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\cntl{ On the Arnold Stability Criterion }
\medskip
\twelverm
\cntl{ Lorenzo Sadun {\tenrm and} Misha Vishik}
\medskip
{\ninerm\cntl{Department of Mathematics}
\cntl{University of Texas
\footnote{}{\no Email addresses: sadun@math.utexas.edu and
vishik@math.utexas.edu. This work was partially
supported by NSF Grant \#9105688 and an NSF Mathematical Sciences
Postdoctoral Fellowship}}
\cntl{Austin, TX 78712}
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\cntl{\today}
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\bigskip
The Arnold stability criterion suggests that a stationary flow of an ideal
incompressible fluid is stable if a certain quadratic form is
definite. We show that, in three or more dimensions, this
quadratic form is never definite. Typically the form is indefinite,
and the spectrum of the associated
Hermitian operator ranges from $-\infty$ to $\infty$.
The exceptional case is where the velocity field is harmonic
(solenoidal and irrotational) in which case the quadratic form
is identically zero.
\bigskip
\no PACS: 47.20-k, also 3.40.Gc
\no 1991 Mathematics Subject Classification 76E99, also 58D30}
\bigskip
\rm \no {\it 1. Background}
V.I. Arnold [1,2] was the first to recognize and develop
a new set of geometric ideas in the hydrodynamics of an ideal fluid.
In particular, his stability criterion [3] has been used in a
great number of later publications on the subject of hydrodynamic stability.
This criterion states that a velocity field that is a steady-state
solution to the hydrodynamic Euler equations
is stable if a certain quadratic form, defined on the space of
infinitesimal deformations to this velocity field, is definite.
Arnold also gave important applications of his stability criterion
for 2 dimensional flows. He essentially proved nonlinear stability
under the conditions of the criterion in a certain metric, using a method
that avoided problems related to the geometry of a
foliated family of orbits in the vicinity of a stationary point [4,5].
We refer the reader to reference [6] for a
discussion of the applications
of the stability criterion to 2D and quasi-2D flows. H.K.~Moffatt [7]
gave an explicit computation of the quadratic form for ABC flows and
found that it is indefinite for these important examples.
In this paper we examine the applicability of the Arnold criterion in
3 or more dimensions. We find that the quadratic form is never
definite. In a few special cases it is identically zero. In
all other cases it is indefinite, and the spectrum of the associated
Hermitian operator is not even bounded from above or below.
It should be emphasized that this does {\it not} imply that all
velocity fields in 3 or more dimensions are unstable. It merely
means that the conditions of Arnold's stability criterion are
never met.
\no {\it 2. The stability criterion}
We begin by reminding the reader of the geometric setting of the stability
criterion [3]. For simplicity, we work at first in 3 dimensions,
and later generalize to higher dimensions.
Let $M$ be a 3-dimensional smooth
($C^\infty$) orientable Riemannian manifold with a smooth volume form
$\mu$, defined by the Riemannian metric. Let
$\Sd$ denote the group of smooth volume-preserving
diffeomorphisms, and let $\sd$ be the corresponding Lie
algebra of solenoidal (divergence-free) vector fields.
The fundamental questions of developing Lie theory for $\Sd$ were
partially addressed by D. Ebin and J. Marsden [8]. Surprising
phenomena related to the differential geometry of $\Sd$ were
discovered by Schnirelman [9,10].
$\Sd$ is the configuration space of a 3-dimensional
ideal fluid (see, however, [9,10]).
For any smooth curve $g_t: I \mapsto \Sd$, where $I$
is an interval of the real line, we define the Eulerian velocity
vector $u(t) \in \sd$ to be
$$ u(t) = R_{g_t^{-1}*} (\dot g_t), \eqno (1)$$
where $R_g$ denotes right-translation by $g$ on the Lie group $\Sd$.
The Euler-Arnold equations [1,2,3] arise from the variational
principle
$$ \delta \left ( \half \int dt \int_M \big ( u(t,x)\cdot u(t,x)\big ) \mu(dx)
\right )=0.
