The continuity equation and Benamour Brenier formula

Introduction

The continuity equation is an important equation in many fields of science, 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 the dynamic 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 the 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, representing the mass-density of fluid, and $v(x,t)$ as a velocity of particle at position $x$, at time $t$. 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. We use the equation in this form in optimal transport.

Continuity equation in optimal transport

The previous discussion assumed that the density function was smooth, which is not true of the general measures we consider in optimal transport. Even when a measure $\mu$ is absolutely continuous with respect to Lebesgue measure, which we write with a mild abuse of notation as $d\mu (x)=\mu (x)dx$, $\mu$ does not have to be smooth. So, we need to state a proper weak formulation of the continuity equation. Smooth functions satisfy all the cases below.

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:

$\partial _{t}\mu _{t}+\nabla \cdot (\mu _{t}v_{t})=0$,

and two notions of solutions.

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

$\int _{0}^{T}\int _{\Omega }[\partial _{t}\phi +\nabla \phi \cdot v_{t}]d\mu _{t}dt=0,$ for all bounded Lipschitz functions $\phi \in C_{c}^{1}((0,T)\times {\overline {\Omega }})$, where $\Omega$ is a bounded domain 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 $\partial \Omega .$ The main goal of the classical optimal transport theory is how to find the least expensive way to move one measure to the another one. For more information, look at Monge Problem.So, we have to impose initial and terminal conditions on measures, for example $\mu _{0}=\mu$, and $\mu _{1}=\nu .$ Then, our equation becomes $\int _{0}^{T}\int _{\Omega }[\partial _{t}\phi +\nabla \phi \cdot v_{t}]d\mu _{t}dt=\int _{\Omega }\phi (T,x)d\nu (x)-\int _{\Omega }\phi (0,x)d\mu (x),$ for all $\phi \in C_{c}^{1}([0,T]\times {\overline {\Omega }}).$ This notion of solution is called a distributional solution.

• Weak solution. 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: $\partial _{t}\int _{\Omega }\psi d\mu _{t}=\int _{\Omega }\nabla \phi \cdot v_{t}d\mu _{t}.$ This kind of solution is called a weak solution.

Proposition 1., (p.124,) Distributional and weak solutions are equivalent. Every weak solution is a distributional solution. On the other hand, every distributional solution admits a representative (a.e. equal), that is weakly continuous and a weak solution.

Proposition 2.,(p.124,) Let $\mu$ be the Lipschitz function in $(t,x)$ and $v$ be the Lipschitz function in $x.$ Suppose that the continuity equation is satisfied in the weak sense. Then it is satisfied in a.e. sense.

Previous two definitions and connections help us to conclude this section by referencing to Theorem of Cauchy-Lipschitz(, p. 184). It is the analogue of the Picard-Lindelof Theorem in ODE theory, and it provides us with the unique solution, which is crucial for finding a proper geodesics in the applications.

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.