# On the restriction of various Laplace operators on submanifolds

Jukka Tuomela
Department of Physics and Mathematics
University of Eastern Finland
P.O. box 111, FI-80101 Joensuu, Finland
<EMAIL_ADDRESS>

###### Abstract

When considering Navier-Stokes equations on Riemannian manifolds one
frequently encounters situations where the manifold is embedded in the ambient
Euclidean space. In this context it is interesting to investigate what is the
precise relationship of the diffusion operator in the ambient space to the
diffusion operator on the manifold. The present paper gives a precise
characterization of this situation for general surfaces in three dimensional
space.

_M_ athematics Subject Classification (2020) 35Q30, 53Z05, 58J99, 76D05

_K_ eywords Navier-Stokes equations, Laplace operators, Riemannian manifolds,
moving frames

##  1 Introduction

In recent years there has been a growing interest on analysing Navier-Stokes
systems on Riemannian manifolds, see for example [1, 5, 6, 9] and many
references therein. An important issue in the analysis has turned out to be
the choice of an appropriate diffusion operator. As discussed in [3, 9]
different choices can lead to very different solutions. It appears that it is
at present somewhat unclear how to choose a correct diffusion operator in a
concrete physical model.

In the present article we will discuss a related problem. Often we have a
three dimensional flow, but we would like to study its restriction to some
invariant submanifold. For example in the large scale atmospheric flow we
could be interested in the flow on some (fictitious) surface. Then whatever
diffusion operator one chooses, one would like to see how the three
dimensional operator on the ambient space and the corresponding two
dimensional operator on the relevant surface are related. In fact it is
sufficient to analyse the Bochner Laplacian because other diffusion operators
can be expressed in terms of it as will be explained below. Hence our results
apply equally well for all diffusion operators.

In [2] this problem was solved in the case of ellipsoid, and the present paper
can be understood as a generalization of it. Hence we refer to [2] and its
references for more background on this problem. In this paper we compute the
restriction of Bochner Laplacian (and hence of any diffusion operator) to an
arbitrary surface in three dimensional space. In fact the ambient space need
not be $\mathbb{R}^{3}$; it can be any three dimensional Riemannian manifold.
This is because the technical tool which we use, namely adapted moving frames
(see [11]), works in the same way also in this general context. We compute
also the normal component of the Bochner Laplacian. As far as we know the
normal component has not been considered previously, but one would think that
it could also be of some interest.

Our approach is thus differential geometric. There has also been articles from
the functional analysis point of view. One of the first papers in this
direction for curved surfaces is [12]; see also [5, 7, 8] for more recent
works. Perhaps combining more closely differential geometric and functional
analytic ideas could lead to a better understanding of the restriction
problem.

In section 2 we introduce the necessary tools of Riemannian geometry and
discuss how Navier-Stokes equations are formulated on Riemannian manifolds. In
section 3 we consider the problem treated in [2] in some detail because
analyzing it in our framework gives some insights about the general case. In
section 4 various types of extension/restriction are discussed, and it is seen
that by the moving frame approach one can easily prove the existence of
various types of extensions. In section 5 the main results are stated and
proved. The results show that in general the diffusion operator restricted to
the submanifold is rather unrelated to the diffusion operator on the
submanifold. Our results, however, give precise formulas for the restriction
which makes it then possible to study under which conditions there is a
reasonable relationship between them.

##  2 Preliminaries

###  2.1 Riemannian geometry

Let us first recall some facts about Riemannian geometry which are needed
later on; for details we refer to [10, 11]. Einstein summation convention is
used whenever convenient. Let $M$ be a smooth Riemannian manifold with metric
$g$. For a function $f\,:\,M\to\mathbb{R}$ its gradient is
$\mathsf{grad}(f)=g^{ij}f_{;j}$ and the divergence of the vector field $u$ is
$\mathsf{div}(u)=u^{i}_{;i}$. The Ricci tensor is denoted by $\mathsf{Ri}$. In
order to simplify the notations we use the standard convention that when the
indices are raised or lowered the notation does not change. Hence $u$ can
denote both covariant and contravariant vector and for example in two
dimensional case we can write $\mathsf{Ri}=\kappa g$ where $\kappa$ is the
Gaussian curvature and also $\mathsf{Ri}\,v=\kappa\,v$ where $v$ is a vector
field. If the clarity requires to specify the type precisely then we use the
musical isomorphisms $\sharp$ and $\flat$.

Let $M_{0}$ be a three dimensional Riemannian manifold with Riemannian metric
$g_{0}$ and covariant derivative $\overline{\nabla}$. Let then $M$ be a two
dimensional manifold with an embedding $f\,:\,M\to M_{0}$ and as usual we
identify $M$ and $f(M)$ when convenient. The Riemannian metric on $M$ is
$g=f^{\ast}g_{0}$ and the covariant derivative is denoted by $\nabla$.

Let us now introduce an adapted orthonormal frame $\\{b^{1},b^{2},b^{3}\\}$ in
the neighborhood of $M$, i.e. $b^{1}$ and $b^{2}$ are tangent along $M$ and
$b^{3}$ is the unit normal. Note that globally such a frame may not exist, but
since all the analysis that follows is local we only need a local frame which
always exists. For simplicity of notation we will not indicate the domain
where the frame is defined. Then the dual frame is given by $\theta^{j}=\flat
b^{j}$ and the associated connection forms are the one forms $\omega_{ij}$
which satisfy the equations

$\displaystyle\omega_{ij}=$ $\displaystyle-\omega_{ji}$ $\displaystyle
d\theta^{i}=$ $\displaystyle-\sum_{k}\omega_{ik}\wedge\theta^{k}\ .$

Note that $\omega_{13}(b^{2})=\omega_{23}(b^{1})$. Similarly we have also an
orthonormal frame
$\\{b^{1}_{z},b^{2}_{z}\\}=\\{f^{\ast}b^{1},f^{\ast}b^{2}\\}$,
$\theta_{z}^{j}=\flat b_{z}^{j}$ and the connection forms $\omega_{ij}^{z}$ on
$M$. The covariant derivative and connection forms are related by the formula

$\overline{\nabla}_{b^{k}}b^{j}=\sum_{i}\omega_{ij}(b^{k})b^{i}\ .$ (2.1)

Then we have the curvature forms $\Omega_{ij}$ defined by

$d\omega_{ij}=-\sum_{k}\omega_{ik}\wedge\omega_{kj}+\Omega_{ij}\ .$ (2.2)

As the name suggests $\Omega_{ij}=0$ if $M_{0}$ is flat. The volume form of
$M_{0}$, denoted $\Xi_{0}$, is given by
$\Xi_{0}=\theta^{1}\wedge\theta^{2}\wedge\theta^{3}$ and similarly
$\Xi=\theta_{z}^{1}\wedge\theta_{z}^{2}$ is the volume form on $M$. On $M$ we
have $d\omega_{12}^{z}=\Omega_{12}^{z}=\kappa\,\Xi$ and hence
$\Omega_{12}^{z}(b_{z}^{1},b_{z}^{2})=\kappa$.

Let then $X$ and $Y$ be vector fields on $M_{0}$ which are tangent along $M$.
Then one can write

$\overline{\nabla}_{X}Y=\nabla_{X}Y+S(X,Y)b^{3}\ .$ (2.3)

These are known as Gauss formulas and the symmetric tensor $S$ is the second
fundamental form. If $X$ is a vector field tangent along $M$, then

$\overline{\nabla}_{X}b^{3}=-S\,X=\omega_{13}(X)b^{1}+\omega_{23}(X)b^{2}\ .$
(2.4)

The first equality is known as Weingarten’s equation. It will be convenient to
define the functions $t_{ij}$ by

$\displaystyle t_{11}=$ $\displaystyle S(b^{1},b^{1})=-\omega_{13}(b^{1})$
$\displaystyle t_{22}=$ $\displaystyle S(b^{2},b^{2})=-\omega_{23}(b^{2})$
$\displaystyle t_{12}=$ $\displaystyle
S(b^{1},b^{2})=-\omega_{13}(b^{2})=-\omega_{23}(b^{1})\ .$

If we consider the equality $t_{ij}=S(b^{i},b^{j})$ then one thinks that the
functions $t_{ij}$ are defined on $M$ and in this case one could also write
$t_{ij}=S(b_{z}^{i},b_{z}^{j})$ ; however, the equality
$t_{ij}=-\omega_{i3}(b^{j})$ allows one to interpret $t_{ij}$ as functions in
the neighborhood of $M$. It turns out that the adjugate of $S$ appears in the
formulas below so we introduce

$S_{\mathsf{adj}}=t_{22}\theta^{1}\otimes\theta^{1}-t_{12}\theta^{1}\otimes\theta^{2}-t_{12}\theta^{2}\otimes\theta^{1}+t_{11}\theta^{2}\otimes\theta^{2}\
.$

It is also convenient to introduce the following notations:

$\alpha_{j}=\omega_{j3}(b^{3})\quad\mathrm{and}\quad\gamma_{j}=\omega_{12}(b^{j})\
.$ (2.5)

Again depending on the context these can be interpreted as functions on $M$ or
in the neighborhood of $M$. Then we have

$\kappa=t_{11}t_{22}-t_{12}^{2}\quad\mathrm{and}\quad\mathcal{H}=\tfrac{1}{2}\,\big{(}t_{11}+t_{22}\big{)}$
(2.6)

where $\mathcal{H}$ is the mean curvature. Note that the sign of $\mathcal{H}$
depends on the choice of orientation of the normal field. Finally we need the
formula

$d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega\big{(}[X,Y]\big{)}$ (2.7)

where $\omega$ is a one form and $X$ and $Y$ are vector fields.

On some occasions we will refer to ”standard results in PDE”; this means that
the corresponding statements can easily be found for example in [4]. When
computing concrete examples we have used the _Differential Geometry_ package
of Maple.111https://www.maplesoft.com/

###  2.2 Navier-Stokes equations

We can write the incompressible Navier-Stokes equations on Riemannian
manifolds as follows:

$\displaystyle u_{t}-Au+\nabla_{u}u+\mathsf{grad}(p)=0$ (2.8)
$\displaystyle\mathsf{div}(u)=0$

Here $A$ is some diffusion operator, and as discussed in [3, 9], there are
basically three candidates for $A$; namely Bochner Laplacian, Hodge Laplacian
and symmetric Laplacian. Let us first set

$\Sigma u=g^{ki}u^{j}_{;i}+g^{ij}u^{k}_{;i}\ .$

Then the various Laplacians are

$\displaystyle\Delta_{B}u=$ $\displaystyle g^{ij}u^{k}_{;ij}$
$\displaystyle\Delta_{H}u=$ $\displaystyle-\sharp(d\delta+\delta d)\flat
u=\Delta_{B}u-\mathsf{Ri}(u)$ $\displaystyle Lu=$
$\displaystyle\mathsf{div}(\Sigma
u)=\Delta_{B}u+\mathsf{grad}(\mathsf{div}(u))+\mathsf{Ri}(u)$

Here $\delta$ is the codifferential and we recall that for vector fields
$\mathsf{div}(u)=-\delta\flat u$. It is seen that for flat manifolds Hodge
Laplacian is the same as Bochner Laplacian, and if in addition $u$ is
divergence free, then the symmetric Laplacian is also equal to them. Note that
for divergence free vector fields all diffusion operators can be written as

$Au=\Delta_{B}u+\beta\,\mathsf{Ri}(u)$

where $\beta=\pm 1$ or $\beta=0$. Let us then recall that

  * (i)

$u$ is parallel, if $\nabla u=0$. Parallel fields are solutions to (2.8) with
$\beta=0$ and $p$ constant.

  * (ii)

$u$ is harmonic, if $\Delta_{H}u=0$. Divergence free harmonic fields are
solutions to (2.8) with $\beta=-1$ and $p=-\tfrac{1}{2}\,g(u,u)$.

  * (iii)

$u$ is Killing, if $\Sigma u=0$. Killing fields are solutions to (2.8) with
$\beta=1$ and $p=\tfrac{1}{2}\,g(u,u)$.

As discussed in [3, 9] different choices of $\beta$ can produce completely
different solutions. In some sense this is very natural, or at least
intuitively plausible, because essentially choosing $\beta=-1$ corresponds to
choosing antisymmetric part of $\Delta_{B}u$ while $\beta=1$ corresponds to
choosing symmetric part of $\Delta_{B}u$. So different choices of $\beta$
correspond to different physical models. From the point of view of continuum
mechanics $L$ would seem a natural choice, but perhaps in some situations
other choices are appropriate. One might speculate that maybe in addition to
above three values other values of $\beta$ could be relevant in some
situations, or that $\beta$ might even be a function. Probably one should in
any case require that $|\beta|\leq 1$ since this condition guarantees that
$-A$ is positive definite for divergence free vector fields.

##  3 Restriction problem

Our goal is to generalize the treatment of the restriction problem given in
[2]. Let us first recall the setting of the problem in [2]. Let
$D_{x}\subset\mathbb{R}^{3}$ be some convenient domain with the standard
Riemannian metric $g_{1}$ and let us denote the coordinates of $D_{x}$ by $x$.
Then let $D_{y}\subset\mathbb{R}^{3}$ be some other domain with coordinates
$y$ and let $\psi\,:\,D_{y}\to D_{x}$ be a diffeomorphism. Finally let
$D_{z}\subset\mathbb{R}^{2}$ be a two dimensional domain with coordinates $z$
and let $f\,:\,D_{z}\to D_{y}$ be an embedding. Since all the analysis that
follow is local it is not necessary to explicitly specify various domains. The
metrics in $D_{y}$ and $D_{z}$ are obtained by pullbacks:
$g_{0}=\psi^{\ast}g_{1}$ and $g=f^{\ast}g_{0}$.

Of course one may view that $D_{y}$ and $D_{x}$ are same as manifolds, and $y$
and $x$ are merely two different coordinate systems. Indeed below when we
consider the general case the ”intermediate” coordinates are not needed.
However, when computing concrete examples it is convenient to specify the
problem in this way.