\eqno (2)$$
This means that the flows of an ideal fluid on $M$ are geodesics of
the right-invariant metric $<\cdot,\cdot>$ on $\Sd$, defined on
the tangent space $T_e\Sd$ as
$$__ = \int_M (u \cdot v) \mu(dx). \eqno (3)
$$
The Euler-Arnold equations then take the form
$$
\dot u = -(u \cdot \del)u - \del p; \qquad \qquad \div u = 0,
\eqno (4)
$$
where $u$ is an Eulerian velocity, $p$ is a pressure function
defined on $M$, chosen such that $\div \dot u=0$,
and $\del$ acts on $u$ via the Levi-Civita
connection on $M$.
Let $u(t)\in\sd$ be a solution to (4). Kelvin's theorem says that
$u(t)$ is restricted to a single orbit of the coadjoint
representation of $\Sd$ on $(\sd)^*$, the dual space of $\sd$. We
identify $\sd$ with $(\sd)^*$
by the metric $<\cdot,\cdot>$.
The tangent space of the coadjoint
orbit at a point $u \in \sd$ is spanned by vectors fields of the form
$ad_\xi^* = -\xi \times (\curl u) - \del p$, where $\xi$ is an arbitrary
solenoidal vector field and $p$ is chosen to make the
entire expression $-\xi \times (\curl u) - \del p$ solenoidal.
The second variation of the energy $H(u) = {1 \over 2}____$ on the
orbit is given by the quadratic form
$$ d^2H(\xi) = \half
<-\xi \times (\curl u) - \del p,-\xi \times (\curl u) - \del p> +
\half <-\xi \times (\curl u) - \del p,\curl(\xi \times u) >.
\eqno (5)
$$
The Arnold stability criterion states that a steady-state solution
to the Euler-Arnold equations (4) is stable if the quadratic form
(5) is positive (or negative) definite, and if the stratification of
$(\sd)^*$ into orbits of the coadjoint representation is regular in a
neighborhood of the steady-state solution $u$.
The corresponding (unbounded) self-adjoint operator is defined from
the expression (5) in $\hil$, the space of all
solenoidal square-integrable
vector fields $\xi$. It is a simple exercise to see that the null-space
of this operator contains the null-space $K$ of the bounded operator
$\bcal: \hil \mapsto \hil$
$$ \bcal(\xi) = - \xi \times (\curl u) - \del p, \eqno (6)
$$
leading to a self-adjoint operator defined in the factor-space
$\hil/K$, the formal tangent space to the orbit.
\bigskip
\no {\it 3. The second variation for 3-dimensional fluid flow}
{\bf Main Theorem} {\it Let $M$ be a 3-dimensional closed manifold.
If $\curl u$ is not identically zero,
then the quadratic form $d^2H$ is indefinite. Moreover, the
spectrum of the corresponding self-adjoint operator is neither
bounded from below nor from above.}
\no{\it Proof:} If $\curl u$ is not identically zero, we can pick
a point $x_0 \in M$ where $u$ and $\curl u$ are both nonzero.
Let $\phi(x)$ be a function on $M$ with $\del \phi \cdot u$
and $\del \phi \cdot (\curl u)$ both being nonzero at $x_0$.
By continuity, $\del \phi \cdot u$
and $\del \phi \cdot (\curl u)$ are both nonzero on a
small neighborhood $N$ of $x_0$. We then pick a smooth vector field
$a_R$, everywhere orthogonal to $\del \phi$, that vanishes outside
of $N$, and define the real vector field
$a_I = (\del \phi) \times a_R/|\del \phi|$ and
the complex vector field $a=a_R + i a_I$.
