generalized flows for the euler model of incompressible

Generalized flows for the Euler model ofincompressible fluids and resolution of the initial

value problem by convex minimization

Yann BrenierCNRS, Centre de mathématiques LAURENT SCHWARTZ,

ECOLE POLYTECHNIQUE, Palaiseau, France

En l’honneur de Patrick CATTIAUX et Christian LEONARDANR STAB, Toulouse, 6-9 Juin 2017

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 1 / 31

OUTLINE OF THE LECTURE:

Geometric interpretation of the Euler equations forincompressible fluids after V.I. Arnold 1966.From combinatorics to generalizedincompressible flows.Generalized least action principles,with probability and convexity tools.(REVIEW: 1989-2012).What about the initial value problem? (NEW!)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 2 / 31

OUTLINE OF THE LECTURE:Geometric interpretation of the Euler equations forincompressible fluids after V.I. Arnold 1966.

From combinatorics to generalizedincompressible flows.Generalized least action principles,with probability and convexity tools.(REVIEW: 1989-2012).What about the initial value problem? (NEW!)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 2 / 31

OUTLINE OF THE LECTURE:Geometric interpretation of the Euler equations forincompressible fluids after V.I. Arnold 1966.From combinatorics to generalizedincompressible flows.

Generalized least action principles,with probability and convexity tools.(REVIEW: 1989-2012).What about the initial value problem? (NEW!)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 2 / 31

OUTLINE OF THE LECTURE:Geometric interpretation of the Euler equations forincompressible fluids after V.I. Arnold 1966.From combinatorics to generalizedincompressible flows.Generalized least action principles,with probability and convexity tools.(REVIEW: 1989-2012).

What about the initial value problem? (NEW!)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 2 / 31

OUTLINE OF THE LECTURE:Geometric interpretation of the Euler equations forincompressible fluids after V.I. Arnold 1966.From combinatorics to generalizedincompressible flows.Generalized least action principles,with probability and convexity tools.(REVIEW: 1989-2012).What about the initial value problem? (NEW!)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 2 / 31

Euler, 200 BC (*)

BC(*) Before Christian...

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 3 / 31

Euler, 200 BC (*)

BC(*) Before Christ

ian...

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 3 / 31

Euler, 200 BC (*)

BC(*) Before Christian...

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 3 / 31

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 4 / 31

GEOMETRY OF THE EULER MODEL OFINCOMPRESSIBLE FLOWS.

According to V.I. Arnold 1966, an incompressiblefluid, confined in a domain D and movingaccording to the Euler equations, just follows a(constant speed) geodesic curve along themanifold of all possible incompressible maps of D.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 5 / 31

GEOMETRY OF THE EULER MODEL OFINCOMPRESSIBLE FLOWS.According to V.I. Arnold 1966, an incompressiblefluid, confined in a domain D and movingaccording to the Euler equations, just follows a(constant speed) geodesic curve along themanifold of all possible incompressible maps of D.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 5 / 31

0

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POLAR FACTORIZATION OF A PERIODIC MAP

Three maps of the (periodized) square: only one is incompressible.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 6 / 31

From a more concrete and computationalviewpiont, it is worth considering the discreteversion of an incompressible motion inside D:

the permutation of N sub-cells of equal volume.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 7 / 31

From a more concrete and computationalviewpiont, it is worth considering the discreteversion of an incompressible motion inside D:the permutation of N sub-cells of equal volume.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 7 / 31

From Combinatorics to Fluids (Thanks to Mirko Rokyta, Charles University)!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 8 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 12

2 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 11

2 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 11

4 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 9

6 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 7

6 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 7

8 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 5

8 10 6 12 4 11 2 9 1 7 3 57 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

Example of a discrete incompressible motionwith 7 time steps and 12 sub-cells (in line)

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

7 time steps have been performed.Time is on vertical axis and space on horizontal axis.The trajectories of 2 selected sub-cells (4 and 5) are drawn in red.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 9 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 12

2 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 11

2 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 11

4 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 9

6 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 7

6 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 7

8 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 5

8 10 6 12 4 11 2 9 1 7 3 5The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.