Now we would like to compare the Laplacian defined on $D_{y}$ and the
Laplacian on $M=f(D_{z})$. Note that for divergence free vector fields the
only difficulty is the Bochner part of the operator, so it is sufficient to
analyse $\Delta_{B}$. Let $u$ be a vector field on $D_{y}$ which is tangent to
$M$. Hence there is a vector field $v$ on $M$ such that $v=u$ on $M$. We want
to compare $\Delta_{B}u$ and $\Delta_{B}v$ somehow. The problem is that while
$\Delta_{B}v$ is a vector field on $M$ by definition, $\Delta_{B}u$ is not.

Now let $p\in M$ and consider $T_{p}M$ as subspace of $T_{p}D_{y}$; then we
can introduce the orthogonal projection $\pi\,:\,T_{p}D_{y}\to T_{p}M$ and we
could compare $\Delta_{B}v$ and $\pi\big{(}\Delta_{B}u\big{)}$. Note that in
[2] authors regarded $\Delta_{B}u$ as a one form and then pulled it back to
$M$; however, this is equivalent to projection since if $X$ is any vector
field then

$f^{\ast}\flat\,\pi(X)=f^{\ast}\flat\,X\ .$

Let $\\{b^{1},b^{2},b^{3}\\}$ be an adapted orthonormal frame and let
$\\{b_{z}^{1},b_{z}^{2}\\}$ be the corresponding orthonormal frame on $M$. Now
we set

$u=u^{1}b^{1}+u^{2}b^{2}+u^{3}b^{3}\quad\mathrm{and}\quad
v=v^{1}b_{z}^{1}+v^{2}b_{z}^{2}=f^{\ast}\big{(}u^{1}b^{1}+u^{2}b^{2}\big{)}\
.$ (3.1)

The derivatives of $u^{k}$ are denoted by subscripts: $u^{k}_{1}=\partial
u^{k}/\partial y_{1}$ etc. Since $u$ should be tangent to $M$ we require that
$u^{3}$ is identically zero on $M$. Let us now consider the explicit problem
in [2] before considering the general case. We feel that this is useful since
it gives a good idea of the more complicated computations of the general case,
and moreover it gives some indications about what the general result should
look like.

The following maps were chosen in [2]:

$\displaystyle\psi(y)=$
$\displaystyle\big{(}ay_{3}\cos(y_{1})\sin(y_{2}),ay_{3}\sin(y_{1})\sin(y_{2}),y_{3}\cos(y_{2})\big{)}$
(3.2) $\displaystyle f(z)=$ $\displaystyle\big{(}z_{1},z_{2},1\big{)}$

Here $a$ is some positive constant. Then setting
$\lambda=\sqrt{a^{2}\cos(y_{2})^{2}+\sin(y_{2})^{2}}$ we can choose for
example the following frames:

$\displaystyle b^{1}=$
$\displaystyle\frac{1}{ay_{3}\sin(y_{2})}\,\partial_{y_{1}}$ $\displaystyle
b^{2}=$ $\displaystyle\frac{1}{y_{3}\lambda}\,\partial_{y_{2}}$ $\displaystyle
b^{3}=$
$\displaystyle\frac{(1-a^{2})\sin(2y_{2})}{2ay_{3}\lambda}\partial_{y_{2}}+\frac{\lambda}{a}\,\partial_{y_{3}}$
(3.3) $\displaystyle b_{z}^{1}=$
$\displaystyle\frac{1}{a\sin(z_{2})}\,\partial_{z_{1}}$ $\displaystyle
b_{z}^{2}=$ $\displaystyle\frac{1}{\lambda}\,\partial_{z_{2}}$

We will denote also the pullback $f^{\ast}\lambda$ by $\lambda$. Then the
computations with formulas (2.3) and (2.6) give

$\kappa=\frac{1}{\lambda^{4}}\quad\mathrm{and}\quad
2\mathcal{H}=-\frac{1}{a\lambda}-\frac{a}{\lambda^{3}}.$

The connection forms are

$\displaystyle\omega_{12}=$
$\displaystyle\frac{\cos(y_{2})}{y_{3}\lambda\sin(y_{2})}\,\theta^{1}$
$\displaystyle\omega_{13}=$ $\displaystyle\frac{1}{ay_{3}\lambda}\,\theta^{1}$
$\displaystyle\omega_{23}=$
$\displaystyle\frac{a}{y_{3}\lambda^{3}}\,\theta^{2}+\frac{(1-a^{2})\sin(2y_{2})}{2y_{3}\lambda^{3}}\,\theta^{3}\
.$

Then we compute

$\pi\big{(}\Delta_{B}u\big{)}=c_{1}b^{1}+c_{2}b^{2}\quad\mathrm{where}\quad
c_{j}=g_{0}(\Delta_{B}u,b^{j})\ .$

Now $c_{j}$ contain derivatives with respect to $y_{3}$ which are not
meaningful intrinsically in $M$. Putting
$\hat{\lambda}=\sqrt{a^{2}\sin(y_{2})^{2}+\cos(y_{2})^{2}}$ we have

$\displaystyle c_{1}=$
$\displaystyle\frac{1}{a^{2}\sin\\!(y_{2})^{2}}\,u^{1}_{11}+\frac{\hat{\lambda}^{2}}{a^{2}}\,u^{1}_{22}+\frac{\left(1-a^{2}\right)\sin\\!\left(2y_{2}\right)}{a^{2}}\,u^{1}_{23}+$
$\displaystyle\frac{\lambda^{2}}{a^{2}}\,u^{1}_{33}+\mathrm{lower\ order\
terms}\,.$

The terms in $c_{2}$ are similar except that there are second order
derivatives of $u^{2}$. So the goal is to somehow identify the terms
containing derivatives with respect to $y_{3}$, and then see what is left.
With the benefit of hindsight one might argue as follows: in $c_{j}$ there are
at most second order derivatives, and all information about these is contained
in the tensor $\overline{\nabla}\,\overline{\nabla}u$. However, we want to
eliminate the derivatives in the normal direction so somehow it would be
natural to consider $\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes
b^{3}$. Then we compute that

$\displaystyle\pi\big{(}\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes
b^{3}\big{)}=$ $\displaystyle d_{1}b^{1}+d_{2}b^{2}\quad\mathrm{where}\quad
d_{j}=g_{0}(\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes b^{3},b^{j})$
$\displaystyle c_{1}-d_{1}=$
$\displaystyle\frac{\sin^{2}\left(2y_{2}\right)\left(a^{2}-1\right)^{2}}{4\lambda^{2}a^{2}}\,u^{1}_{3}+\mathrm{other\
terms}$ $\displaystyle c_{2}-d_{2}=$
$\displaystyle\frac{\sin^{2}\left(2y_{2}\right)\left(a^{2}-1\right)^{2}}{4\lambda^{2}a^{2}}\,u^{2}_{3}+\mathrm{other\
terms}$

where the other terms do not contain derivatives with respect to $y_{3}$.
Hence the ”offending” second order derivatives in $c_{j}$ have disappeared.
Somewhat unexpectedly the coefficients of the main terms in $c_{j}-d_{j}$ are
equal. To get rid of these terms we try the Lie bracket; this gives

$\displaystyle\pi\big{(}\big{[}u,b^{3}\big{]}\big{)}=$ $\displaystyle
e_{1}b^{1}+e_{2}b^{2}$ $\displaystyle\hat{v}=$
$\displaystyle(c_{1}-d_{1}-2\mathcal{H}\,e_{1})b^{1}+(c_{2}-d_{2}-2\mathcal{H}\,e_{2})b^{2}$

Now $\hat{v}$ contains only terms which make sense on $M$ so that we can
consider the corresponding vector field on $M$, still denoted by $\hat{v}$. Of
course it was not a priori clear that one must multiply $e_{j}$ by
$2\mathcal{H}$, but once one tries the Lie bracket then it is easy to compute
that this is the right choice. Now we can compare $\hat{v}$ to Bochner
Laplacian, and simply computing we obtain that

$\hat{v}=\Delta_{B}v+\kappa v\ .$

Let us summarize the results so far. In the problem specified by (3.2) we have
obtained using the adapted frame (3.3) that

$\pi\big{(}\Delta_{B}u\big{)}=\Delta_{B}v+\kappa
v+\pi\Big{(}\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes
b^{3}+2\mathcal{H}[u,b^{3}]\Big{)}\ .$ (3.4)

It is curious that both Gaussian and mean curvature appear in the result. Now
note that everything in the formula (3.4) makes sense on an arbitrary surface,
so it is natural to assume that the result holds in general. This is indeed
almost true and one could actually prove it directly in a purely computational
way, following the same steps as above. The word almost appears because in the
general case there are terms which are automatically zero for the problem
(3.2). Anyway the symmetric Laplacian for divergence free fields
$Lv=\Delta_{B}v+\kappa v$ will always be present.

However, it seems that it would be better give a more conceptual proof, and
moreover it would be interesting to compute also the normal component. Also
note that the properties of $\mathbb{R}^{3}$ were not really used; all
computations were done in terms of the moving frame and these are possible
also when the ambient manifold is an arbitrary three dimensional Riemannian
manifold. Recall that all the tools introduced in section 2.1 are valid in
this general case. So instead of proving the formula (3.4) directly let us now
move on to the general case.

##  4 General formulation of the problem

Now we use the tools and notations outlined in section 2 to formulate the
problem in the general case. Let us introduce vector fields $u$ and $v$ as in
(3.1) such that $u^{3}$ is identically zero on $M$. Before analyzing the
decomposition of $\Delta_{B}u$ let us first discuss the divergence, bracket
and curl.

In the framework of our problem we may view $u$ as an extension of $v$, or $v$
as a restriction of $u$. Let us consider how the properties of $u$ and $v$ are
related. Now in the flow problems the important situation is where the flow is
incompressible and it would be natural to require that both $u$ and $v$ are
incompressible. This is convenient to consider as an extension problem: given
an incompressible flow $v$ can we find an incompressible extension $u$? In
fact an even more general result is true.

######  Theorem 4.1

Let $v$ be a vector field on $M$. Then there is a vector field $u$ which
extends $v$ such that $\mathsf{div}(u)$ restricted to $M$ is equal to
$\mathsf{div}(v)$.

We will call this kind of extension a compatible extension.

Proof.  Note that
$\mathsf{div}(b^{1})\Xi_{0}=d\big{(}\theta^{2}\wedge\theta^{3}\big{)}$ and
similarly for other $b^{j}$ and $b_{z}^{j}$. Then we compute, using the
notation (2.5), that

$\displaystyle\mathsf{div}(u)=$ $\displaystyle
u^{1}\mathsf{div}(b^{1})+u^{2}\mathsf{div}(b^{2})+u^{3}\mathsf{div}(b^{3})+b^{1}(u^{1})+b^{2}(u^{2})+b^{3}(u^{3})$
$\displaystyle=$
$\displaystyle-(\gamma_{2}+\alpha_{1})u^{1}+(\gamma_{1}-\alpha_{2})u^{2}-(t_{11}+t_{22})u^{3}+b^{1}(u^{1})+b^{2}(u^{2})+b^{3}(u^{3})$
$\displaystyle\mathsf{div}(v)=$ $\displaystyle
v^{1}\mathsf{div}(b_{z}^{1})+v^{2}\mathsf{div}(b_{z}^{2})+b_{z}^{1}(v^{1})+b_{z}^{2}(v^{2})$
$\displaystyle=$
$\displaystyle-\gamma_{2}v^{1}+\gamma_{1}v^{2}+b_{z}^{1}(v^{1})+b_{z}^{2}(v^{2})$

Hence $\mathsf{div}(u)$ restricted to $M$ is equal to $\mathsf{div}(v)$, i.e.
$f^{\ast}\mathsf{div}(u)=\mathsf{div}(v)$, if

$b^{3}(u^{3})=\alpha_{1}u^{1}+\alpha_{2}u^{2}\ .$ (4.1)

Now we can choose extensions $u^{1}$ and $u^{2}$ arbitrarily and if we require
that (4.1) is satisfied in the neighborhood of $M$ then we have a linear first
order PDE for $u^{3}$ which is noncharacteristic because $b^{3}$ is normal to
$M$. Hence a local solution with initial condition $u^{3}=0$ on $M$ exists by
standard theorems. $\square$

Note that (4.1) is not the only way to extend $u^{3}$; one could for example
also choose

$b^{3}(u^{3})=\alpha_{1}u^{1}+\alpha_{2}u^{2}+Cu^{3}$

where $C$ is a linear operator which contains only tangential derivatives.

In case of compressible problems it is perhaps not so clear how
$\mathsf{div}(v)$ and $\mathsf{div}(u)$ should be related. However, it seems
that in incompressible problems compatibility would be a natural requirement.
Since the incompressible case is so fundamental we will in the sequel consider
both general and compatible extensions. Note that if one thinks from the point
of view of restrictions, then typically restrictions are not compatible.

The conclusion of the previous Theorem is still quite weak in the sense that
if $v$ is divergence free and then the compatible extension constructed above
is not necessarily divergence free in the neighborhood of $M$. However, by
modifying the argument slightly we obtain

######  Corollary 4.2

Let $v$ be a vector field on $M$ such that $\mathsf{div}(v)=0$. Then there is
a compatible extension $u$ such that $\mathsf{div}(u)=0$ in the neighborhood
of $M$.