Next we construct deformations $\xi_\eps$ for which $d^2H(\xi_\eps)$
is arbitrarily positive or negative. Let
$$ \xi_\eps = \eps \curl \left ( \left ( {1 \over |\del \phi|}
\right ) a e^{i \phi / \eps} \right )
= a e^{i \phi / \eps} + O(\eps). \eqno (7)
$$
The quantity $d^2H(\xi_\eps)$ can be expanded in a power series in
$\eps$. To leading order,
$$ \eqalign{
d^2H(\xi_\eps) = & \half
<-\xi_\eps \times (\curl u) - \del p,-\xi_\eps \times (\curl u) - \del p>_c
\cr & \qquad +
\half <-\xi_\eps \times (\curl u) - \del p,\curl(\xi_\eps \times u) >_c \cr
= & {-i \over 2 \eps} <-\xi_\eps \times (\curl u) - \del p,
\del \phi \times (\xi_\eps \times u)>_c + O(1) \cr
= & {-1 \over \eps} \int (u \cdot \del \phi)(a_R \times a_I\cdot(\curl u))
\mu(dx) + O(1) \cr
= & {-1 \over \eps} \int {|a_R|^2 \over |\del \phi|} (u \cdot \del \phi)
((\curl u) \cdot \del \phi) \mu(dx) + O(1),} \eqno (8)
$$
where $<\cdot,\cdot>_c$ is the Hermitian inner product, linear in the
first argument and anti-linear in the second, that extends the real
inner product $<\cdot,\cdot>$.
Since $(u \cdot \del \phi)(\curl u \cdot \del \phi)$ is nonzero
on $N$, the integral over $N$ is nonzero. By choosing the sign of
$\eps$ and picking $\eps$ sufficiently close to zero, we can make
$d^2H(\xi_\eps)$ as negative (or positive) as we wish, so $d^2H$ cannot
be a definite form.
As $\eps \to 0$, the $L^2$ norm of $\xi_\eps$ remains bounded, as does the
$L^2$ norm of $-\xi_\eps \times (\curl u)$. Since
$-\xi_\eps \times (\curl u) - \del p$ is the $L^2$-orthogonal
projection of $-\xi_\eps \times (\curl u)$
onto the solenoidal vector fields, the norm of
$-\xi_\eps \times (\curl u) - \del p$ is also bounded. Since we can get
arbitrarily large (positive or negative) values of $d^2H(\xi_\eps)$ with
a bounded set of $\xi_\eps$'s, the self-adjoint operator associated to
$d^2H$ is bounded neither above nor below. Its spectrum runs, possibly
with gaps, from
$-\infty$ to $\infty$. \qed
\no {\it 4. Generalization to higher dimensions}
We now briefly describe the generalization of the
main theorem to higher dimensions. Let $M$ be an $n$-dimensional smooth manifold
($n \ge 3$), with a smooth volume form $\mu$ that does not necessarily
come from the Riemannian metric [11]. For each vector $v \in TM$, we
let $\tilde v$ be the 1-form obtained by ``lowering indices''.
That is, $\tilde v(w)= v \cdot w$.
In $n$ dimensions, the Euler-Arnold equations take the form
$$
\dot u = -(u \cdot \del)u - \del p; \qquad \qquad {\rm div}_\mu u = 0,
\eqno (9)
$$
where $u \in \sd$ and $\del$ acts on vectors
by the Levi-Civita connection. The condition ${\rm div}_\mu u = 0$
can be written invariantly as $d (i_u \mu)=0$. Alternatively,
if $\mu$ is $\rho(x)$ times the Riemannian volume form, then this
condition reduces to $\div (\rho u)=0$, or equivalently $d^*(\rho \tilde u)=0$.
As before, the pressure $p$ is chosen such that ${\rm div}_\mu \dot u = 0$.
In this more general case, our main theorem reads as follows
\no {\bf Theorem} {\it Let $u$ be a smooth steady-state solution to
the equations (9). If $d \tilde u=0$ everywhere,
then the quadratic form $d^2H$ is identically zero.
Otherwise, the quadratic form $d^2H$ is indefinite. Moreover, the
spectrum of the corresponding self-adjoint operator is neither
bounded from below nor from above.}
\no {\it Proof:} In this case the quadratic form $d^2H$ is given by
$$ d^2H(\xi) = {1 \over 2} < i_\xi d\tilde u - df,i_\xi d\tilde u - df>
+ {1\over 2} , \eqno (10)
$$
where the function $f$ is chosen to make $i_\xi d\tilde u - df$
correspond to a $\mu$-solenoidal vector field. When $d\tilde u$
vanishes, $d^2H$ is identically zero. When $d\tilde u$ does not vanish
we construct, as before, highly oscillatory circularly polarized vector fields
$\xi_\eps$ that make $d^2H$ arbitrarily positive or negative.
