# Glimpse into the work of Alessio Figalli

### Klas Modin

Seminar series about Fields Medalists 2018

Alessio Figalli

Nationality: Italian

Born: 1984

Master's degree 2006 (SNS Pisa)

PhD 2007 (SNS Pisa and ENS Lyon)

Supervisors:

Luigi Ambrosio (SNS Pisa)

Cedric Villani (ENS Lyon)

Affiliations:

CNRS 2007

École polytechnique 2008-2009

University of Texas in Austin 2009-2016

ETH Zürich 2016-today

Awarded the Fields Medal for his contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry, and probability

## Research fields

Optimal mass transport (OMT)

Fluid dynamics (Euler equations, etc)

Weak KAM theory

Geometric measure theory

Stochastic analysis

Random matrices

etc.

Very productive 120+ papers, 2 books, 25 lecture notes, ...

Focus of today

## Optimal mass transport (OMT)

Question (Monge 1781): Cheapest way to transport one mass distribution to another?

\mu_0
\mu_1

Mathematical formulation:

Additional requirement: $$T$$ should minimize

\displaystyle E(T) =
\displaystyle\int_\Omega c(T(\mathbf x),\mathbf x)d \mathbf x
\displaystyle\int_\Omega |T(\mathbf x)-\mathbf x|^2d \mathbf x

## Example of transport map

\mu_0 = d\mathbf x
\text{Domain}\; \Omega = \mathbb T^2
\mu_1 =
T:\Omega\to\Omega, \; T_*\mu_0 = \mu_1

Transport map

## Fundamental result for $$L^2$$ OMT

Theorem (Brenier 1991):

$$\mu_0,\mu_1$$ prob measures on $$\mathbb R^n$$ with finite second moment, $$\mu_0=\rho_0d \mathbf x$$
Then $$(*)$$ has unique solution $T = \nabla P$

where $$P:\mathbb R^n \to \mathbb R$$ is convex

If $$\mu_1 = \rho_1 d\mathbf x$$ then (formally) $$P$$ solves the Monge-Ampere equation $\det(D^2P) = \frac{\rho_0}{\rho_1\circ\nabla P} \iff \text{MA}(P) = \frac{\rho_0}{\rho_1\circ\nabla P}$

\displaystyle \min_{T_*\mu_0=\mu_1} \int_{\mathbb R^n} |T(\mathbf x)-\mathbf x|^2 d\mu_0\qquad (*)

## Sobolev result for $$L^2$$ OMT

Theorem (De Philippis and Figalli 2013):

If $$\Omega\subset \mathbb R^n$$ is convex and bounded, $$P:\Omega\to\mathbb R$$ is convex, and $0 < \lambda \leq \det(D^2 P) \leq \Lambda < \infty$

then $$P \in W^{2,1}_{\text{loc}}(\Omega)$$

Corollary

The solution $$T = \nabla P$$ to $$(*)$$  belong to $$W^{1,1}_{\text{loc}}(\Omega)$$

\displaystyle \min_{T_*d\mathbf x=\rho d\mathbf x} \int_{\Omega} |T(\mathbf x)-\mathbf x|^2 d\mathbf x\qquad (*)

## Incompressible hydrodynamics

\displaystyle\frac{\partial \mathbf v}{\partial t} + \mathbf v\cdot \nabla \mathbf v = -\nabla p\qquad (*)

Domain $$\Omega\subset \mathbb R^3$$, velocity field $$\mathbf v = \mathbf v(t,\mathbf x)$$

Discovery by Arnold 1966:

Solutions to $$(*)$$ correspond to geodesics on the infinite-dimensional  manifold $$\mathrm{SDiff}(\Omega)$$ with Riemannian $$L^2$$-metric $\langle V,V\rangle_{S} = \int_\Omega |V|^2 d \mathbf x$ where $\mathbf v = V\circ S^{-1}$

\displaystyle \nabla \cdot\mathbf v = 0

pressure

## OMT $$\leftrightarrow$$ hydrodynamics

\displaystyle \pi: F\mapsto F_*d\mathbf{x}

(Benamou and Brenier 2000, Otto 2001)

\displaystyle \mathrm{Diff}(\Omega)
\displaystyle \mathrm{id}
\displaystyle \mathrm{SDiff}(\Omega)
\displaystyle F
\displaystyle \nabla P
\displaystyle S
\displaystyle \mathcal{P}^\infty(\Omega) \simeq \mathrm{Diff}(\Omega)/\mathrm{SDiff}(\Omega)
\displaystyle d\mathbf{x}
\displaystyle \rho\, d\mathbf{x}

Brenier's polar

factorization: $$F = \nabla P\circ S$$

Remember:

$$T = \nabla P$$ solves OMT problem with $$\mu_0 = d\mathbf x$$ and $$\mu_1 = \rho\,d\mathbf x$$

hydrodynamics

OMT

## Atmospheric hydrodynamics

\displaystyle\frac{\partial \mathbf v}{\partial t} + \mathbf v\cdot \nabla \mathbf v = -\nabla p + \begin{pmatrix}v_2 \\ -v_1 \\ 0\end{pmatrix} + \mathbf{g}
\displaystyle \nabla \cdot\mathbf v = 0

