An integrable example of gradient flow based on optimal transport of differential forms

Abstract

Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed as gradient flows of volume forms (or, in other words, 0-currents) according to suitable transportation metrics. In this paper, we present an example of gradient flow for closed $(d-1)$-differential forms, or, more appropriately, to closed 1-currents, which can be identified to divergence-free vector fields, in the Euclidean space $\mathbb {R}^d$. In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed, in a suitable sense, as an Eulerian version of the heat equation for loops in the Euclidean space. We analyze this system in terms of “relative entropy” and “dissipative solutions” and provide global existence and weak–strong uniqueness results.

Appendix 1: Direct recovery of Eq. (2.1)

Let loop X(t, s) be a solution to the heat equation (1.3). For any smooth vector field $b^*$, the relative entropy

$$\begin{aligned} \mathcal {E}(t) =\int _{\mathbb {R/Z}}\frac{{|\partial _s X(t,s)-b^*(t,X(t,s))|}^2}{2}ds \end{aligned}$$

can be written as

$$\begin{aligned} \mathcal {E}(t)=\int _{\mathbb {R/Z}}\left( \frac{|\partial _s X|^2}{2} -\partial _s X\cdot b^*(t,X) +\frac{|b^*(t,X)|^2}{2}\right) ds=\mathcal {E}_1 (t)+\mathcal {E}_2 (t)+\mathcal {E}_3 (t). \end{aligned}$$

For $\mathcal {E}_1 (t)=\int |\partial _s X|^2/2$, we have

$$\begin{aligned} \frac{d}{dt}\mathcal {E}_1(t)=\int \partial _s X\cdot \partial _{ts} X =-\int \partial _{ss}X \cdot \partial _t X=-\int |\partial _t X|^2. \end{aligned}$$

(since $\partial _t X=\partial _{ss} X$). For any smooth vector field $v^*$, we have

$$\begin{aligned}&-\int |\partial _t X|^2= \int -|\partial _t X-v^*(t,X)|^2 + |v^*(t,X)|^2-2\partial _t X\cdot v^*(t,X)\\&\quad =\int -|\partial _t X-v^*(t,X)|^2 + |v^*(t,X)|^2-\partial _t X\cdot v^*(t,X)-\partial _{ss} X\cdot v^*(t,X). \end{aligned}$$

In coordinates, we have

$$\begin{aligned}&-\int \partial _{ss} X^i v^*_i(t,X)= \int \partial _s X^i \partial _{j}v^*_i(t,X)\partial _s X^j\\&\quad =\frac{1}{2}\int \big (\partial _s X^i-{b^*}^i(t,X)\big )\big (\partial _s X^j-{b^*}^j(t,X)\big )(\partial _{j}v^*_i + \partial _{i}v^*_j)(t,X)\\&\qquad +\int \partial _sX^i{b^*}^j(t,X)(\partial _{j}v^*_i + \partial _{i}v^*_j)(t,X) - \big ({b^*}^i{b^*}^j\partial _jv^*_i\big )(t,X). \end{aligned}$$

So we have,

$$\begin{aligned} \frac{d}{dt}\mathcal {E}_1(t)= & {} \int -|\partial _t X-v^*(t,X)|^2 + \mathrm {L}'_1(t,X)+\partial _s X\cdot \mathrm {L}'_2(t,X)+ \partial _t X\cdot \mathrm {L}'_3(t,X) \\&+\frac{1}{2}\int \big (\partial _s X^i-{b^*}^i(t,X)\big )\big (\partial _s X^j-{b^*}^j(t,X)\big )(\partial _{j}v^*_i + \partial _{i}v^*_j)(t,X), \end{aligned}$$

where

$$\begin{aligned} \mathrm {L}'_1=|v^*|^2-{b^*}^i{b^*}^j\partial _jv^*_i,\;\;\;(\mathrm {L}'_2)_i=(\partial _{j}v^*_i + \partial _{i}v^*_j){b^*}^j,\;\;\; (\mathrm {L}'_3)_i=-v^*_i . \end{aligned}$$

