}_{2})\times\mathbb{R}$. The period is equal to
$|z_{0}-z^{\prime}_{0}|$ if $p_{0}=p^{\prime}_{0}$, otherwise the period is
equal to $2|z_{0}-z^{\prime}_{0}|$.

An example of $z$-periodic Gavrilov flow is presented on Figure 1. The
solution $f(p,z)=f(p,z;\alpha_{0},\beta_{0},\gamma_{0},f_{0},\varepsilon)$ to
the IVP (7.16), (9.1) was computed for
$\alpha_{0}=1,\beta_{0}=0.01,\gamma_{0}=0.5,f_{0}=0.97,\varepsilon=1$. The
solution to the Cauchy problem (7.21)–(7.22), (9.2) exists for $-0.07<p<7.48$.
For $0<p<p_{c}\approx 0.25$, the flow turns out to be $z$-periodic with the
period approximately equal to $2.8$. Graphs $\Gamma_{i}$ of functions
$r=f(p_{i},\,z)$ for $p_{i}=0.02\,i$ and $i=0,\dots,8$ are drawn on the left-
hand side of Figure 1. Each curve $\Gamma_{i}$ is the generatrix of the
isobaric surface $M_{p_{i}}$. The velocity vector field $u$ is drawn on the
right picture for $\Gamma_{0}$ and $\Gamma_{8}$. We used (8.14) for computing
physical coordinates of $u$. The two-dimensional vector field
$u_{r}e_{r}+u_{z}e_{z}$ is tangent to $\Gamma_{0}$ and $\Gamma_{8}$, and the
vector field $u_{\theta}e_{\theta}$ (the swirl) is orthogonal to the plane of
the picture. The latter vector field is drawn by vectors orthogonal to
$\Gamma_{0}$ and $\Gamma_{8}$ in order to avoid 3D pictures. In order to make
a nice picture, both fields $u_{r}e_{r}+u_{z}e_{z}$ and $u_{\theta}e_{\theta}$
are drawn in the scale $1:4$, i.e., $|u|=0.25$ on the picture. But we remember
that actually $|u|=1$. As is seen from the picture, for $p$ close to the
critical value $p_{c}$, the flow is close to the $z$-independent solution
(3.2): isobaric surfaces are close to circular cylinders and particle
trajectories are close to spiral lines intersecting parallels approximately at
the angle of $\pi/4$. The hypothesis $f^{\prime\prime}_{zz}(p_{0},z_{0})\neq
0$ of Lemma 9.1 is violated at $p_{0}=p_{c}$. Therefore our flow does not need
to be periodic with the same period $\approx 2.8$ for $p>p_{c}$.

Figure 1. Periodic axisymmetric Gavrilov flow.  Figure 2. Gavrilov flow in a
neighborhood of a minimum point of the pressure.

The Gavrilov flow discussed in Section 8 is illustrated on Figure 2. In our
calculations we use the 5th order segment of the Taylor series of the function
$\psi$, i.e., we ignore the remainder denotes by dots on the right-hand side
of (8.13). Six isolines $\Gamma_{i}=\\{(r,z):\psi(r,z)=\psi_{i}=0.04\,i\\}\
(i=1,\dots,6)$ are drawn on the left-hand side of Figure 2. Each of the curves
$\Gamma_{1},\dots,{\Gamma}_{5}$ consists of two connected components while
$\Gamma_{6}$ has one component. Besides the minimum point $(1,0)$, the
function $\psi(r,z)$ has the saddle point at $(r,z)=(1/3,0)$. The same curves
$\Gamma_{i}$ are isolines of the pressure function, see (8.2). We set
$C=-\frac{1}{3}\ln\psi_{1}$, thus the formula (8.2) becomes
$p(r,z)=\frac{1}{3}\ln\frac{\psi(r,z)}{\psi_{1}}$. Each curve $\Gamma_{i}$ is
the generatrix of the isobaric surface $M_{p_{i}}$, where
$p_{i}=\frac{1}{3}\ln\frac{\psi_{i}}{\psi_{1}}$. In our case,
$p_{i}=\frac{1}{3}\ln i$. The velocity vector field $u$ is drawn on the right
picture for $\Gamma_{3}$ and $\Gamma_{5}$.