Semi-geostrophic wind:

Coriolis force

\displaystyle \mathbf v^g = \begin{pmatrix}\partial_2 p \\ -\partial_1 p \end{pmatrix}

Aim: equation for $$\mathbf v^g$$

## Semigeostrophic equation (SGE)

\displaystyle\frac{\partial \nabla p}{\partial t} + (\mathbf v\cdot \nabla) \nabla p = -J\nabla p - \mathbf v
\displaystyle \nabla \cdot\mathbf v = 0

where

Now 2D: $$\mathbf v = (v_1,v_2)$$ and replace l.h.s. by semi-geostrophic wind

J = \begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}

Notice: no time-derivative on $$\mathbf v$$

As written, SGE is hard to analyze!

Rewrite in simpler form?

(Hoskins and Bretherton 1972)

## Brief derivation of dual SGE

\displaystyle \mathbf u = \frac{\partial T}{\partial t}\circ T^{-1} = \frac{\partial \nabla P}{\partial t}\circ \nabla P^*
\displaystyle P(t,\mathbf x) = p(t,\mathbf x) + \frac{|\mathbf x|^2}{2}

Change of variables:

(assume $$P$$ convex)

\displaystyle \rho\,d\mathbf x = (\nabla P)_*d\mathbf x

Key: Think of $$T=\nabla P$$ as transport map

Apply Arnold's approach:

right translation of $$\partial_t T$$ gives vector field

\displaystyle \mathbf u = J(\nabla P^*-\mathbf x)

By construction

\displaystyle \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf u) = 0

## Dual SGE

\displaystyle \rho = \det(D^2 P^*) \iff \rho = \text{MA}(P^*)
\displaystyle \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf u) = 0
\displaystyle \rho\,d\mathbf x = (\nabla P)_*d\mathbf x = (\nabla P^*)^*d\mathbf x
\displaystyle \mathbf u = J(\nabla P^*-\mathbf x)

Close relation to vorticity equation

\displaystyle \frac{\partial \omega}{\partial t} + \nabla\cdot(\omega \mathbf u) = 0
\displaystyle \mathbf u = J\nabla \psi
\displaystyle \omega = \Delta\psi
\displaystyle \rho \sim \omega + 1
\displaystyle P^* \sim \psi+\frac{|\mathbf x|^2}{2}
\displaystyle \text{MA}(P^*) = \det(I + D^2\psi)
\displaystyle \approx 1 + \Delta\psi

## Problem with existence for SGE

Theorem (Benamou and Brenier 1998):

Dual SGE admits global weak solution

\displaystyle\frac{\partial \nabla p}{\partial t} + (\mathbf v\cdot \nabla) \nabla p = -J\nabla p - \mathbf v
\displaystyle \nabla \cdot\mathbf v = 0

Recall: SGE

Does weak dual SGE solution give solution to SGE?

How is $$p,\mathbf v$$ reconstructed from $$\rho,P^*$$ ?

\displaystyle p = P -\frac{|\mathbf x|^2}{2}
\displaystyle \mathbf v = - D^2P^*\circ\nabla P\Big(\partial_t\nabla P+ J(\nabla P - \mathbf x) \Big)

Easy!

Problem: no meaning a priori

## Existence theorem for SGE

Theorem (Ambrosio, Colombo, De Philippis, Figalli 2012):

$$(\rho,P^*)$$ weak solution to dual SGE on $$\mathbb T^2$$ with initial data fulfilling

• $$P_0^*$$ convex
• $$0 < \lambda \leq \rho_0 \leq \Lambda < \infty$$

Then $\mathbf v = - D^2P^*\circ\nabla P\Big(\partial_t\nabla P+ J(\nabla P - \mathbf x) \Big)$ is well-defined weakly and $$(\mathbf v,p=P -\frac{|\mathbf x|^2}{2})$$ is weak solution to SGE

Proof uses:

• Condition $$(*)$$ retained through time ($$\rho$$ is transported)
• $$\lambda \leq \det(D^2P) \leq \Lambda \Rightarrow P\in W^{2,1}$$ by De Philippis and Figalli

$$(*)$$

# THANKS!

References

• G. De Philippis, A. Figalli
$$W^{2,1}$$ regularity for solutions of the Monge-Ampere equation
Invent. Math. (2013)

• L. Ambrosio, M. Colombo, G. De Philippis, A. Figalli
Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case
Comm. Partial Differential Equations (2012)

Slides available at: slides.com/kmodin

By Klas Modin

# Glimpse into the work of Alessio Figalli

Seminar series about Fields Medalists 2018 at the Department of Mathematical Sciences at Chalmers and GU.

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