Proof.  We let $u^{1}$ and $u^{2}$ be $v^{1}$ and $v^{2}$ on $M$ and then we
extend them in some way. Then the equation $\mathsf{div}(u)=0$, given above,
can be written as

$b^{3}(u^{3})-(t_{11}+t_{22})u^{3}=F$ (4.2)

where $F$ is some known function. This together with the initial condition
$u^{3}=0$ on $M$ is a standard problem so the local solution exists. $\square$

Let us now see how this looks with the example (3.2). Choosing for example
$v^{1}=az_{1}/\lambda$ and $v^{2}=-z_{2}/\sin(z_{2})$ we have
$\mathsf{div}(v)=0$. Then putting simply $u^{1}=ay_{1}/\lambda$ and
$u^{2}=-y_{2}/\sin(y_{2})$ we get

$\frac{(1-a^{2})\sin(2y_{2})}{2ay_{3}\lambda}u^{3}_{2}+\frac{\lambda}{a}\,u^{3}_{3}+\frac{a^{2}+\lambda^{2}}{a\lambda^{3}y_{3}}\,u^{3}=\frac{\left(a^{2}-1\right)\cos\\!\left(y_{2}\right)y_{2}}{\lambda^{3}}\
.$

Let us then consider the vorticity of the flow and its restriction or
extension. Let $v$ be a vector field on $M$ and $u$ be a vector field on
$M_{0}$; their vorticities are given by formulas

$\mathsf{rot}(v)=\ast d\,v\quad\mathrm{and}\quad\mathsf{curl}(u)=\sharp\ast
d\,u$

where $\ast$ is the Hodge star.

######  Lemma 4.3

Let $u$ be a vector field on $M_{0}$ which is tangent along $M$ and let $v$ be
its restriction on $M$. Then $\mathsf{rot}(v)$ is essentially the pullback of
$\mathsf{curl}(u)$:

$\mathsf{rot}(v)\Xi=f^{\ast}du=f^{\ast}\big{(}\ast\,\flat\,\mathsf{curl}(u)\big{)}\
.$ (4.3)

In particular if $u$ is harmonic and divergence free, then
$\mathsf{rot}(v)=0$.

Proof.  This follows because pullback commutes with exterior derivative:
$df^{\ast}u=f^{\ast}du$. Then recall that if $u$ is harmonic and
$\mathsf{div}(u)=-\delta u=0$ then also $du=0$. $\square$

If a planar flow is interpreted as a three dimensional flow invariant with a
certain direction, then the vorticity of the planar flow can be interpreted as
a vector orthogonal to the plane. In the present context we could then ask if
there is an extension of $v$ such that

$\mathsf{curl}(u)=\mathsf{rot}(v)b^{3}\ .$

It is intuitively rather clear that all extensions do not satisfy this
requirement. However, the previous lemma implies that the normal component of
$\mathsf{curl}(u)$ is $\mathsf{rot}(v)b^{3}$.

######  Corollary 4.4

Let $v$ be a vector field on $M$ and let $u$ be its extension. Then
$g_{0}(\mathsf{curl}(u),b^{3})=\mathsf{rot}(v)$.

One interesting class of flows is the Beltrami flow where vorticity and flow
itself are linearly dependent, i.e. there is a function $\mu$ such that
$\mathsf{curl}(u)=\mu u$. Hence if $u$ is tangent along $M$, then the
vorticity is also tangent to $M$ and in this case $\mathsf{rot}(v)=0$. If we
require in addition that $v$ is divergence free, then $v$ would be in fact
harmonic.

Proof.  Recall that

$\theta^{3}\wedge du=g_{0}(\mathsf{curl}(u),b^{3})\Xi_{0}\ .$

Then using (4.3) we have

$(\theta^{3}\wedge
du)(b^{1},b^{2},b^{3})=du(b^{1},b^{2})=dv(b_{z}^{1},b_{z}^{2})=\mathsf{rot}(v)\
.$

$\square$

Then it is natural to ask if there is an extension such that
$\mathsf{curl}(u)=\mathsf{rot}(v)b^{3}$.

######  Theorem 4.5

Let $v$ be a vector field on $M$. Then there is an extension $u$ such that

$\mathsf{curl}(u)=\mathsf{rot}(v)b^{3}\quad\mathrm{and}\quad\mathsf{div}(u)=\mathsf{div}(v)\quad\mathrm{on}\
M.$

Moreover, if $\mathsf{div}(v)=0$, then there is an extension such that
$\mathsf{div}(u)=0$ in the neighborhood of $M$.

Note that a priori we have four equations for three unknowns, but since
according to Corollary 4.4 one condition is automatically satisfied, then it
is rather natural to expect that there might be a solution.

Proof.  Computing and evaluating on $M$ we obtain

$\displaystyle\theta^{1}\wedge du=$
$\displaystyle\theta^{1}\wedge\Big{(}-u^{1}(\omega_{12}\wedge\theta^{2}+\omega_{13}\wedge\theta^{3})-u^{2}\omega_{23}\wedge\theta^{3}+du^{2}\wedge\theta^{2}+du^{3}\wedge\theta^{3}\Big{)}$
$\displaystyle\theta^{2}\wedge du=$
$\displaystyle\theta^{2}\wedge\Big{(}-u^{1}\omega_{13}\wedge\theta^{3}+u^{2}(\omega_{12}\wedge\theta^{1}-\omega_{23}\wedge\theta^{3})+du^{1}\wedge\theta^{1}+du^{3}\wedge\theta^{3}\Big{)}$
$\displaystyle g_{0}(\mathsf{curl}(u),b^{1})=$ $\displaystyle
t_{12}u^{1}+t_{22}u^{2}-du^{2}(b^{3})$ $\displaystyle
g_{0}(\mathsf{curl}(u),b^{2})=$ $\displaystyle-
t_{11}u^{1}-t_{12}u^{2}+du^{1}(b^{3})$

This leads to a linear system of first order PDE:

$\displaystyle b^{3}(u^{1})=$ $\displaystyle t_{11}u^{1}+t_{12}u^{2}$ (4.4)
$\displaystyle b^{3}(u^{2})=$ $\displaystyle t_{12}u^{1}+t_{22}u^{2}$
$\displaystyle b^{3}(u^{3})=$ $\displaystyle\alpha_{1}u^{1}+\alpha_{2}u^{2}$

By standard theorems there is a local solution. Then if $\mathsf{div}(v)=0$
and we want that $\mathsf{div}(u)=0$ in the neighborhood of $M$ then we can
simply replace the third equation in (4.4) by the equation (4.2). $\square$

Here also one gets other extensions by adding appropriate terms $C_{j}u^{3}$
where the operators $C_{j}$ contain derivatives only in tangential directions.
In the case of example (3.2) $t_{12}=0$ so that the new equations are
decoupled:

$\displaystyle\frac{(1-a^{2})\sin(2y_{2})}{2ay_{3}\lambda}u^{1}_{2}+\frac{\lambda}{a}\,u^{1}_{3}=$
$\displaystyle-\frac{1}{a\lambda y_{3}}\,u^{1}$
$\displaystyle\frac{(1-a^{2})\sin(2y_{2})}{2ay_{3}\lambda}u^{2}_{2}+\frac{\lambda}{a}\,u^{2}_{3}=$
$\displaystyle-\frac{a}{\lambda^{3}y_{3}}\,u^{2}\ .$

Theorem 4.1 has also another interesting corollary.

######  Corollary 4.6

If $\mathsf{div}(u)$ restricted to $M$ is equal to $\mathsf{div}(v)$ then
$[u,b^{3}]$ and $\overline{\nabla}_{b^{3}}u$ are tangent along $M$.

Proof.  Simply computing using (2.1) we obtain

$\displaystyle\overline{\nabla}_{b^{3}}u=$ $\displaystyle
u^{1}\overline{\nabla}_{b^{3}}b^{1}+u^{2}\overline{\nabla}_{b^{3}}b^{2}+u^{3}\overline{\nabla}_{b^{3}}b^{3}+b^{3}(u^{1})b^{1}+b^{3}(u^{2})b^{2}+b^{3}(u^{3})b^{3}$
$\displaystyle=$
$\displaystyle\big{(}b^{3}(u^{1})+\gamma_{3}u^{2}+\alpha_{1}u^{3}\big{)}b^{1}+\big{(}b^{3}(u^{2})-\gamma_{3}u^{1}+\alpha_{2}u^{3}\big{)}b^{2}+\big{(}b^{3}(u^{3})-\alpha_{1}u^{1}-\alpha_{2}u^{2}\big{)}b^{3}\
.$

Hence $\overline{\nabla}_{b^{3}}u$ is tangent along $M$ if (4.1) holds. Since
$\overline{\nabla}_{u}b^{3}$ is always tangent along $M$ by (2.4) this implies
that $[u,b^{3}]$ is then also tangent along $M$. $\square$

In particular if the restriction/extension is compatible, then there is no
need to project the bracket term in the formula (3.4). However, the bracket,
while seemingly natural here, does not appear in the decomposition below
because it is necessarily extrinsic to $M$. Hence we must in any case analyze
separately $\overline{\nabla}_{b^{3}}u$ and $\overline{\nabla}_{u}b^{3}$.

Before starting the decomposition let us first establish some notation which
will facilitate the computations below. The main issue is how to define
$\overline{\nabla}_{b^{3}}u$ in the neighborhood of $M$. We will consider two
cases: (1) the general case where the extension of $u$ is arbitrary, and (2)
divergence free extension where we extend $u^{3}$ such that
$\mathsf{div}(u)=0$ in the neighborhood of $M$.

Both cases can then be divided to two subcases, depending if we want the
extension/restriction to be compatible or not. Let us then set
$X_{3}=\overline{\nabla}_{b^{3}}b^{3}=\alpha_{1}b^{1}+\alpha_{2}b^{2}$ and

$\rho=b^{3}(u^{3})-\alpha_{1}u^{1}-\alpha_{2}u^{2}=b^{3}(u^{3})-g_{0}(X_{3},u)\
.$

The vector field $X_{3}$ will appear frequently in various formulas. When we
evaluate it on $M$ we can also write
$X_{3}=\alpha_{1}b_{z}^{1}+\alpha_{2}b_{z}^{2}$, without changing the
notation. Let us also introduce $q=b^{3}(u^{1})b^{1}+b^{3}(u^{2})b^{2}$. With
this notation one can write

$\overline{\nabla}_{b^{3}}u=q+u^{3}X_{3}+\gamma_{3}(u^{2}b^{1}-u^{1}b^{2})+\rho
b^{3}$ (4.5)

in the neighborhood of $M$. In the compatible case $\rho=0$ on $M$, but not
(necessarily) in the neighborhood of $M$. Then we compute using (2.2) and
(2.7) that

$\displaystyle b^{1}(t_{12})=$ $\displaystyle
b^{2}(t_{11})-\Omega_{13}(b^{1},b^{2})+(t_{11}-t_{22})\gamma_{1}+2t_{12}\gamma_{2}$
(4.6) $\displaystyle b^{2}(t_{12})=$ $\displaystyle
b^{1}(t_{22})+\Omega_{23}(b^{1},b^{2})-2t_{12}\gamma_{1}+(t_{11}-t_{22})\gamma_{2}$

##  5 Decomposition of Bochner Laplacian

We are now ready to start the decomposition of $\Delta_{B}u$. The basis of the
decomposition is the identity

$\displaystyle\Delta_{B}u=$
$\displaystyle\overline{\nabla}\,\overline{\nabla}u\Big{(}b^{1}\otimes
b^{1}+b^{2}\otimes b^{2}+b^{3}\otimes b^{3}\Big{)}$ $\displaystyle=$
$\displaystyle\overline{\nabla}_{b^{1}}\overline{\nabla}_{b^{1}}u+\overline{\nabla}_{b^{2}}\overline{\nabla}_{b^{2}}u-\overline{\nabla}_{X}u+\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes
b^{3}$

where $X=X_{1}+X_{2}$ and $X_{j}=\overline{\nabla}_{b^{j}}b^{j}$. Note that we
first compute with vector field $u$ and then evaluate the final result on $M$
so in the end we have the vector $v$ and its derivatives, but also terms
containing the normal derivatives of $u$.

Let us first consider the term $\overline{\nabla}_{X}u$.

######  Lemma 5.1

$\overline{\nabla}_{X}u$ on $M$ is given by

$\overline{\nabla}_{X}u=\nabla_{Kw}v+2\mathcal{H}\,(q+\gamma_{3}Kv)+\big{(}2\mathcal{H}\,\rho+S(Kw,v)\big{)}b^{3}$

where
$K=\theta_{z}^{1}\otimes\theta_{z}^{2}-\theta_{z}^{2}\otimes\theta_{z}^{1}$
and $w=\gamma_{1}b_{z}^{1}+\gamma_{2}b_{z}^{2}$.

Proof.  On $M$ we can write $X_{j}=Y_{j}+t_{jj}b^{3}$ where
$Y_{j}=\nabla_{b_{z}^{j}}b_{z}^{j}$ so that using (4.5) we obtain

$\overline{\nabla}_{X_{j}}u=\nabla_{Y_{j}}v+S(Y_{j},v)b^{3}+t_{jj}\overline{\nabla}_{b^{3}}u=\nabla_{Y_{j}}v+S(Y_{j},v)b^{3}+t_{jj}(q+\gamma_{3}Kv+\rho
b^{3})\ .$

Since $Kw=Y_{1}+Y_{2}$, adding gives the result. $\square$

It turns out to be useful to define tensors

$\displaystyle T_{0}=$
$\displaystyle\Big{(}2\,b_{z}^{1}\big{(}\mathcal{H}\big{)}+\Omega_{23}(b_{z}^{1},b_{z}^{2})\Big{)}\theta_{z}^{1}+\Big{(}2\,b_{z}^{2}\big{(}\mathcal{H}\big{)}-\Omega_{13}(b_{z}^{1},b_{z}^{2})\Big{)}\theta_{z}^{2}\quad\mathrm{and}$
$\displaystyle Q=$ $\displaystyle b_{z}^{1}(v^{1})b_{z}^{1}\otimes
b_{z}^{1}+b_{z}^{1}(v^{2})b_{z}^{1}\otimes
b_{z}^{2}+b_{z}^{2}(v^{1})b_{z}^{2}\otimes
b_{z}^{1}+b_{z}^{2}(v^{2})b_{z}^{2}\otimes b_{z}^{2}$

Recall that for flat manifolds $\Omega_{ij}=0$. Since $2\mathcal{H}$ is the
trace of $S$ one could also express
$2\,b_{z}^{1}\big{(}\mathcal{H}\big{)}\theta_{z}^{1}+2\,b_{z}^{2}\big{(}\mathcal{H}\big{)}\theta_{z}^{2}$
using $\nabla S$; however, this does not seem to make the formulas any
simpler.