##  10\. Some open questions

In our opinion, the main open question is: are there Gavrilov flows on
${\mathbb{R}}^{3}$ which are not axisymmetric? From an analytical point of
view, this is a question on the consistency conditions of the system
(5.8)–(5.11). The most expected answer to the question is “yes”. However, the
question is not easy because of the following. For an axisymmetric Gavrilov
flow, the constants $c$ and $C$ in the Bernoulli law (2.1) can be expressed
through each other. In other words, all particles living on an isobaric
surface $M_{p}$ move with the same speed in the case of an axisymmetric
Gavrilov flow. Most likely, this statement is not true for a general (not
axisymmetric) Gavrilov flow, at least we cannot prove it by local reasoning.
But maybe simple global arguments will help answer the question. For instance,
if there exists a particle trajectory dense in $M_{p}$, then
$|u|=\mbox{const}$ on $M_{p}$.

Except of Sections 3 and 4, we studied Gavrilov flows locally in a
neighborhood of a regular point. The most interesting questions relate to
Gavrilov flows with regular compact isobaric hypersurfaces
$M_{p}\subset{\mathbb{R}}^{n}$. As mentioned in Introduction, application of
the Gavrilov localization to such flow gives a compactly supported Gavrilov
flow on the whole of ${\mathbb{R}}^{n}$. Since a compact regular isobaric
hypersurface $M_{p}$ is endowed with a non-vanishing tangent vector field $u$,
the Euler characteristic of $M_{p}$ is equal to zero. In the most important
3D-case this means that $M_{p}$ is diffeomorphic to the two-dimensional torus
${\mathbb{T}}^{2}$. The restriction of $u$ onto $M_{p}$ is a non-vanishing
geodesic vector field. There are Riemannian metrics on ${\mathbb{T}}^{2}$
admitting a non-vanishing geodesic vector field, the corresponding example can
be found in the class of so called double-twisted products [3]. But we are
interested in metrics on ${\mathbb{T}}^{2}$ induced from the Euclidean metric
by an embedding $i:{\mathbb{T}}^{2}\subset{\mathbb{R}}^{3}$. Apart from
surfaces of revolution, we do not know any example related to the following
problem.

######  Problem 10.1.

Classify triples $(i,u,\lambda)$, where
$i:{\mathbb{T}}^{2}\subset{\mathbb{R}}^{3}$ is an embedding of the torus, $u$
is a nonzero geodesic vector field on ${\mathbb{T}}^{2}$ endowed with the
Riemannian metric induced from the Euclidean metric of ${\mathbb{R}}^{3}$ by
embedding $i$, and $\lambda>0$ is a smooth function on ${\mathbb{T}}^{2}$
satisfying the equations $($obtained from (2.3) and (2.8) by setting
$\lambda=|\nabla p\,|\,)$

$\mbox{\rm div}\,u=u(\log\lambda),\quad II(u,u)=-\lambda,$ (10.1)

where $II$ is the second quadratic form of $\,{\mathbb{T}}^{2}$.

Equations (10.1) are obtained from (2.3) and (2.8) by setting $\lambda=|\nabla
p\,|$.

In this paper, we did not discuss the behavior of the Gavrilov flow near a
critical point of the pressure. Such a discussion could be of great interest.
For instance, it makes sense to study the Gavrilov flow $(u,p)$ with the Morse
function $p$, i.e., all critical points of $p$ are non-degenerate. For such a
flow, $M_{p}$ is still a regular hypersurface for a regular value of the
pressure; but $M_{p}$ undergoes a Morse surgery when $p$ changes near a
critical value of the pressure. Which Morse surgeries are compatible with the
Euler equations?

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