This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

"transportation cost" to reach the finalpermutation

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 5

The "cost" is obtained by adding up the squares of alldisplacements at all steps. Here: 12+10+12+42+10+12+10=108.This is the "cost" to reach the final permutation in 7 steps. Noticethat step 4 costs a lot!

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 10 / 31

Obviously, there is at least a solution leading to thefinal permutation at the lowest possible cost, among

the... (12!)6 ∼ 1052 possible candidates!

This is the discrete version of a minimizing geodesicalong the semi-group of all volume preserving maps.

Presumably, passing to the limit (in the number ofcubes and steps), we should recover the motion of

an incompressible fluid obeying the Euler equations.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 11 / 31

Obviously, there is at least a solution leading to thefinal permutation at the lowest possible cost, among

the... (12!)6 ∼ 1052 possible candidates!This is the discrete version of a minimizing geodesicalong the semi-group of all volume preserving maps.

Presumably, passing to the limit (in the number ofcubes and steps), we should recover the motion of

an incompressible fluid obeying the Euler equations.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 11 / 31

Obviously, there is at least a solution leading to thefinal permutation at the lowest possible cost, among

the... (12!)6 ∼ 1052 possible candidates!This is the discrete version of a minimizing geodesicalong the semi-group of all volume preserving maps.

Presumably, passing to the limit (in the number ofcubes and steps), we should recover the motion of

an incompressible fluid obeying the Euler equations.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 11 / 31

Exercise: let us try to find a discrete geodesicleading to permutation 12-11-10-9-8-7-6-5-4-3-2-1using twelve steps

1 2 3 4 5 6 7 8 9 10 11 12

12 11 10 9 8 7 6 5 4 3 2 1

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 12 / 31

LET US TRY TO MOVE BY EXCHANGINGNEIGHBORS...

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 11

12 11 10 9 8 7 6 5 4 3 2 1

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 13 / 31

FINALLY ARRIVED...AFTER 12 STEPS.

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 94 6 2 8 1 10 3 12 5 11 7 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 510 8 12 6 11 4 9 2 7 1 5 310 12 8 11 6 9 4 7 2 5 1 312 10 11 8 9 6 7 4 5 2 3 112 11 10 9 8 7 6 5 4 3 2 1

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 14 / 31

LET US FOLLOW THE TRAJECTORIES OF TWONEIGHBOURS: 4 AND 5

1 2 3 4 5 6 7 8 9 10 11 122 1 4 3 6 5 8 7 10 9 12 112 4 1 6 3 8 5 10 7 12 9 114 2 6 1 8 3 10 5 12 7 11 94 6 2 8 1 10 3 12 5 11 7 96 4 8 2 10 1 12 3 11 5 9 76 8 4 10 2 12 1 11 3 9 5 78 6 10 4 12 2 11 1 9 3 7 58 10 6 12 4 11 2 9 1 7 3 510 8 12 6 11 4 9 2 7 1 5 310 12 8 11 6 9 4 7 2 5 1 312 10 11 8 9 6 7 4 5 2 3 1

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 15 / 31

Is it really the lowest possible cost?

◦ ◦ ◦ O O ◦ ◦ ◦ ◦ ◦ ◦ ◦

◦ ◦ O ◦ ◦ O ◦ ◦ ◦ ◦ ◦ ◦

◦ O ◦ ◦ ◦ ◦ O ◦ ◦ ◦ ◦ ◦

O ◦ ◦ ◦ ◦ ◦ ◦ O ◦ ◦ ◦ ◦

O ◦ ◦ ◦ ◦ ◦ ◦ ◦ O ◦ ◦ ◦

◦ O ◦ ◦ ◦ ◦ ◦ ◦ ◦ O ◦ ◦

◦ ◦ O ◦ ◦ ◦ ◦ ◦ ◦ ◦ O ◦

◦ ◦ ◦ O ◦ ◦ ◦ ◦ ◦ ◦ ◦ O◦ ◦ ◦ ◦ O ◦ ◦ ◦ ◦ ◦ ◦ O◦ ◦ ◦ ◦ ◦ O ◦ ◦ ◦ ◦ O ◦

◦ ◦ ◦ ◦ ◦ ◦ O ◦ ◦ O ◦ ◦

◦ ◦ ◦ ◦ ◦ ◦ ◦ O O ◦ ◦ ◦

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 16 / 31

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’fort.12’

