The continuity equation

Introduction

The continuity equation is an important equation in many science fields, for example, electromagnetism, computer vision, fluid dynamics etc. However, in the field of optimal transport, the formulation from fluid dynamics is of a large significance. This form helps to explain dynamics formulation of special cases of Wasserstein metric, and we will focus in this direction. For more general information about the continuity equation, look at the article Continuity equation.

Continuity equation in fluid dynamics

First, because of the intuition, we will introduce definition of the continuity equation in fluid mechanics. The exposition in this section will follow the book by Chorin and Marsden.

For the starting point, we need to introduce length of the curve in our metric space $(X,d)$.

Definition. A length of the curve $\omega :[0,1]\rightarrow X$ is defined by

                  $L(\omega )=\sup\{\sum _{j=0}^{n-1}d(\omega (t_{j}),\omega (t_{j+1}))|\quad n\geq 2,\quad 0=t_{0}<t_{1}<...<t_{n-1}=1\}$

Secondly, we use the definition of length of a curve to introduce a geodesic curve.

Definition. A curve $c:[0,1]\rightarrow X$ is said to be geodesic between $x$ and $y$ in $X$ if it minimizes the length $L(\omega )$ among all the curves $\omega :[0,1]\rightarrow X$
such that $x=\omega (0)$ and $y=\omega (1)$.

Since we have a definition of a geodesic in the general metric space, it is natural to think of Riemannian structure. It can be formally defined. More about this topic can be seen in the following article Formal Riemannian Structure of the Wasserstein_metric.

Now, we proceed with necessary definitions in order to be able to understand Wasserstein metric in a different way.

Definition. A metric space $(X,d)$ is called a length space if it holds

                    $d(x,y)=\inf\{L(\omega )|\quad \omega \in AC(X),\quad \omega (0)=x\quad \omega (1)=y\}.$

A space $(X,d)$ is called geodesic space if the distance $d(x,y)$ is attained for some curve $\omega$.

Definition. In a length space, a curve $\omega :[0,1]\rightarrow X$ is said to be constant speed geodesic between $\omega (0)$ and $\omega (1)$ in $X$ if it satisfies

                    $d(\omega (s),\omega (t))=|t-s|d(\omega (0),\omega (1))$ for all $t,s\in [0,1]$

It is clear that constant-speed geodesic curve $\omega$ connecting $x$ and $y$ is a geodesic curve. This is very important definition since we have that every constant-speed geodesic $\omega$ is also in $AC(X)$ where $|\omega '(t)|=d(\omega (0),\omega (1))$ almost everywhere in $[0,1]$.
In addition, minimum of the set $\{\int _{0}^{1}|c'(t)|^{p}dt|c:[0,1]\rightarrow X,c(0)=x,c(1)=y\}$ is attained by our constant-speed geodesic curve $\omega .$ Last fact is important since it is connected to Wasserstein $p$ metric. For more information, please take a look at Wasserstein metric.

For more information on constant-speed geodesics, especially how they depend on uniqueness of the plan that is induced by transport and characterization of a constant-speed geodesic look at the book by L.Ambrosio, N.Gilgi, G.Savaré or the book by Santambrogio.

Continuity equation in optimal transport

Finally, we can rephrase Wasserstein metrics in dynamic language as mentioned in the Introduction.

Whenever $\Omega \subseteq {\mathcal {R}}^{d}$ is convex set, $W_{p}(\Omega )$ is a geodesic space. Proof can be found in the book by Santambrogio.

Theorem. Let $\mu ,\nu \in {\mathcal {P}}_{2}(R^{d})$. Then

      $W_{p}^{p}(\mu ,\nu )=\inf _{(\mu (t).\nu (t))}\{\int _{0}^{1}|v(,t)|_{L^{p}(\mu (t))}^{p}dt\quad |\quad \partial _{t}\mu +\nabla \cdot (v\mu )=0,\quad \mu (0)=\mu ,\quad \mu (1)=\nu \}.$

In special case, when $\Omega$ is compact, infimum is attained by some constant-speed geodesic.

Applications

There are many ways to generalize this fact. We will talk about a special case $p=2$ and a displacement convexity. Here we follow again book by Santambrogio.

In general, the functional $\mu \rightarrow W_{2}^{2}(\mu ,\nu )$ is not a displacement convex. We can fix this by introducing a generalized geodesic.

Definition. Let $\rho \in {\mathcal {P}}(\Omega )$ be an absolutely continuous measure and $\mu _{0}$ and $\mu _{1}$ probability measures in ${\mathcal {P}}(\Omega )$. We say that $\mu _{t}=((1-t)T_{0}+tT_{1})\#\rho$
is a generalized geodesic in ${\mathcal {W}}_{2}(\Omega )$ with base $\rho$, where $T_{0}$ is the optimal transport plan from $\rho$ to $\mu _{0}$ and $T_{1}$ is the optimal transport plan from $\rho$ to $\mu _{1}$.

By calculation, we have the following $W_{2}^{2}(\mu _{t},\rho )\leq (1-t)W_{2}^{2}(\mu _{0},\rho )+tW_{2}^{2}(\mu _{1},\rho ).$

Therefore, along the generalized geodesic, the functional $t\rightarrow W_{2}^{2}(\mu _{t},\rho )$ is convex.

This fact is very important in establishing uniqueness and existence theorems in the geodesic flows.