Let $x_0 \in M$ be a point where $u$ and $d \tilde u$ are both nonzero.
Let $\phi(x)$ be a function on $M$ with $u \cdot \del \phi$ and $d\phi \wedge
d \tilde u$ both nonzero on a neighborhood $N$ of $x_0$.
Pick smooth vector fields
$a_R$ and $a_I$, everywhere orthogonal to $\del \phi$, that vanish outside
of $N$, for which $d\tilde u (a_R, a_I) \ge 0$, and for which
$d\tilde u (a_R, a_I) > 0$ in a smaller neighborhood $N' \subset N$.
This is possible since $d\phi \wedge d\tilde u$ is smooth and
nonzero on $N$. Let $a=a_R + i a_I$, and let
$$\tilde \xi_\eps = {i \eps \over \rho} d^*\left (
{\rho {d\phi} \wedge \tilde a \over |d\phi|^2} e^{i\phi/\eps} \right )
= \tilde a e^{i \phi/\eps} + O(\eps). \eqno (11)
$$
The leading term in the expansion of $d^2H(\xi_\eps)$ is
$$ \eqalign{
& {1 \over 2\eps} < i_{\xi_\eps} d \tilde u - df,
i (u \cdot \del \phi) \tilde a
e^{i \phi/\eps}>_c
=
{-i \over 2\eps} \int_M (u \cdot \del \phi) d\tilde u(a,\bar a) \mu(dx) \cr
= & {-1 \over \eps} \int_M (u \cdot \del \phi) d\tilde u(a_R, a_I) \mu(dx).}
\eqno(12) $$
By assumption, $u \cdot \del \phi$ is nonzero,
and hence of definite sign, on $N$,
while $d\tilde u(a_R, a_I)$ is positive on $N'$ and is never negative.
The integral over $M$ is therefore nonzero, and we can make $d^2H(\xi_\eps)$
arbitrarily large or small by taking $\eps$ sufficiently close to zero
and of appropriate sign. \qed
\smallskip
\no{\it 5. Manifolds with boundary}
Finally, we consider the Arnold stability criterion for fluid flow on
a manifold with boundary. As before, if $u$ is irrotational, then
$d^2H$ is identically zero. Otherwise, we construct the vector fields
$\xi_\eps$ as before, in an arbitrarily small neighborhood of a point. In
particular, we can choose $\xi_\eps$ to vanish near the boundary,
so our conclusions
about $d^2H$ being indefinite are unaffected by the presence of the
boundary. Translating
this conclusion into a statement about the spectrum of the
self-adjoint operator associated to $d^2H$ is
more difficult, however, since it is not immediately clear what the
domain of this operator would be. In this section we construct this
operator.
For simplicity, we consider $M$ to be a compact smooth domain in ${\bf R}^3$,
with the standard metric and volume form. We work on the space
$$ \hil = \left \{ u \in L^2(M)^3 | \div u =0, (u \cdot n)|_{\partial M}=0
\right \}. \eqno (13)
$$
Let $V = C_0^\infty(M) \cap \hil$. $V$ is a dense subspace of $\hil$,
consisting of smooth solenoidal vector fields with compact support in
the interior of $M$. We define an unbounded self-adjoint
operator $\acal: \hil \to \hil$
on the following domain $D(\acal) \supset V$.
$$ D(\acal) = \left \{ \xi \in \hil | -u \times \div((\curl u)\times \xi
- \del p) - \del g \in \hil \right \}, \eqno(14)
$$
where the inclusion in $\hil$ is in the following sense.
The vector $\bcal(\xi)=(\curl u)\times \xi- \del p$ is in $\hil$,
since multiplication by $\curl u$ and $L^2$-projection onto solenoidal vectors
are bounded operations. The vector $-u \times \div((\curl u)\times \xi
- \del p)$ must be understood in the distributional sense as an element
of ${\cal D}'(M^\circ)^3$.