######  Lemma 5.2

Let $\sigma=S_{ij}Q^{ij}$; then

$\displaystyle\overline{\nabla}_{b^{1}}\overline{\nabla}_{b^{1}}u+\overline{\nabla}_{b^{2}}\overline{\nabla}_{b^{2}}u=$
$\displaystyle\Delta_{B}v+\kappa\,v-2\mathcal{H}\,Sv+\nabla_{Kw}v+$
$\displaystyle\Big{(}2\,\sigma+T_{0}v+S(Kw,v)+2S(w,Kv)\Big{)}b^{3}\ .$

Somewhat strangely the tensor $P=\kappa\,g-2\mathcal{H}\,S$ which appears
above has a name: $-P$ is the _third_ fundamental form. Note that $P$ is
negative definite.

Proof.  We have first

$\overline{\nabla}_{b^{1}}u=\nabla_{b_{z}^{1}}v+S(v,b_{z}^{1})b^{3}\ .$

Then differentiating the tangential component gives

$\overline{\nabla}_{b^{1}}\nabla_{b_{z}^{1}}u=\nabla_{b_{z}^{1}}\nabla_{b_{z}^{1}}v+S(\nabla_{b_{z}^{1}}v,b_{z}^{1})b^{3}=\nabla\nabla
v\,b_{z}^{1}\otimes
b_{z}^{1}+\nabla_{Y_{1}}v+S(\nabla_{b_{z}^{1}}v,b_{z}^{1})b^{3}\ .$

Computing similarly with $\overline{\nabla}_{b^{2}}\nabla_{b_{z}^{2}}u$ and
adding we obtain

$\displaystyle\overline{\nabla}_{b^{1}}\nabla_{b^{1}}u+\overline{\nabla}_{b^{2}}\nabla_{b^{2}}u=$
$\displaystyle\Delta_{B}v+\nabla_{Kw}v+\Big{(}S(\nabla_{b_{z}^{1}}v,b_{z}^{1})+S(\nabla_{b_{z}^{2}}v,b_{z}^{2})\Big{)}b^{3}$

Then it is straightforward to check that

$S(\nabla_{b_{z}^{1}}v,b_{z}^{1})+S(\nabla_{b_{z}^{2}}v,b_{z}^{2})=\sigma+S(w,Kv)\
.$

Then from the normal component we get using (4.6) that

$b^{1}_{z}(S(v,b_{z}^{1}))+b^{2}_{z}(S(v,b_{z}^{2}))=\sigma+T_{0}v+S(Kw,v)+S(w,Kv)\
.$

Then

$\displaystyle\sum_{j=1}^{2}S(v,b_{z}^{j})\overline{\nabla}_{b^{j}}b^{3}=$
$\displaystyle-(t_{11}v^{1}+t_{12}v^{2})(t_{11}b_{z}^{1}+t_{12}b_{z}^{2})-(t_{12}v^{1}+t_{22}v^{2})(t_{12}b_{z}^{1}+t_{22}b_{z}^{2})$
$\displaystyle=$
$\displaystyle(t_{11}t_{22}-t_{12}^{2})v+(t_{11}+t_{22})\Big{(}(t_{11}v^{1}+t_{12}v^{2})b_{z}^{1}+(t_{12}v^{1}+t_{22}v^{2})b_{z}^{2}\Big{)}$
$\displaystyle=$ $\displaystyle\kappa\,v-2\mathcal{H}\,Sv\ .$

Now adding everything gives the result. $\square$

Putting together previous Lemmas gives

$\displaystyle\overline{\nabla}\,\overline{\nabla}u\Big{(}b^{1}\otimes
b^{1}+b^{2}\otimes b^{2}\Big{)}=$
$\displaystyle\Delta_{B}v+\kappa\,v-2\mathcal{H}\Big{(}\,Sv+q+\gamma_{3}Kv\Big{)}+$
(5.1)
$\displaystyle\Big{(}2\,\sigma+T_{0}v+2S(w,Kv)-2\mathcal{H}\,\rho\Big{)}b^{3}$

Then we have to analyze

$\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes
b^{3}=\overline{\nabla}_{b^{3}}\overline{\nabla}_{b^{3}}u-\overline{\nabla}_{X_{3}}u=\overline{\nabla}_{b^{3}}\overline{\nabla}_{b^{3}}u-\nabla_{X_{3}}v-S(X_{3},v)b^{3}\
.$

Let us first set

$\displaystyle T_{1}=$ $\displaystyle
b^{3}(\alpha_{1})\theta_{z}^{1}+b^{3}(\alpha_{2})\theta_{z}^{2}$
$\displaystyle T_{2}=$ $\displaystyle\gamma_{3}\,KX_{3}-T_{1}$ $\displaystyle
E_{0}=$ $\displaystyle\flat X_{3}\otimes\flat X_{3}$ $\displaystyle E_{1}=$
$\displaystyle E_{0}+b^{3}(\gamma_{3})K-\gamma_{3}^{2}g\ .$

######  Lemma 5.3

Let $\overline{\nabla}_{b^{3}}u$ be given in a neighborhood of $M$ by (4.5).
Then $\overline{\nabla}_{b^{3}}\overline{\nabla}_{b^{3}}u$ evaluated on $M$ is
given by

$\displaystyle\overline{\nabla}_{b^{3}}\overline{\nabla}_{b^{3}}u=$
$\displaystyle\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\ b^{3}\otimes
b^{3}+X_{3}(v^{1})\Big{)}b^{1}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{2}\
b^{3}\otimes b^{3}+X_{3}(v^{2})\Big{)}b^{2}+2\gamma_{3}Kq+$ $\displaystyle
2\rho X_{3}+E_{1}v+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{3}\
b^{3}\otimes b^{3}+T_{2}v-2g(X_{3},q)\Big{)}b^{3}\ .$

Proof.  We compute

$\overline{\nabla}_{b^{3}}b^{3}(u^{1})b^{1}=b^{3}(u^{1})\overline{\nabla}_{b^{3}}b^{1}+b^{3}\big{(}b^{3}(u^{1})\big{)}b^{1}=-\gamma_{3}b^{3}(u^{1})b^{2}-\alpha_{1}b^{3}(u^{1})b^{3}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\
b^{3}\otimes b^{3}+X_{3}(v^{1})\Big{)}b^{1}\ .$

Computing similarly with $\overline{\nabla}_{b^{3}}b^{3}(u^{2})b^{2}$ and
adding gives

$\overline{\nabla}_{b^{3}}q=\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\
b^{3}\otimes
b^{3}+X_{3}(v^{1})\Big{)}b^{1}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{2}\
b^{3}\otimes b^{3}+X_{3}(v^{2})\Big{)}b^{2}+\gamma_{3}Kq-g(X_{3},q)b^{3}\ .$

Then $\overline{\nabla}_{b^{3}}u^{3}X_{3}=b^{3}(u^{3})X_{3}=\rho X_{3}+E_{0}v$
because we evaluate on $M$. Next we have

$\displaystyle\overline{\nabla}_{b^{3}}\gamma_{3}(u^{2}b^{1}-u^{1}b^{2})=$
$\displaystyle
b^{3}(\gamma_{3})Kv+\gamma_{3}\big{(}Kq+u^{2}\overline{\nabla}_{b^{3}}b^{1}-u^{1}\overline{\nabla}_{b^{3}}b^{2}\big{)}$
$\displaystyle=$
$\displaystyle\gamma_{3}Kq+b^{3}(\gamma_{3})Kv-\gamma_{3}^{2}v+\gamma_{3}g(KX_{3},v)b^{3}\
.$

Finally we compute

$\displaystyle\overline{\nabla}_{b^{3}}\rho b^{3}=$ $\displaystyle
b^{3}(\rho)b^{3}+\rho X_{3}\quad\mathrm{where}$ (5.2) $\displaystyle
b^{3}(\rho)=$ $\displaystyle\overline{\nabla}\,\overline{\nabla}u^{3}\
b^{3}\otimes b^{3}-b^{3}(\alpha_{1})v^{1}-b^{3}(\alpha_{2})v^{2}-g(X_{3},q)$
$\displaystyle=$ $\displaystyle\overline{\nabla}\,\overline{\nabla}u^{3}\
b^{3}\otimes b^{3}-T_{1}v-g(X_{3},q)\ .$

$\square$

This gives immediately

######  Lemma 5.4

Putting $E_{2}=E_{1}-g(X_{3},w)K$ we have

$\displaystyle\overline{\nabla}\,\overline{\nabla}u\,b^{3}\otimes b^{3}=$
$\displaystyle\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\ b^{3}\otimes
b^{3}\Big{)}b^{1}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{2}\
b^{3}\otimes b^{3}\Big{)}b^{2}+2\gamma_{3}Kq+$ $\displaystyle 2\rho
X_{3}+E_{2}v+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{3}\ b^{3}\otimes
b^{3}+T_{2}v-2g(X_{3},q)-S(X_{3},v)\Big{)}b^{3}\ .$

Proof.  One has to check that

$X_{3}(v^{1})b^{1}+X_{3}(v^{2})b^{2}-\nabla_{X_{3}}v=-g(X_{3},w)Kv\ .$

$\square$

This gives

######  Theorem 5.5

Let us write $\Delta_{B}u=B_{t}+B_{n}b^{3}$ where $B_{t}$ is the tangential
component and let $T=T_{0}+T_{2}$,

$N=2\gamma_{3}K-2\mathcal{H}\,g\quad\mathrm{and}\quad
E=E_{2}-2\mathcal{H}\,\gamma_{3}K\ .$

Then we have

$\displaystyle B_{t}=$
$\displaystyle\Delta_{B}v+\kappa\,v-2\mathcal{H}\,Sv+Ev+2\rho X_{3}+$ (5.3)
$\displaystyle\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\ b^{3}\otimes
b^{3}\Big{)}b^{1}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{2}\
b^{3}\otimes b^{3}\Big{)}b^{2}+Nq$ $\displaystyle B_{n}=$ $\displaystyle
2\,\sigma+2S(w,Kv)-S(X_{3},v)+Tv+\overline{\nabla}\,\overline{\nabla}u^{3}\
b^{3}\otimes b^{3}-2\,g(X_{3},q)-2\mathcal{H}\,\rho\ .$

In the compatible case we have $\rho=0$.

Note that the sign of $S$ and $\mathcal{H}$ depend on the choice of
orientation of $b^{3}$, but the sign of $B_{n}b^{3}$ does not depend on this
choice.

Proof.  We simply combine the formula (5.1) and Lemma 5.4. $\square$

In the divergence free case this yields

######  Theorem 5.6

Let us set $\ell=b^{3}(\gamma_{1})b_{z}^{1}+b^{3}(\gamma_{2})b_{z}^{2}$. If
$\mathsf{div}(u)=0$ in the neighborhood of $M$ we obtain

$\displaystyle B_{t}=$
$\displaystyle\Delta_{B}v+\kappa\,v-2\mathcal{H}\,Sv+Ev-2\,\mathsf{div}(v)X_{3}+$
(5.4) $\displaystyle\Big{(}\overline{\nabla}\,\overline{\nabla}u^{1}\
b^{3}\otimes
b^{3}\Big{)}b^{1}+\Big{(}\overline{\nabla}\,\overline{\nabla}u^{2}\
b^{3}\otimes b^{3}\Big{)}b^{2}+Nq$ $\displaystyle B_{n}=$ $\displaystyle
2\,\sigma+\gamma_{3}\big{(}b_{z}^{2}(v^{1})-b_{z}^{1}(v^{2})\big{)}+2S(w,Kv)+S_{\mathsf{adj}}(X_{3},v)+T_{\mathsf{div}}v$
$\displaystyle-\overline{\nabla}\,\overline{\nabla}u^{1}\,(b^{1}\otimes
b^{3})-\overline{\nabla}\,\overline{\nabla}u^{2}\,(b^{2}\otimes
b^{3})+g(Kw,q)$

where

$T_{\mathsf{div}}=T_{0}+K\big{(}\ell+\gamma_{3}X_{3}\big{)}\ .$

In the compatible case one just substitutes $\mathsf{div}(v)=0$ to $B_{t}$ and
for $B_{n}$ one obtains

$\displaystyle B_{n}=$ $\displaystyle
2\hat{\sigma}+\gamma_{3}\big{(}b^{2}(v^{1})-b^{1}(v^{2})\big{)}+2S(w,Kv)+S_{\mathsf{adj}}(X_{3},v)+T_{\mathsf{div,c}}v$
$\displaystyle-\overline{\nabla}\,\overline{\nabla}u^{1}\,(b^{1}\otimes
b^{3})-\overline{\nabla}\,\overline{\nabla}u^{2}\,(b^{2}\otimes
b^{3})+g(Kw,q)$

where

$\displaystyle\hat{\sigma}=$
$\displaystyle(t_{22}-t_{11})b^{2}(v^{2})+t_{12}\big{(}b^{2}(v^{1})+b^{1}(v^{2})\big{)}\quad\mathrm{and}$
$\displaystyle T_{\mathsf{div,c}}=$ $\displaystyle
T_{0}+K\big{(}\ell+\gamma_{3}X_{3}+2t_{11}w\big{)}\ .$

Proof.  Recall that if $\mathsf{div}(u)=0$ then

$b^{3}(u^{3})=-b^{1}(u^{1})-b^{2}(u^{2})+(t_{11}+t_{22})u^{3}+\alpha_{1}u^{1}+\alpha_{2}u^{2}+\gamma_{2}u^{1}-\gamma_{1}u^{2}\
.$

Hence on $M$ we have $\rho X_{3}=-\mathsf{div}(v)X_{3}$ which gives the
formula for $B_{t}$. Then we have to compute

