- 93-35 Sadun L., Vishik M.
- On the Arnold Stability Criterion
(19K, plain TeX)
Feb 19, 93
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Abstract. 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.