The statement that
$-u \times \div((\curl u)\times \xi - \del p) - \del g \in \hil$ means that
there exists $w \in \hil$ such that, for all $v \in V$,
$$ < -u \times \div((\curl u)\times \xi - \del p)-w,v> = 0, \eqno(15)$$
in which case we define $\acal(\xi)=w$.
It is easy to check that, for any $\xi$, $\zeta \in D(\acal)$,
$<\xi, \acal(\zeta)>=<\acal(\xi),\zeta>$ and that
${1 \over 2}<\acal(\xi),\xi> +{1 \over 2}<\bcal(\xi),\bcal(\xi)> = d^2H(\xi)$.
The quadratic forms $d^2H$ and $<\acal(\xi),\xi>$ are clearly defined on
the factor space $\hil/K$, where $K$ is the kernel of the bounded operator
$\bcal$. Since $\bcal$ is a bounded operator, unboundedness of the
quadratic form $<\acal(\xi),\xi>$ both from above and below implies the
same statement for $d^2H$ on $\hil/K$ and similarly for the self-adjoint
operator ${1 \over 2} (\acal + \bcal^*\bcal)$ mod $K$.
\no{\it 6. Conclusions}
The quadratic form $d^2H$ ((5) or, more generally (10))
contains two terms, a positive-definite first
term and an indefinite second term. The second term contains one more
derivative of $\xi$ than the first term, and so should be expected
to dominate when $\xi$ is highly oscillatory. When $\xi$ is a circularly
polarized transverse wave this is in fact the case. At any point
where $u$ and $\curl u$ are both nonzero, there exist directions where
right (or left) circularly polarized waves give large positive values
of $d^2H$ and left (or right) circularly polarized waves give large
negative values. In 3 or more dimensions this makes $d^2H$ unbounded and
indefinite.
In two dimensions, however, circular polarization does not exist.
One can only construct linearly polarized waves, for which the
$O(1/\eps)$ contribution to $d^2H$ is identically zero. So in two dimensions
$d^2H$ is sometimes definite, and the Arnold criterion is a useful
measure of stability. In 3 or more dimensions, $d^2H$ is never definite, and the Arnold criterion can never be applied.
\bigskip
\no{\it Acknowledgements:}
L.S. is partially supported by an NSF Mathematical Sciences
Postdoctoral Fellowship. M.V. is partially supported by
NSF Grant \#9105688.
\bigskip \cntl{REFERENCES}
\item{1.} V.I.~Arnold, Ann. Inst. Fourier {\bf 16}, 316--361 (1966)
\item{2.} V.I.~Arnold, Usp. Math. Nauk. {\bf 24}, no. 3, 225--226 (1969)
({\it in Russian})
\item{3.} V.I.~Arnold, ``Mathematical Methods of
Classical Mechanics'', 2nd. Ed., Springer-Verlag, Berlin (1989)
\item{4.} V.I.~Arnold, Dokl. Acad. Nauk SSSR {\bf 162}, no. 5, 975--978 (1965)
(translated in Sov. Math. {\bf 6}, 773--777)
\item{5.} V.I.~Arnold, Izv. Vyssh. Ucheb. Zaved. Mathematika {\bf 54}, 3--5 (1966)(translated in Amer. Math. Soc. Trans. Ser. 2 {\bf 79}, 267--269)
\item{6.} M. McIntyre and T. Shepherd, J. Fluid Mech. {\bf 181},
527--565 (1987)
\item{7.} H.K. Moffatt, J. Fluid Mech. {\bf 166}, 359--378 (1986)
\item{8.} D. Ebin and J. Marsden, Ann. Math. {\bf 22}, 102-163 (1970)
\item{9.} A.I. Schnirelman, Matem. Sb.{\bf 128}, no. 1 (1985)
(translated in Math. USSR Sbornik {\bf 56} no. 1, 79--105 (1987))
\item{10.} A.I. Schnirelman, {\it Attainable Diffeomorphisms}, Tel-Aviv
University preprint (1992)
\item{11.} V.I.~Arnold and B.A. Khesin, Ann. Rev. Fluid Mech. {\bf 24},
145--166 (1992)
\vfill\eject
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