$\overline{\nabla}\,\overline{\nabla}u^{3}\ b^{3}\otimes
b^{3}=b^{3}(b^{3}(u^{3}))=b^{3}\Big{(}-b^{1}(u^{1})-b^{2}(u^{2})+(t_{11}+t_{22})u^{3}+\alpha_{1}u^{1}+\alpha_{2}u^{2}+\gamma_{2}u^{1}-\gamma_{1}u^{2}\Big{)}\
.$

We compute first

$\displaystyle-b^{3}\big{(}b^{1}(u^{1})+b^{2}(u^{2})\big{)}=$
$\displaystyle-\overline{\nabla}\,\overline{\nabla}u^{1}\,(b^{1}\otimes
b^{3})-\overline{\nabla}\,\overline{\nabla}u^{2}\,(b^{2}\otimes
b^{3})+\gamma_{3}\,\big{(}b_{z}^{2}(v^{1})-b_{z}^{1}(v^{2})\big{)}+g(X_{3},q)$
$\displaystyle b^{3}\big{(}(t_{11}+t_{22})u^{3}\big{)}=$ $\displaystyle
2\mathcal{H}\,b^{3}(u^{3})=2\mathcal{H}\,(\rho+g(X_{3},v))\ .$

Note that $2\mathcal{H}\,\rho$ cancels directly the corresponding term in
(5.3) and moreover

$2\mathcal{H}\,g(X_{3},v)-S(X_{3},v)=S_{\mathsf{adj}}(X_{3},v)\ .$

Then using (5.2) we have

$b^{3}(\alpha_{1}u^{1}+\alpha_{2}u^{2})=g(X_{3},q)+T_{1}v\ .$

Finally

$b^{3}(\gamma_{2}u^{1}-\gamma_{1}u^{2})=g(Kw,q)+g(K\ell,v)\ .$

If $\mathsf{div}(v)=0$ we can solve

$b^{1}(v^{1})=g(Kw,v)-b^{2}(v^{2})$

which gives

$\sigma=(t_{22}-t_{11})b^{2}(v^{2})+t_{12}\big{(}b^{2}(v^{1})+b^{1}(v^{2})\big{)}+t_{11}g(Kw,v)\
.$

$\square$

In [2, Theorem 1.1] $\flat B_{t}$ was computed for the problem (3.2) in the
divergence free and compatible case.

There are several interesting points about the formulas in above Theorems. For
definiteness let us concentrate on Theorem 5.6, but similar remarks of course
apply to formulas in Theorem 5.3. First it is clear that $B_{t}$ can be rather
arbitrary. For example one may view

$\displaystyle\overline{\nabla}\,\overline{\nabla}u^{1}\ b^{3}\otimes
b^{3}-2\mathcal{H}\,b^{3}(u^{1})+2\gamma_{3}b^{3}(u^{2})=h_{1}$
$\displaystyle\overline{\nabla}\,\overline{\nabla}u^{2}\ b^{3}\otimes
b^{3}-2\mathcal{H}\,b^{3}(u^{2})-2\gamma_{3}b^{3}(u^{1})=h_{2}$

as a system of PDE for $u^{1}$ and $u^{2}$ in the neighborhood of $M$ and
solve it, given some $h_{j}$ and some appropriate boundary conditions. Even on
a more elementary level it is clear that choosing extensions appropriately one
can make the terms $\overline{\nabla}\,\overline{\nabla}u^{j}\ b^{3}\otimes
b^{3}$ arbitrarily big compared to $\Delta_{B}v$. So the overall conclusion is
that $\pi(\Delta_{B}u)$ and $\Delta_{B}v$ are unrelated in general. To have a
meaningful relationship the normal derivatives of $u^{j}$ should be ”small”,
in other words only the first lines in the formulas for $B_{t}$ and $B_{n}$ in
(5.4) should be ”big”. In this case we could write

$\displaystyle B_{t}\approx$
$\displaystyle\,\Delta_{B}v+\kappa\,v-2\mathcal{H}\,Sv+Ev-2\mathsf{div}(v)X_{3}$
$\displaystyle B_{n}\approx$
$\displaystyle\,2\,\sigma+\gamma_{3}\big{(}b_{z}^{2}(v^{1})-b_{z}^{1}(v^{2})\big{)}+2S(w,Kv)+S_{\mathsf{adj}}(X_{3},v)+T_{\mathsf{div}}v\
.$

The symmetric Laplacian $Lv=\Delta_{B}v+\kappa\,v$ (for divergence free $v$)
appears in some sense naturally in the formula of $B_{t}$. Note that the
operator $L$ is intrinsically defined. The tensor $2\mathcal{H}\,S$ depends on
the embedding, but this is perfectly fine in the present context; the
important thing is that it does not depend on the chosen frame. The presence
of mean curvature in the formulas is intriguing. It suggests that minimal
surfaces are somehow special here, but it is not clear how this should or
could be interpreted from the point of view of physical model.

However, unfortunately the tensor $E$ does depend on the chosen frame. Let us
simply illustrate the difficulties without resolving them. Let us choose $a=1$
in (3.2) and compare our computations to the ones in [1]. In this case we have
$\kappa=1$ and $X_{3}=\gamma_{3}=0$. Choosing further that $u^{3}$ is
identically zero and $u^{1}$ and $u^{2}$ do not depend on $y_{3}$ one obtains

$\pi(\Delta_{B}u)=\Delta_{B}v+\kappa
v-2\mathcal{H}\,Sv=\Delta_{B}v-v=\Delta_{H}v\ .$

In [1, p. 344] the authors obtained using a similar extension
$\hat{u}=u^{1}\partial_{y_{1}}+u^{2}\partial_{y_{2}}$ that

$\pi(\Delta_{B}\hat{u})=\Delta_{B}\hat{v}+\hat{v}=L\hat{v}\ .$

The authors considered the divergence free case, but one obtains the same
result also in general, because the extensions considered here are in any case
compatible. The above results are not contradictory because in our case
$u=u^{1}b^{1}+u^{2}b^{2}$ and $b^{j}$ do depend on $y_{3}$ although $u^{j}$ do
not. Also the frame
$\\{\partial_{y_{1}},\partial_{y_{2}},\partial_{y_{3}}\\}$, used in [1, 2], is
not orthonormal. However, this illustrates clearly how seemingly minor
differences in the extension or the frame can have dramatic consequences for
the restriction even in this simple case.

Let us then give an example how the tensor $E$ depends on the frame even in
the case that the frames are orthonormal. Writing $E$ explicitly gives

$E=\flat X_{3}\otimes\flat
X_{3}-\gamma_{3}^{2}g+\big{(}b^{3}(\gamma_{3})-2\mathcal{H}\,\gamma_{3}-g(X_{3},w)\big{)}K\
.$

Note that $w$ is in fact well-defined: one way to see this is to observe that
$w=[b^{1}_{z},b^{2}_{z}]$. The problem is with $\gamma_{3}$ and $X_{3}$. Let
us then for the final time consider the problem (3.2). With the frame given in
(3.3) we have $\gamma_{3}=g(X_{3},w)=0$ and

$E=\flat X_{3}\otimes\flat
X_{3}=\frac{(1-a^{2})^{2}\sin^{2}(2z_{2})}{4\lambda^{6}}\,\theta_{z}^{2}\otimes\theta_{z}^{2}\
.$

Let us then choose the following frame:

$\displaystyle\tilde{b}^{1}=$
$\displaystyle\frac{1}{\sqrt{\mu_{0}}}\,\partial_{y_{1}}+\frac{y_{3}-1}{\sqrt{\mu_{0}}}\,\partial_{y_{3}}$
$\displaystyle\tilde{b}^{2}=$
$\displaystyle\frac{(1-a^{2})(y_{3}-1)\sin(2y_{2})}{2a\sqrt{\mu_{0}\mu_{1}}}\,\partial_{y_{1}}+\frac{\sqrt{\mu_{0}}}{ay_{3}\sqrt{\mu_{1}}}\,\partial_{y_{2}}+\frac{(1-a^{2})(y_{3}-1)^{2}\sin(2y_{2})}{2a\sqrt{\mu_{0}\mu_{1}}}\,\partial_{y_{3}}$
$\displaystyle\tilde{b}^{3}=$
$\displaystyle-\frac{y_{3}-1}{ay_{3}\sin(y_{2})\sqrt{\mu_{1}}}\,\partial_{y_{1}}+\frac{(1-a^{2})\sin(2y_{2})\sin(y_{2})}{2a\sqrt{\mu_{1}}}\,\partial_{y_{2}}+\frac{y_{3}\lambda^{2}\sin(y_{2})}{a\sqrt{\mu_{1}}}\,\partial_{y_{3}}$
$\displaystyle\mu_{0}=$
$\displaystyle(y_{3}-1)^{2}\hat{\lambda}^{2}+a^{2}y_{3}^{2}\sin^{2}(y_{2})$
$\displaystyle\mu_{1}=$
$\displaystyle(y_{3}-1)^{2}+y_{3}^{2}\lambda^{2}\sin^{2}(y_{2})$

Note that evaluated on $M$ we have $\tilde{b}^{j}=b^{j}$ so that $b^{j}_{z}$
and hence $\theta^{j}_{z}$ are the same as before. However, when we simply
compute we obtain that

$\gamma_{3}=\frac{(1-a^{2})\cos(z_{2})}{a^{2}}\quad\mathrm{and}\quad\alpha_{1}\gamma_{1}+\alpha_{2}\gamma_{2}=g(X_{3},w)=-\frac{\cos(z_{2})}{a\lambda\sin^{2}(z_{2})}$

when we evaluate on $M$ and further

$\displaystyle E=$
$\displaystyle\frac{\lambda^{4}-(1+a^{2})\lambda^{2}+2a^{2}}{a^{4}\sin^{2}(z_{2})}\,\theta_{z}^{1}\otimes\theta_{z}^{1}+\frac{(a^{2}-1)\cos(z_{2})(3\lambda^{2}-a^{2})}{a^{3}\lambda}\,\theta_{z}^{1}\otimes\theta_{z}^{2}+$
$\displaystyle\frac{(a^{2}-1)\cos(z_{2})(2a^{2}+\lambda^{2}a^{2}-3\lambda^{4})}{a^{3}\lambda^{3}}\,\theta_{z}^{2}\otimes\theta_{z}^{1}+\frac{(\lambda^{2}-1)\big{(}(1-a^{2})\lambda^{6}-a^{4}\lambda^{2}+a^{6}\big{)}}{a^{4}\lambda^{6}}\,\theta_{z}^{2}\otimes\theta_{z}^{2}\
.$

Similarly the normal component $B_{n}b^{3}$ depends on the chosen frame.
Incidentally it seems that the properties of the normal component has not been
analyzed previously. One would think that the normal component is also
important when assessing the relationship between the three dimensional flow
and its restriction.

In conclusion we always have

$\pi\big{(}\Delta_{B}u\big{)}=\Delta_{B}v+\kappa v-2\mathcal{H}\,Sv+\
\mathrm{other\ terms}$

where the other terms depend on the extension of $v$ and the frame. While it
remains unclear how this frame dependence might be resolved, Theorems 5.3 and
5.6 provide general and convenient formulas for exploring this problem
further.

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# Existence of weak solutions for the kinetic models of motion of myxobacteria
with alignment and reversals

Patrick Murphy  Department of Mathematics and Statistics
San Jose State University  ,  Oleg Igoshin  Department of Bioengineering
Rice University  ,  Misha Perepelitsa  Department of Mathematics
University of Houston<EMAIL_ADDRESS>and  Ilya Timofeyev  Department
of Mathematics
University of Houston

###### Abstract.

In this paper, we consider three non-linear kinetic partial differential
equations that emerge in the modeling of motion of rod-shaped cells such as
myxobacteria. This motion is characterized by nematic alignment with
neighboring cells, orientation reversals from cell polarity switching,
orientation diffusion, and transport driven by chemotaxis. Our primary
contribution lies in establishing the existence of weak solutions for these
equations. Our analytical approach is based on the application of the
classical averaging lemma from the kinetic theory, augmented by a novel
version where the transport operator is substituted with a uni-directional
diffusion operator.

##  1\. Introduction

Many rod-shaped bacteria are known to move on surfaces and self-organize into
various spatial patterns due to mechanical interactions and biochemical
signaling between cells. For instance, myxobacteria - bacteria studied for
their multicellular interactions can form dynamic patterns of clusters,
streams, swirls, and aggregates. Motivated by the problem of modeling the
motion of a single layer of myxobacteria on a flat surface, we consider a
mean-filed equation for $f(x,\theta,t)$-the probability distribution functions
for bacteria position $x\in\mathbb{R}^{2},$ and orientation angle of bacteria
longer axis, $\theta\in(-\pi,\pi]:$

(1.1) $\partial_{t}f{}+{}e(\theta)\cdot\nabla
f{}=-{}\partial_{\theta}(f\Psi(v_{f})){}+{}\partial^{2}_{\theta\theta}f-f+\tau_{\pi}f,$

where $e(\theta)=(\cos\theta,\sin\theta),$ as bacteria primary move along
their long axis.

(1.2)
$v_{f}(x,t){}={}\int_{-\pi}^{\pi}f(x,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1},$

$\Psi$ is Lipschitz continuous and bounded, and $\tau_{\pi}$ is the
translation operator in $\theta$ variable:

$\tau_{\pi}f(x,\theta,t){}={}f(x,\theta+\pi,t),$

The right-hand side of equation (1.1) accounts for changes in the angle
(passive part) that combine nematic alignment, orientation diffusion, and cell
reversals to the opposite direction. The left-hand side represents the active
part: transport in the direction of orientation of the cell’s longer axis. The
model is based on the kinetic model of Peruani et al. [19] for the motion of
self-propelled rods with nematic alignment described by Maier-Saupe
interaction potential from the theory of liquid crystals.