Now let’s look at $\mathcal {E}_2(t)=-\int \partial _s X \cdot b^*(t,X)$. In coordinates, we have,

$$\begin{aligned} \frac{d}{dt}\mathcal {E}_2(t)= & {} \int -\partial _{st} X^i \cdot b^*_i(t,X)-\partial _s X^i(\partial _t b^*_i)(t,X)-\partial _s X^i\partial _t X^j (\partial _j b^*_i)(t,X)\\= & {} \int -\partial _s X^i\partial _t X^j(\partial _j b^*_i-\partial _i b^*_j)(t,X) -\partial _s X^i(\partial _t b^*_i)(t,X) \\= & {} -\int \big (\partial _s X^i-{b^*}^i(t,X)\big )\big (\partial _t X^j-{v^*}^j(t,X)\big )(\partial _j b^*_i-\partial _i b^*_j)(t,X)\\&+ \int \mathrm {L}''_1(t,X)+\partial _s X\cdot \mathrm {L}''_2(t,X)+ \partial _t X\cdot \mathrm {L}''_3(t,X), \end{aligned}$$

where

$$\begin{aligned}&\mathrm {L}''_1=(\partial _j b^*_i-\partial _i b^*_j){b^*}^i{v^*}^j,\;\;(\mathrm {L}''_2)_i=-\partial _t b^*_i-(\partial _j b^*_i-\partial _i b^*_j){v^*}^j,\\&(\mathrm {L}''_3)_i=(\partial _j b^*_i-\partial _i b^*_j){b^*}^j. \end{aligned}$$

For the last term $\mathcal {E}_3(t)=\int |b^*(t,X)|^2/2$, we have,

$$\begin{aligned} \frac{d}{dt}\mathcal {E}_3(t)= \int \big (\partial _t b^*_i(t,X) + \partial _t X^j\partial _j b^*_i(t,X)\big ){b^*}^i(t,X). \end{aligned}$$

So, in summary, we have

$$\begin{aligned}&\frac{d\mathcal {E}}{dt}=\frac{1}{2}\int \big (\partial _s X^i-{b^*}^i(t,X)\big )\big (\partial _s X^j-{b^*}^j(t,X)\big )(\partial _{j}v^*_i + \partial _{i}v^*_j)(t,X)\\&\qquad -\int \big (\partial _s X^i-{b^*}^i(t,X)\big )\big (\partial _t X^j-{v^*}^j(t,X)\big )(\partial _j b^*_i-\partial _i b^*_j)(t,X)\\&\int -|\partial _t X-v^*(t,X)|^2 + \mathrm {L}_1(t,X)+\partial _s X\cdot \mathrm {L}_2(t,X)+ \partial _t X\cdot \mathrm {L}_3(t,X), \end{aligned}$$

where

$$\begin{aligned}&\mathrm {L}_1=\mathrm {L}'_1{+}\mathrm {L}''_1 {+} \partial _t\left( \frac{{b^*}^2}{2}\right) ={v^*}^2+D_t^* \left( \frac{{b^*}^2}{2}\right) -(b^*\cdot \nabla )(b^*\cdot v^*), \;\;D_t^*=(\partial _t+v^*\cdot \nabla )\\&\mathrm {L}_2=\mathrm {L}'_2+\mathrm {L}''_2=-D_t^* b^*+(b^*\cdot \nabla )v^*+\nabla (b^*\cdot v^*)\\&\mathrm {L}_3=\mathrm {L}'_3+\mathrm {L}''_3 + \nabla \left( \frac{{b^*}^2}{2}\right) =-v^*+(b^*\cdot \nabla )b^*. \end{aligned}$$

Since

$$\begin{aligned} \int \partial _s X\cdot \big [\nabla (b^*\cdot v^*)\big ](t,X)=\int \partial _{s}\big (b^*(t,X)\cdot v^*(t,X)\big )=0, \end{aligned}$$

we can remove the gradient term $\nabla (b^*\cdot v^*)$ from $\mathrm {L}_2$ and finally get (2.1).