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.

Suppose that mass of our fluid is conserved, through time. Denote $\rho (x,t)$ as a density function and $u(x,t)$ as a particle velocity. Then, for any subspace $W$ of $R^{3}$ we have:

    $\partial _{t}[\int _{W}\rho (x,t)dV]=-\int _{\partial W}\rho u\cdot ndS.$

In this section, we assume both density function and particle velocity are smooth enough. Hence, after differentiating under the integral and applying the Divergence Theorem, we get:

     $\int _{W}\partial _{t}\rho (x,t)dV=-\int _{W}\nabla \cdot (\rho u)dV.$

Finally, we conclude that:

     $\int _{W}\partial _{t}\rho +\nabla \cdot (\rho u)dV,$

which implies, since $W$ is arbitrary, that:

     $\partial _{t}\rho +\nabla \cdot (\rho u)=0.$

The last equation is the continuity equation in fluid dynamics, written in the differential form. Equation of this form we use in the optimal transport.

Continuity equation in optimal transport

Reason why we can not apply the previous equation immediately in the optimal transport theory is a regularity of measures we work with. By abuse of notation, $\mu =\mu (x)dx$, a function $\mu (x)$ does not have to be smooth. So, we need to find a proper space for our measures.

Sometimes in the literature, authors use continuity equation, and transport equation as synonyms. On the other hand, in the optimal transport we differentiate these two and the standard Cauchy problem. Here, we will present definitions and reasoning from the book by L.Ambrosio, E.Brué, and D.Semola.

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 $y=\omega (1)$.

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 $t,s\in [0,1]$

Applications

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

      $W_{2}^{2}(\mu ,\nu )=\min _{(\mu (t),\nu (t))}\{\int _{0}^{1}|v(\cdot ,t)|_{L^{2}(\mu (t))}^{2}dt\quad |\quad \partial _{t}\mu +\nabla \cdot (v\mu )=0,\quad \mu (0)=\mu ,\quad \mu (1)=\nu \}.$

This formula is important for defining Riemannian structure. You can see more at Formal Riemannian Structure of the Wasserstein metric.

In addition, using the continuity equation we can describe geodesics in the Wasserstein space. For more details look at Geodesics and generalized geodesics.