Alongside equation (1.1) we consider (i) a kinetic model obtained in the
asymptotic “fast reversal” regime (diffusion limit) and (ii) a model where the
reversals are biased by a concentration of local field $s$, e.g. chemotactic
signal. In the first case, the kinetic equation is

(1.3) $\partial_{t}f{}-{}\,\text{div}\,\left(e(\theta)\otimes e(\theta)\nabla
f\right){}=-{}\partial_{\theta}(f\Psi(v_{f})){}+{}\partial^{2}_{\theta\theta}f.$

In this equation, the “active” part is the spacial diffusion, with the
diffusion matrix $e(\theta)\otimes e(\theta),$ which results in diffusion only
in the direction of $e(\theta)$ In this model, the kinetic function $f$ is
nematically symmetric: $\tau_{\pi}f{}={}f.$

In the second case, with $s(x,t)$ denoting the concentration of a
chemoattractant, the equation is given by

(1.4) $\partial_{t}f+2\,\text{div}\,([e(\theta)\cdot\nabla
s]e(\theta)f){}-{}\,\text{div}\,(e(\theta)\otimes e(\theta)\nabla
f){}=-{}\partial_{\theta}(f\Psi(v_{f})){}+{}\partial^{2}_{\theta\theta}f.$

As in the classical Keller-Segel model, chemotaxis is expressed as a transport
term to the “active” part of the kinetic equation.

In this paper, we focus on proving the global (in time) existence of weak
solutions to Cauchy problem for equations (1.1), (1.3), and (1.4) (with given
$s(x,t)$), under the assumption that the rate of alignment $\Psi(v_{f})$ is a
bounded function.

The equations listed above have unique classical solutions locally in time for
any $\Psi\in C^{1}(\mathbb{R}).$ This can be proved by, for example, by a
suitable operator-splitting method. It is, however, unknown if and how the
classical solutions blow up in finite time. Such analysis is complicated by
the fact that the only known Lyapunov functional for these equations is the
total mass. On the other hand, when the rate of alignment is bounded, the
orientational diffusion is sufficient to control the alignment term and there
is a priori estimate of solutions in $L^{2}$ norm (equation (2.17)). This is
the case we consider in this paper and the weak solutions we construct have
the regularity ascertained by such “energy” estimate.

The main difficulty of constructing weak solutions is the nonlinearity of the
alignment in which the interaction operator $v_{f}$ is localized to a point.
This situation arises in an asymptotic regime of the vanishing ratio of cell
length $l$ to a macroscopic scale $L$ (domain size) because the orientational
alignment occurs when the cells collide, and thus its range is proportional to
$l,$ see section 2.1. This property makes the equations similar to the
Boltzmann and Bhatnagar-Gross-Krook (BGK) equations of gas dynamics, where the
interaction operators are also localized. Consequently, we look for the
analytical approaches developed for such equations and try to apply them here.
The key tool in our analysis is the theory kinetic averaging (DiPerna and
Lions [8], Perthame [18], Gerard [11], Golse et al[12], Bouchut et al [4]).

Proving the existence of weak solutions requires showing that one can pass to
the limit in the non-linear term $f_{\epsilon}\Psi(v_{f_{\epsilon}})$ on a
sequence of suitable approximate solutions $f_{\epsilon}.$ Using a kinetic
averaging lemma of Gerard [11] we are able to show the strong compactness of
moments $v_{f_{\epsilon}}$ and with that the existence of weak solutions to
(1.1). The situation is slightly different with equations (1.3) and (1.4)
because the kinetic density propagates in space by a diffusion process.
However, since the direction of the diffusion $e(\theta)$ varies, we still
expect improved regularity for $\theta$-moments of $f$ compared to the
regularity of $f.$ This we prove in lemma 1 by showing that moments belong to
a fractional Sobolev space $H^{\frac{2}{7},\frac{1}{7}}_{x,t},$ locally in
$(x,t)\in\mathbb{R}^{2}\times\mathbb{R}_{+}.$ Such regularity implies the
strong compactness of moments and the existence of weak solutions.

Kinetic models for the motion of myxobacteria, or more generally the models of
self-propelled rods with alignment, have been considered in the framework of
mean-field type interactions [2, 1, 6, 7, 10] and in the Boltzmann framework
[3, 13, 17, 16]. Most works deal with the formal derivation of macroscopic
equations in various asymptotic regimes, such as singular relaxation limits.
Resulting equations are typically difficult to treat analytically. For
example, in some models, the macroscopic equations turn out to be hyperbolic
systems in non-conservative form, [6, 13, 16]. In these works, the
mathematical analysis has been mostly restricted to space homogeneous
equations, addressing questions such as the existence of solutions, phase
transition, and convergence to a stationary solution in the long run.

Finally, we mention that the interaction operators similar to (1.2) appear in
Smoluchowski equation (Doi [9]) from the theory of liquid crystals and in the
Kuramoto model of interacting oscillators (Kuramoto [14]).

##  2\. Models of cell motion with nematic alignment and reversals

###  2.1. Model I: liquid crystal-type alignment and reversals

We start with the agent-based model of the motion of self-propelled rods (rod-
shape cells) that alignment describes as in Peruani et al. [19]. The alignment
part of the model describes the change of orientation of rods according to
Maier-Saupe of alignment of liquid crystals. The model is described by the
equations for positions $X_{i}$ and orientations $\Theta_{i}$ of N cells:

(2.1) $\displaystyle\frac{dX_{i}}{dt}{}$
$\displaystyle={}v_{0}e(\theta_{i}),\quad
e(\theta)=(\cos\theta,\sin\theta),\;v_{0}>0,$ (2.2)
$\displaystyle\frac{d\Theta_{i}}{dt}{}$
$\displaystyle={}\gamma\sum_{j\,:\,|X_{j}-X_{i}|<r}\sin(2(\Theta_{j}-\Theta_{i})){}+{}\eta_{i}(t),$

$i=1\dots,N.$ Here, $\gamma>0,$ and $\eta_{i}(t)$ are stochastic processes
such that: (i) $\eta_{i}(t)$ is uniformly distributed on interval $[-s,s],$
where $s=\sqrt{3}\sigma,$ with $\sigma>0;$ (ii) for all $i,j$ and
$t,t^{\prime},$

$\langle\eta_{i}(t)\rangle{}={}0,\quad\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\sigma^{2}\delta_{i,j}\delta_{t,t^{\prime}}.$

For myxobacteria, which are known to periodically change their polarity and
move in the opposite direction, We add cell reversals to the model, assuming
that in addition to the continuous variation according to the above equations,
each $\Theta_{i}$ jumps to value $\Theta_{i}+\pi,$ with the timing of changes
following independent Poisson’s processes with rate $\lambda.$

The summation term in the rate of alignment in (2.2) has a natural bound based
on the fact that cells occupy a finite space and cells (typically) do not
overlap. To account for this, we introduce the following modification. Let $l$
and $w$ denote cells typical length and width, then

$|\sum_{j\,:\,|X_{j}-X_{i}|<r}\sin(2(\Theta_{j}-\Theta_{i}))|{}\leq{}\frac{\pi
r^{2}}{lw},$

This inequality expresses the fact that the left-hand side can not be larger
than the number of cells inside a ball of radius $r$ when cells do not
overlap. This is a common experimental condition. We, therefore, introduce a
cutoff function

$\llbracket r\rrbracket_{a}=\left\\{\begin{array}[]{rr}-a,&r<-a,\\\ r,&|r|\leq
a,\\\ a,&r>a.\end{array}\right.$

and use it in the ODE for $\Theta_{i}$ to enforce the bound on the speed of
alignment:

(2.3)
$\frac{d\Theta_{i}}{dt}{}={}\gamma\left\llbracket\sum_{j\,:\,|X_{j}-X_{i}|<r}\sin(2(\Theta_{j}-\Theta_{i}))\right\rrbracket_{\frac{\pi
r^{2}}{lw}}{}+{}\eta_{i}(t).$

Alternatively, the rate of alignment can be written using the mean nematic
current at position $X_{i}:$

$\mathbf{J}_{i}=\sum_{j\,:\,|X_{j}-X_{i}|<r}e^{2\mbox{\rm i}\Theta_{j}}$

and the mean nematic director $\overline{\Theta}_{i}\in(-\pi/2,\pi/2]$ which
is defined when $\mathbf{J}_{i}\not=0$ as

$\overline{\Theta}_{i}{}={}\frac{1}{2}\mbox{\rm
Arg}\frac{\mathbf{J}_{i}}{|\mathbf{J}_{i}|}.$

In this way,

$\sum_{j\,:\,|X_{j}-X_{i}|<r}\sin(2(\Theta_{j}-\Theta_{i})){}={}|\mathbf{J}_{i}|\sin(2(\overline{\Theta}_{i}-\Theta_{i})).$

Thus, a different way to limit the rate of alignment is to set it to be equal
to

(2.4) $\gamma\left\llbracket|\mathbf{J}_{i}|\right\rrbracket_{\frac{\pi
r^{2}}{lw}}\sin(2(\overline{\Theta}_{i}-\Theta_{i})).$

In this paper, we will use the expression for the nematic alignment in the
form appearing in (2.3), but all our results also apply to the model based on
(2.4).

Denoting by $F(x,\theta,t)$ the distribution function for the number of cells
in $(x,\theta)$ space, one obtains a kinetic equation for evolution of $F,$

(2.5) $\partial_{t}F{}+{}v_{0}e(\theta)\cdot\nabla
F{}+{}\gamma\partial_{\theta}(F\left\llbracket
V^{r}_{F}\right\rrbracket_{\frac{\pi
r^{2}}{lw}}){}-{}\mu_{a}\partial^{2}_{\theta\theta}F{}={}-\lambda
F+\lambda\tau_{\pi}F,$

where

$V^{r}_{F}(x,\theta,t){}={}\int_{|y-x|<r}\int_{\mathbb{T}}F(y,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}dy,\quad\mathbb{T}=(-\pi,\pi),$

and $\mu_{a}{}={}\frac{\sigma^{2}}{2}.$ Equation (2.5) (without reversals and
limited alignment rate) was derived in [19]. We rescale variables with some
length and time scales $L$ and $T:$

$x=L\overline{x},\quad t=T\overline{t},\quad
F(x,\theta,t)=NL^{-2}\overline{f}(\overline{x},\theta,\overline{t}),$

where $\overline{f}$ is a probability density function. We write equation
(2.5) in new non-dimensional variables and drop bars for brevity:

(2.6) $\partial_{t}f{}+{}\frac{v_{0}T}{L}e(\theta)\cdot\nabla
f{}+{}(T\gamma)(\frac{N\pi r^{2}}{L^{2}})\partial_{\theta}(f\llbracket
V^{\frac{r}{l}}_{f}\rrbracket_{\frac{L^{2}}{Nlw}}){}-{}T\mu_{a}\partial^{2}_{\theta\theta}f{}={}-T\lambda
f+T\lambda\tau_{\pi}f,$

with

$V_{f}^{\frac{r}{L}}(x,\theta,t){}={}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{|y-x|<\frac{r}{L}}\int_{\mathbb{T}}f(y,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}dy,$

In the first asymptotic regime that we consider in this paper, we assume that
$r/L=\epsilon\ll 1,$ while considering all other coefficients to be of the
finite order. In particular, this means that $N$ is large but $Nr^{2}L^{-2}$
is finite. For the bacterial motion, one typically has $r\sim l,$ and $w\sim
0.1l,$ see [20]. With these relations, the coefficient in the cutoff function
$V_{f},$ $L^{2}(Nlw)^{-1}$ is of the finite order.

For notational simplicity, we set all finite order coefficients to be equal 1
and consider the equation

(2.7) $\partial_{t}f{}+{}e(\theta)\cdot\nabla
f{}+{}\partial_{\theta}(f\llbracket
V_{f}^{\epsilon}\rrbracket_{1}){}-{}\partial^{2}_{\theta\theta}f{}={}-f+\tau_{\pi}f,$
(2.8) $V_{f}^{\epsilon}(x,\theta,t){}={}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{|y-x|<\epsilon}\int_{\mathbb{T}}f(y,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}dy.$

The formal $\epsilon\to 0$ limit of this equation is the equation with the
interactions localized to a point:

(2.9)
$V_{f}^{0}{}={}\int_{\mathbb{T}}f(x,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1},$

We will use notation $v_{f}$ for $V_{f}^{0}.$

###  2.2. Model II: macroscopic diffusion limit

Now we return to PDE (2.6) and consider a different asymptotic regime. Let
$\tau=\frac{1}{\lambda}$ be the meantime traveled by a cell between reversals.
For the motion of myxobacteria cell, one typically has

$l\sim v_{0}\tau,$

see for example Cotter et al. [5] and Murphy et al. [15]. Thus, the
macroscopic limit $l/L=\epsilon\ll 1$ can be expressed as
$v_{0}\tau/L\sim\epsilon\ll 1.$ As in the previous model, we assume

$r\sim l,\quad\frac{Nlw}{L^{2}}\sim 1.$

Microscopic reversal-type motion, at the macroscopic level, is described as
diffusion along the lines parallel to the cell orientation $e(\theta),$ with
the diffusion coefficient

$d{}={}\frac{(\tau v_{0})^{2}}{\tau}{}={}\tau v_{0}^{2}.$

To observe this diffusion at the macro-level we select time scale $T$ it must
be selected such that

$\frac{L^{2}}{T}{}={}d.$

This implies that

$\frac{T}{\tau}{}\sim{}\frac{1}{\epsilon^{2}}.$

Concerning the alignment and orientational rates $\gamma$ and $\mu,$ we assume
that

$\gamma T\sim 1,\quad\mu T\sim 1.$

In other words, we are discussing a regime with fast reversals and slow
alignment and orientation diffusion. Using the above relations in (2.6) we
arrive at the equation

(2.10) $\partial_{t}f+\frac{1}{\epsilon}e(\theta)\cdot
f{}+{}\partial_{\theta}(f\llbracket
V_{f}^{\epsilon}\rrbracket_{1})-\partial_{\theta\theta}^{2}f{}={}\frac{1}{\epsilon^{2}}\left(-f(x,\theta,t)+f(x,\theta+\pi,t)\right),$

with $V_{f}^{\epsilon}$ as in (2.8). We will also need equation for
$f(x,\theta,t)+f(x,\theta+\pi,t):$

(2.11)
$\partial_{t}(f(\theta)+f(\theta+\pi)){}+{}\frac{1}{\epsilon}e(\theta)\cdot\nabla(f(\theta)-f(\theta+\pi)){}+{}\partial_{\theta}((f(\theta)+f(\theta+\pi))\llbracket
V_{f}^{\epsilon}\rrbracket_{1})\\\
{}-{}\partial_{\theta\theta}^{2}(f(\theta)+f(\theta+\pi)){}={}0.$

Consider now the singular limit of $\epsilon\to 0.$ Writing $f=f_{0}+\epsilon
f_{1}+o(\epsilon)$ we find from (2.10) that
$f_{0}(x,\theta+\pi,t)=f_{0}(x,\theta,t)$ and

$f_{1}(x,\theta+\pi,t)-f_{1}(x,\theta,t)=e(\theta)\cdot\nabla f_{0}.$

Using these relations in (2.11), we find the leading order equation for
$f_{0}:$

(2.12) $\partial_{t}f{}-{}\,\text{div}\,\left(e(\theta)\otimes e(\theta)\nabla
f\right){}+{}\partial_{\theta}(f\llbracket
v_{f}\rrbracket_{1}){}-{}\partial^{2}_{\theta\theta}f{}={}0.$

###  2.3. Model III: uni-directional chemotaxis

A model in which reversals are biased by a chemical signal can be described by
setting the rate of reversals $\lambda$ to depend on the chemoattractant
concentration $S(x,t),$ for example, as in the following equation

$\lambda=\lambda_{0}-\beta\Phi(l^{3}_{ch}e(\theta)\cdot\nabla S),$

where $\Phi$ is increasing function with $\Phi(0)=0,$ $\Phi^{\prime}(0)=1,$
and $\sup_{r}|\Phi(r)|<\frac{\lambda_{0}}{\beta}.$ Here $\beta>0$ is the
chemotaxis intensity (rate), and $l_{ch}$ is the characteristic length scale
of signal sensing (distance over which the changes in $S$ are sensed by a
cell). $\lambda_{0}$ is the unbiased reversal rate.

