# First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ over $\Omega$) which satisfies $q\cdot \nu =0$ over $\partial \Omega$.

By first integrals of $q$, we mean any function $w\in H^1(\Omega)$, $w$ is periodic and $q\cdot\nabla w=0$.

Obviously a function $w$ which is a first integral is going to be constant over the streamlines of $q$. These first integrals turn out to be important in studying some qualitative properties of the KPP minimal speed within large drift etc. Existence of nontrivial first integrals (other than the constant functions) is the question in the 3 dimensional case. One easy situation, is the case where we have a shear flow (the streamlines of $q$ are unidirectional). In this particular case, we know that nontrivial first integrals exist.

While looking at this kind of problems, one can think of another particular situation which is also interesting: the case where the vector field $q$ vanishes outside a cylindrical component, $V$, and inside, this vector field exhibits an ergodic character. That is, there is a streamline of $q$ which traverses every point in $V$. In such case, it could be proved that any first integral $w$ must be constant over the ergodic component $V$ (the proof is not straight forward though).

Now that we have introduced all these things,

Q. Do we know or can we, give an explicit example of an incompressible field which vanishes outside a cylindrical component $V$, satisfies $q\cdot\nu=0$ on $\partial V$, while $V$ is an ergodic component for $q$.

Construction of such flows was done in papers of Prof. Pesin but it involved a lot of complicated steps-- the result led to more than ergodicity (in fact, Pesin constructed Bernoulli flows over compact manifolds of dimension $\geq 3$). As ergodicity over a component for a vector field is not as sophisticated as Bernoulli, one still hopes to have an explicit example of an incompressible field which admits an ergodic component $V$ (cylindrical) and vanishes outside $V$ as well as on $\partial V$?

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The ABC flow (Arnold Beltrami Childress), which admits ergodic components, in which a streamlines are volume filling, turns out to be a good example of those. It took a while but now, we have plenty of examples.

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