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 $v(x,t)$ as a particle velocity. Then, for any subspace $W$ of $\mathbb {R} ^{3}$ we have:

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

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:

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

Finally, we conclude that:

     $\int _{W}\partial _{t}\rho +\nabla \cdot (\rho v)dV=0,$

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

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

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 book by F.Santambrogio.

From this point, we are looking at the following equation: $\partial _{t}\mu _{t}+\nabla \cdot (\mu _{t}v_{t})=0$.

All the measures we are interested in satisfy $\int _{0}^{1}||v_{t}||_{L^{1}(\mu _{t})}<\infty$, and solve continuity equation in a distributional sense, namely

   $\int _{0}^{T}\int _{\Omega }(\partial _{t}\phi )d\mu _{t}dt+\int _{0}^{T}\int _{\Omega }\nabla \phi \cdot v_{t}d\mu _{t}dt=0,$  

for all $\phi \in C_{c}^{1}((0,T)X\Omega )$, where $\Omega$ is a compact set or the whole space $\mathbb {R} ^{d}$, and $0<T<1$. We assume no-flux condition in this case, namely $\mu _{t}v_{t}\cdot n=0$ on the boundary $\partial \Omega .$

As it is always in the optimal transport, we want to move one pile of dirt to another one. So, we have to impose initial and terminal conditions, for example $\mu _{0}=\mu$, and $\mu _{1}=\nu .$ Then, our equation becomes

    $\int _{0}^{T}\int _{\Omega }(\partial _{t}\phi )d\mu _{t}dt+\int _{0}^{T}\int _{\Omega }\nabla \phi \cdot v_{t}d\mu _{t}dt=\int _{\Omega }\phi (T,x)d\nu (x)-\int _{\Omega }\phi (0,x)d\mu (x),$

for all $\phi \in C_{c}^{1}([0,T]X\Omega ).$

Another way to interpret solutions to the continuity equation is to assume that function $t\rightarrow \int _{\Omega }\psi d\mu _{t}$ is absolutely continuous, and for a.e. $t$ it holds:

 $\partial _{t}\int _{\Omega }\psi d\mu _{t}=\int _{\Omega }\nabla \phi \cdot v_{t}.$

This kind of solution is called a weak solution.

Proposition 4.2., on the page 124. in the book by Santambrogio states that these solutions are basically the same.

Applications

The following theorem can be found at the book by L.Ambrosio, E.Brué, and D.Semola.

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