Switching to the non-dimensional variables

(2.13) $S(x,t)=NL^{-2}s(x/L,t/T),$

and

$\lambda T=\lambda_{0}T-\beta T\Phi(\frac{Nl_{ch}^{3}}{L^{3}}e(\theta)\cdot
s).$

For the asymptotic regime, in addition to assumptions made in the previous
model, we postulate

$\beta T\sim\lambda_{0}T\sim\frac{1}{\epsilon^{2}},\quad l_{ch}\sim l.$

The last assumption implies that

$\frac{Nl_{ch}^{3}}{L^{3}}{}\sim{}\frac{Nl^{2}}{L^{2}}\frac{l}{L}\sim\epsilon.$

Setting all finite order coefficients to 1, for simplicity, we obtain the
expression for the non-dimensional reversal rate

$T\lambda=\frac{1}{\epsilon^{2}}{}-{}\frac{1}{\epsilon}e(\theta)\cdot\nabla
s+O(1).$

Retaining only the first two terms, we arrive at the equation

$\displaystyle\partial_{t}f+\frac{1}{\epsilon}e(\theta)\cdot\nabla
f+\partial_{\theta}(f\llbracket
V_{f}^{\epsilon}\rrbracket_{1})-\partial^{2}_{\theta\theta}f{}={}$
$\displaystyle-\frac{1}{\epsilon^{2}}(1-\epsilon e(\theta)\cdot\nabla
s)f(\theta)$ $\displaystyle{}+{}\frac{1}{\epsilon^{2}}(1+\epsilon
e(\theta)\cdot\nabla s)f(\theta+\pi).$

The formal asymptotic limit of $\epsilon\to 0$ leads to the following equation
for $\pi$-periodic in $\theta$ function $f.$

(2.14) $\displaystyle\partial_{t}f+\,\text{div}\,(2[e(\theta)\nabla
s]e(\theta)f){}-{}\,\text{div}\,(e(\theta)\otimes e(\theta)\nabla
f){}+{}\partial_{\theta}(f\llbracket
v_{f}\rrbracket_{1}){}-{}\partial^{2}_{\theta\theta}f{}={}0.$

Here, $v_{f}$ as in (2.9).

Finally, we mention a coupled system of chemotaxis, when the chemoattractant
is secreted by cells and spreads by diffusion

(2.15) $\partial_{t}S-\mu_{S}\Delta S=\alpha n-\frac{1}{\tau_{S}}S,\quad
n(x,t)=\int_{\mathbb{T}}F(x,\theta,t)\,d\theta,$

where $\mu_{S},$ $\alpha$, and $\tau_{S}$ stay for the chemoattractant
diffusion coefficient, the rate of production, and the characteristic decay
time (relaxation time). Assuming that the coefficients verify relations

$\alpha T\sim 1,\quad\frac{T}{\tau_{S}}\sim 1,\quad\frac{T\mu_{S}}{L^{2}}\sim
1,$

the non-dimensional concentration $s,$ defined in (2.13), verifies equation

(2.16) $\partial_{t}s-\Delta s=n-s,\quad
n(x,t)=\int_{\mathbb{T}}f(x,\theta,t)\,d\theta.$

We postpone to the future the analysis of the coupled system (2.14), (2.16).

###  2.4. Existence theory: model I

In our first theorem, we prove the existence of a weak solution to the
initial-value problem for equation (2.7), (2.8) and then we pass to the limit
of $\epsilon\to 0$ to obtain solutions with alignment rate (2.9).

All equations are considered on domain
$\mathbb{R}^{2}\times\mathbb{R}\times(0,+\infty),$ with solutions $f$ being
periodic over $\Pi=(0,1)\times(0,1)$ in spacial variable $x$ and over
$\mathbb{T}{}={}(-\pi,\pi)$ in $\theta$ variable. Initial conditions are given
as $f(x,\theta,0)=f_{0}(x,\theta).$ We will use notation $L^{2}_{x,\theta}$
for the space $L^{2}(\Pi\times\mathbb{T}),$ with the standard inner product
$\langle\cdot,\cdot\rangle_{L^{2}_{x,\theta}},$ and $H^{1}_{x,\theta}$ for
$H^{1}(\Pi\times\mathbb{T}).$ Let $Q_{T}=\Pi\times\mathbb{T}\times(0,T).$

We will work with weak solutions that verify the inequality

(2.17)
$\sup_{[0,T]}\|f(\cdot,t)\|^{2}_{L^{2}_{x,\theta}}{}+{}\|\partial_{\theta}f\|^{2}_{L^{2}(0,T;L^{2}_{x,\theta})}{}+{}\|f(\theta)-f(\theta+\pi)\|^{2}_{L^{2}(0,T;L^{2}_{x,\theta})}{}\leq{}2e^{T}\|f_{0}\|^{2}_{L^{2}_{x,\theta}},$

obtained by integrating the equation against solution $f.$

######  Theorem 1:

Let $f_{0}\in L^{2}_{x,\theta}\cap L^{1}_{x,\theta},$ periodic in
$(x,\theta),$ $f_{0}\geq 0$ a.e. and have unit mass. There is a unique, global
in time, periodic, weak solution of (2.7). The solution is non-negative, has
unit mass for all $t>0,$ and such that for any $T>0,$

(2.18) $\displaystyle f{}\in{}L^{\infty}(0,T;L^{2}_{x,t})\cap
C([0,T];L^{2}_{x,\theta}-weak)$ (2.19) $\displaystyle\partial_{\theta}f\in
L^{2}(0,T;L^{2}_{x,\theta})$ (2.20) $\displaystyle\partial_{t}\langle
f,w\rangle_{L^{2}_{x,\theta}}\in L^{2}(0,T),\quad\forall w\in
H^{1}_{x,\theta},$

and (2.17) holds.

###### Proof.

The weak solution can be constructed by a parabolic approximation, by adding
$x$-diffusion $\delta\Delta_{x}f$ to the equation and passing to the limit
$\delta\to 0.$ A theory of linear parabolic equations and a fixed point
argument can be used to obtain such a solution (we omit the details).
Solutions, call them $f_{\delta},$ verify estimate (2.17) and thus, have a
subsequence $f_{\delta_{k}}$ that converges to some $f$ – the element of
spaces listed in (2.18)–(2.20), in the following topologies:

(2.21) $\displaystyle f_{\delta_{k}}\to f\quad\mbox{\rm weakly
in}\;L^{2}(0,T;L^{2}_{x,\theta})$ (2.22)
$\displaystyle\partial_{\theta}f_{\delta_{k}}\to\partial_{\theta}f\quad\mbox{\rm
weakly in}\;L^{2}(0,T;L^{2}_{x,\theta})$ (2.23) $\displaystyle\langle
f_{\delta_{k}},w\rangle_{L^{2}_{x,\theta}}\to\langle
f,w\rangle_{L^{2}_{x,\theta}}\quad\mbox{\rm in}\;C([0,T]),\quad\forall w\in
L^{2}_{x,\theta}$

The only nonlinear term in the equation (2.7) is operator

(2.24) $A(f)=\partial_{\theta}(f\llbracket
V_{f}^{\epsilon}\rrbracket_{1})=\partial_{\theta}f\llbracket
V_{f}^{\epsilon}\rrbracket_{1}{}-{}2f\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{|y-x|<\epsilon}\int_{\mathbb{T}}f(y,\theta_{1},t)\cos(2(\theta_{1}-\theta))\,d\theta_{1}dy\chi_{\\{|V_{f}^{\epsilon}(x,\theta,t)|\leq
1\\}}.$

Given the above compactness properties of $f_{\delta_{k}},$
$V^{\epsilon}_{f_{\delta_{k}}}$ and the other integral term in (2.24) converge
uniformly on compact subset in $(x,\theta,t)$ domain, and consequently,

$A(f_{\delta_{k}})\to A(f),\quad\mbox{\rm weakly in}\quad
L^{2}(0,T;L^{2}_{x,\theta}).$

For the uniqueness, given $f_{1}$ and $f_{2}$ in the regularity class of the
theorem, we can estimate for any $t>0,$

$\|A(f_{1}(t))-A(f_{2}(t))\|_{L^{2}_{x,\theta}}{}\leq
C\|\partial_{\theta}f_{1}(t)-\partial_{\theta}f_{2}(t)\|_{L^{2}_{x,\theta}}{}+{}C_{\epsilon}\left(\|f_{1}(t)\|_{L^{2}_{x,\theta}}{}+{}\|\partial_{\theta}f_{1}(t)\|_{L^{2}_{x,\theta}}\right)\|f_{1}(t)-f_{2}(t)\|_{L^{2}_{x,\theta}},$

for some $C>0,$ and $C_{\epsilon}>0.$ The uniqueness then follows from the
“energy” inequality for $f_{1}-f_{2}$ and the Gronwall’s lemma. ∎

######  Theorem 2:

Let $f_{\epsilon}$ be the family solutions (2.7) from theorem 1 corresponding
to initial data $f_{0},$ as in theorem 1. The set of moments

$\left\\{\int_{\mathbb{T}}f_{\epsilon}\phi(\theta)\,d\theta\right\\}\quad\mbox{\rm
is pre-compact in
}L^{2}_{loc}(\mathbb{R}^{2}\times(0,+\infty)),\quad\forall\phi\in
C^{1}(\overline{\mathbb{T}}).$

On a suitable subsequence $\epsilon_{n}\to 0$, $f_{\epsilon_{n}}$ converges to
a weak solution (2.7), (2.9), $f,$ with properties (2.18)–(2.20).

###### Proof.

We start with weak solutions $f_{\epsilon}$ of (2.7) that verify the energy
inequality (2.17). Given the energy estimates, we only need to establish the
strong compactness property of moments $V_{f_{\epsilon}}^{\epsilon}$, for a
solution to exist. For that, we use a version of the result of Gerard [11]
from Bouchut et al [4], Theorem 1.5. We write (2.7) as a transport equation

$\partial_{t}f_{\epsilon}{}+{}e(\theta)\cdot\nabla
f_{\epsilon}{}={}-f_{\epsilon}(\theta)+f_{\epsilon}(\theta+\pi)-\partial_{\theta}(f\llbracket
V^{\epsilon}_{f_{\epsilon}}\rrbracket_{1})+\partial^{2}_{\theta\theta}f_{\epsilon}:=g_{\epsilon}.$

For $\phi\in C^{1}([-\pi,\pi]),$ one has
$\int_{-\pi}^{\pi}g_{\epsilon}\phi\,d\theta$ bounded in
$L^{2}(0,T;L^{2}(\mathbb{R}^{2}))$ uniformly in $\epsilon.$ This implies that

$\left\\{\int_{\mathbb{T}}g_{\epsilon}\phi\,d\theta\right\\}\quad\mbox{\rm is
pre-compact in }\;H^{-1}_{loc}(\mathbb{R}^{2}\times\mathbb{R}_{+}).$

Moreover, $f_{\epsilon}$ are bounded in $L^{2}(0,T;L^{2}_{x,v})$ for any
$T>0,$ and the transport velocity verifies the property

$\mbox{\rm
meas}\left\\{\theta\,:\,e(\theta)\cdot\xi{}={}u\right\\}{}={}0,\quad\forall\xi\in\mathbb{R}^{2},\,|\xi|=1,\;\forall
u\in\mathbb{R}.$

These properties imply the compactness of the moments of $f_{\epsilon}$ stated
in the theorem.

Based on that, there a sequence $\epsilon_{n}$ and $f$ such that for any
$T>0,$ $f_{\epsilon_{n}}\to f$ in the weak topology of
$L^{2}(0,T;L^{2}_{x,v}),$ and

$\int_{\mathbb{T}}f_{\epsilon_{n}}(x,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}\to\int_{\mathbb{T}}f(x,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}$

a.e. $(x,\theta,t)$ and in $L^{2}_{loc}(Q_{T}).$ A subsequence can be found
such that,

$V^{\epsilon_{n}}_{f_{\epsilon_{n}}}{}={}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{|y-x|<\epsilon_{n}}\int_{\mathbb{T}}f_{\epsilon_{n}}(y,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}dy\to\int_{\mathbb{T}}f(y,\theta_{1},t)\sin(2(\theta_{1}-\theta))\,d\theta_{1}=v_{f},$

a.e. $(x,\theta,t)$ and in $L^{2}_{loc}(Q_{T}).$ With this, one can pass to
the limit in the equation (2.7) to obtain the equation with the localized
alignment rate (2.9). ∎

###  2.5. Existence theory: model II

Solutions to this model will be obtained from a model with added diffusion in
$x$-direction,

(2.25) $\partial_{t}f{}-{}\,\text{div}\,\left(e(\theta)\otimes e(\theta)\nabla
f\right){}+{}\partial_{\theta}(f\llbracket
v_{f}\rrbracket_{1}){}-{}\partial^{2}_{\theta\theta}f-\epsilon\Delta f{}={}0,$

in the zero limit of the diffusion coefficient $\epsilon.$ For the solutions
of the later equation, one has the energy inequality

(2.26)
$\sup_{[0,T]}\|f(\cdot,t)\|^{2}_{L^{2}_{x,\theta}}{}+{}\|\partial_{\theta}f\|^{2}_{L^{2}(0,T;L^{2}_{x,\theta})}{}+{}\|e(\theta)\cdot\nabla
f\|^{2}_{L^{2}(0,T;L^{2}_{x,\theta})}{}+{}\epsilon\|\nabla
f\|^{2}_{L^{2}(0,T;L^{2}_{x,\theta})}{}\leq{}2e^{T}\|f_{0}\|^{2}_{L^{2}_{x,\theta}}.$

We will assume that weak solutions with this regularity exist for any non-
negative $f_{0}\in L^{2}_{x,v}\cap L^{1}_{x,v},$ $\Pi$-periodic in $x$ and
$\pi$-periodic in $\theta,$ and with properties (2.18)-(2.20). This can be
proved by the argument parallel to that of the proof of Theorem 1. We omit the
proof.

Our main result is the following variant of the averaging lemma proved in
[12].

######  Lemma 1.

Let $f_{\epsilon}$ be a weak solution of (2.25) with $\epsilon\in(0,1],$ with
initial data $f_{0}.$ Let $u_{\epsilon}(x,t)=\psi(x,t)f_{\epsilon},$ for
$\psi\in C^{\infty}_{0}(\mathbb{R}^{2}\times\mathbb{R}_{+}),$ and be extended
by zero to all of $\mathbb{R}^{3}.$ Denote a $\theta$-moment function

$\rho^{\phi}_{\epsilon}{}={}\int_{\mathbb{T}}u_{\epsilon}\phi(\theta)\,d\theta,\quad\phi\in
C^{1}(\overline{\mathbb{T}}).$

Then,
$\rho^{\phi}_{\epsilon}\in{H}^{\frac{2}{7},\frac{1}{7}}(\mathbb{R}^{2}\times\mathbb{R}),$
and

$\|\rho^{\phi}_{\epsilon}\|_{{H}^{\frac{2}{7},\frac{1}{7}}}{}\leq{}C_{\psi,\phi}\|f_{0}\|_{L^{2}_{x,\theta}},$

for some $C_{\psi,\phi},$ independent of $\epsilon.$ In particular, set
$\\{\rho^{\phi}_{\epsilon}\\}_{\epsilon\in(0,1]}$ is pre-compact in
$L^{2}(\mathbb{R}^{3}).$

######  Remark 1.

${H}^{\frac{2}{7},\frac{1}{7}}$ estimate for $\rho^{\phi}_{\epsilon}$ of the
lemma equally applies to a family of solutions of (2.12) verifying the energy
balance (2.26).

###### Proof.

Let $\psi$ and $u_{\epsilon}$ be as in the statement of the lemma. We will
write $u$ for $u_{\epsilon}$ to simplify the notation. $u$ verifies equation

(2.27) $\partial_{t}u{}-{}\,\text{div}\,\left(D(\theta)\nabla
u\right){}-\epsilon\Delta u{}={}q_{1}+\partial_{\theta}q_{2},$

where

$q_{1}{}={}-(e(\theta)\cdot\nabla
f_{\epsilon})e(\theta)\cdot\nabla\psi+2\epsilon\nabla f_{\epsilon}\cdot\nabla
u+\epsilon f_{\epsilon}\Delta\psi,$ $q_{2}{}={}-(u\llbracket
v_{f_{\epsilon}}\rrbracket_{1}){}+{}\partial_{\theta}u,$

with $q_{1},$ $q_{2}$ uniformly in $\epsilon$ bounded in
$L^{2}(\mathbb{R}^{2}\times\mathbb{T}\times\mathbb{R})$ due to (2.26). For
function $\hat{u}(\xi,\theta,\tau)$ – the Fourier transform of $u$ in $(x,t),$
we get

$(-i\tau+(e(\theta)\cdot\xi)^{2}+\epsilon|\xi|^{2})\hat{u}{}={}\hat{q}_{1}+\partial_{\theta}\hat{q}_{2}.$

Writing $\xi=|\xi|e(\theta_{1}),$ for the polar angle $\theta_{1},$ we can
express $e(\theta)\cdot\xi{}={}|\xi|\cos(\theta_{1}-\theta).$ Let
$\omega(\theta)$ be a smooth cutoff function for the set

$\\{\theta\,:\,|\cos(\theta_{1}-\theta)|<\delta\\},$

for small $\delta>0,$ that is,

$\omega(\theta){}={}\left\\{\begin{array}[]{ll}0,&|\cos(\theta_{1}-\theta)|\leq\delta/2\\\
1,&|\cos(\theta_{1}-\theta)|\geq\delta\end{array}\right.$

with $\max_{\theta}|\omega|=1,$ and
$\max_{\theta}|\partial_{\theta}\omega|\leq C\delta^{-1}.$ Then, we can write

$\displaystyle\left(\int_{\mathbb{T}}\hat{u}\phi\,d\theta\right)^{2}{}\leq{}$
$\displaystyle
2C\left(\int_{\mathbb{T}}|\hat{u}||1-\omega|\,d\theta\right)^{2}{}+{}2\left(\int_{\mathbb{T}}\frac{(\hat{q}_{1}+\partial_{\theta}\hat{q}_{2})\phi(\theta)\omega(\theta)\,d\theta}{-i\tau+(|\xi|\cos(\theta_{1}-\theta))^{2}+\epsilon|\xi|^{2}}\right)^{2}$
$\displaystyle{}\leq{}$ $\displaystyle
2C\delta\int_{\mathbb{T}}|\hat{u}|^{2}\,d\theta{}+{}C\int_{\mathbb{T}}\frac{|\hat{q}_{1}|^{2}\,d\theta}{|-i\tau+\delta^{2}|\xi|^{2}+\epsilon|\xi|^{2}|^{2}}$
$\displaystyle{}+{}C\int_{\mathbb{T}}\frac{|\hat{q}_{2}|^{2}|\partial_{\theta}(\phi\omega)|^{2}\,d\theta}{|-i\tau+\delta^{2}|\xi|^{2}+\epsilon|\xi|^{2}|^{2}}{}+{}C\int_{\mathbb{T}}\frac{|\hat{q}_{2}|^{2}|\xi|^{4}\cos^{2}(\theta_{1}-\theta)\,d\theta}{|-i\tau+(|\xi|\cos(\theta_{1}-\theta))^{2}+\epsilon|\xi|^{2}|^{4}}$
$\displaystyle{}\leq{}$ $\displaystyle
2C\delta\int_{\mathbb{T}}|\hat{u}|^{2}\,d\theta{}+{}C\delta^{-2}\frac{1}{(\tau^{2}+\delta^{4}|\xi|^{4})}\int_{\mathbb{T}}|\hat{q}_{1}|^{2}+|\hat{q}_{2}|^{2}\,d\theta$
$\displaystyle{}+{}C\frac{|\xi|^{2}}{(\tau^{2}+\delta^{4}|\xi|^{4})^{3/2}}\int_{\mathbb{T}}|\hat{q}_{2}|^{2}\,d\theta$
$\displaystyle{}\leq{}$ $\displaystyle
2C\delta\int_{\mathbb{T}}|\hat{u}|^{2}\,d\theta{}+{}C\delta^{-6}\frac{1}{(\tau^{2}+|\xi|^{4})}\int_{\mathbb{T}}|\hat{q}_{1}|^{2}+|\hat{q}_{2}|^{2}\,d\theta.$

Choosing

$\delta=(\tau^{2}+|\xi|^{4})^{1/7},$

we find that

$(\tau^{2}+|\xi|^{4})^{1/7}\left(\int_{\mathbb{T}}\hat{u}\phi\,d\theta\right)^{2}{}\leq{}C\left(\int_{\mathbb{T}}|\hat{u}|^{2}{}+{}|\hat{q}_{1}|^{2}+|\hat{q}_{2}|^{2}\,d\theta\right),$

with $C$ independent of $\epsilon.$ We also have

$\left(\int_{\mathbb{T}}\hat{u}\phi\,d\theta\right)^{2}{}\leq{}C\int_{\mathbb{T}}|\hat{u}|^{2}\,d\theta.$

Then, we obtain the estimate of the lemma, by integrating the last two
inequalities in $(\xi,\tau)$ and using the Parseval’s inequality.

∎

With this lemma, we can now prove

######  Theorem 3:

Let $f_{0}\in L^{2}_{x,\theta}\cap L^{1}_{x,\theta},$ $f_{0}\geq 0$ a.e.,
$\pi$-periodic in $\theta$ and have unit mass. There is a global in time, weak
solution of (2.12). The solution is non-negative, $\pi$-periodic in $\theta,$
has unit mass for all $t>0,$ and such that for any $T>0,$ (2.26) and
(2.18)–(2.20) hold.

######  Remark 2.

The same result applies to equation (2.14) with given chemoattractant
concentration $s,$ when $s\in L^{\infty}(0,T;W^{1,\infty}(\Pi)).$

###### Proof.

We start with solutions $f_{\epsilon}$ of (2.25) with initial data $f_{0},$
which verify the energy inequality (2.26) and properties (2.18)–(2.20). A
sequence $f_{\epsilon_{n}},$ $\epsilon_{n}\to 0$ can be found such that
$f_{\epsilon_{n}}\to f$ in $C([0,T];L^{2}_{x,\theta}-weak).$ To pass to the
limit in the nonlinear term of the equation, we use lemma 1. According to it,
there is a subsequence, that we still label $\epsilon_{n},$ such that
$v_{f_{\epsilon_{n}}}$ is pre-compact in
$L^{2}_{loc}(\mathbb{R}^{2}\times\mathbb{R}_{+})$ and converges to $v_{f}.$ A
further subsequence can be extracted such that $v_{f_{\epsilon_{n}}}\to v_{f}$
a.e. $(x,t).$ With that, one can pass to the limit in the alignment term of
the equation. ∎

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relation from Churchill et al. (2020). Surprisingly, the statistics of the
mock CIV 1551Å absorption spectra also seem to agree with the observed MgII
relation, and the HD-MHD differences are wider as expected, but to the lower
number of unsaturated mock spectra, it is harder to draw concrete conclusions.

This apparent robustness of the observed relation might hint towards a more
fundamental origin of the relation, like the clump distribution. But, we leave
it to future studies to investigate this further.

Figure 22: Same as Fig. 18 but for high resolution MgII 2796 Å absorption
spectra with $d\lambda=0.01$Å Figure 23: Same as Fig. 18 but for high
resolution CIV Å absorption spectra with $d\lambda=0.01$Å

##  Appendix D Effect of stochasticity

The stochastic nature of the turbulence can cause variations in the evolution
of quantities in a turbulent environment. Gronke et al. (2022) found that this
stochasticity affects the cold gas mass evolution in hydrodynamic turbulent
boxes with an intermediate-sized initial cold cloud. In this regime, they saw
both survival and destruction of the cold cloud for different choices of
random seeds for turbulent driving. This was attributed to the higher
significance of the exact turbulent velocity field in the intermediate regime
between cloud survival and destruction.

We repeat this test for our simulations, with and without magnetic fields. We
run turbulent box simulations with 3 different random seeds at
$\mathcal{M}=0.5$ and introduce clouds of different sizes to check for the
effect of stochasticity of the turbulence. We use a $L_{\rm box}/R_{\rm
cl}=20$, instead of 40, due to its lower computational costs.

Fig. 24 shows the cold gas mass evolution for the different cases. We find
that the cold gas mass growth/destruction rate for cold gas clouds in
intermediate and destruction regimes is sensitive to the exact choice of the
random seed. We also find that this is true for both HD and MHD and with no
clear order of growth rate between the HD and MHD counterparts. This high
dependence on stochasticity in these regimes is due to the lack of cold gas
mass. This results in a very stochastic sampling of turbulence, hence making
the evolution very stochastic in nature.

On the other hand, in the survival regime, the MHD simulations seem to have a
slightly lower growth rate, compared to their HD counterparts, although still
a much lower difference compared to the order of magnitude difference observed
in TRML simulations. We attribute this minor difference to some unavoidable
systematic differences between the HD and MHD simulations. The biggest of them
is the difference in dissipation rate between MHD and HD, due to the extra
dissipation of magnetic energy via numerical resistivity. This higher
dissipation results in a slightly hotter medium in a fully developed turbulent
box, in turn resulting in a slightly deviated density distribution. These
slight deviations affect the evolution via a slight difference in overdensity,
mixed gas temperature, etc.

Still, as Fig. 24 shows, this difference is minor and it gets even more
trivial when we take the spread due to stochasticity into account.

Figure 24: Cold gas mass evolution for simulations with the same parameters
but different random instances of turbulence. The different panels refer to
different $R_{\rm cl}/l_{\rm shatter}$ in a turbulent medium with
$\mathcal{M}=0.5$. The different colours denote simulations with varying
random seeds for turbulent driving. The solid and dashed lines show the
evolution of simulations with and without magnetic fields, respectively.