ANYWAY, IT IS EASY TO "PASS TO THE LIMIT"

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 17 / 31

AS A MATTER OF FACT, THIS IS NOT THE BESTSOLUTION. THE COST CAN BE REDUCED BYFACTOR π2/12 ∼ 0.8225

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 18 / 31

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CALCUL APPROCHE AVEC 4000 CUBES ET 16 ETAPES

’fort.10’

NUMERICS WITH 4000 CUBES AND 16 STEPS

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 19 / 31

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SOLUTION EXACTE

’fort.12’

EXACT SOLUTION (30 AC)

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 20 / 31

Incompressible fluids: a probabilist description

A generalized incompressible flow on a compactdomain D is defined as a probability measure µ onpaths t ∈ [0,T ]→ ξt ∈ D, such that:

i) µ has finite energy (*): Eµ∫ T

012 |

dξtdt |

2dt < +∞,ii) (incompressibility) for each t ∈ [0,T ], the projectionµt is the (normalized) Lebesgue measure LD on D.

(*) Of course, such measures are very different from Wiener measures.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 21 / 31

Incompressible fluids: a probabilist descriptionA generalized incompressible flow on a compactdomain D is defined as a probability measure µ onpaths t ∈ [0,T ]→ ξt ∈ D, such that:

i) µ has finite energy (*): Eµ∫ T

012 |

dξtdt |

2dt < +∞,

ii) (incompressibility) for each t ∈ [0,T ], the projectionµt is the (normalized) Lebesgue measure LD on D.

(*) Of course, such measures are very different from Wiener measures.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 21 / 31

Incompressible fluids: a probabilist descriptionA generalized incompressible flow on a compactdomain D is defined as a probability measure µ onpaths t ∈ [0,T ]→ ξt ∈ D, such that:

i) µ has finite energy (*): Eµ∫ T

012 |

dξtdt |

2dt < +∞,ii) (incompressibility) for each t ∈ [0,T ], the projectionµt is the (normalized) Lebesgue measure LD on D.

(*) Of course, such measures are very different from Wiener measures.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 21 / 31

Generalized solutions to the Euler model

We say that a generalized incompressible flow (GIF) µsolves the Euler model if there is a scalar field pdefined on ]0,T [×D, sufficiently smooth, such that,µ−a.s., every path ξ satisfies

d2ξt

dt2 = −(∇p)(t , ξt), ∀t ∈]0,T [.

Of course, each classical solution V (t , x) generates a generalized solution µ by

∀Φ, EµΦ[t → ξt ] =

∫D

Φ[t → Xt (a)]da; ∂tXt (a) = V (t ,Xt (a)), X0(a) = a, ∀a ∈ D.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 22 / 31

Generalized solutions to the Euler modelWe say that a generalized incompressible flow (GIF) µsolves the Euler model if there is a scalar field pdefined on ]0,T [×D, sufficiently smooth, such that,µ−a.s., every path ξ satisfies

d2ξt

dt2 = −(∇p)(t , ξt), ∀t ∈]0,T [.

Of course, each classical solution V (t , x) generates a generalized solution µ by

∀Φ, EµΦ[t → ξt ] =

∫D

Φ[t → Xt (a)]da; ∂tXt (a) = V (t ,Xt (a)), X0(a) = a, ∀a ∈ D.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 22 / 31

Generalized solutions to the Euler modelWe say that a generalized incompressible flow (GIF) µsolves the Euler model if there is a scalar field pdefined on ]0,T [×D, sufficiently smooth, such that,µ−a.s., every path ξ satisfies

d2ξt

dt2 = −(∇p)(t , ξt), ∀t ∈]0,T [.

Of course, each classical solution V (t , x) generates a generalized solution µ by

∀Φ, EµΦ[t → ξt ] =

∫D

Φ[t → Xt (a)]da; ∂tXt (a) = V (t ,Xt (a)), X0(a) = a, ∀a ∈ D.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 22 / 31

The generalized least action principle (31 AC)

Let D be a convex body and (µ,p) be a generalizedsolution to the Euler model.Assume D2

xp(t , x) ≤ rId uniformly for some finite r .

Then, for all 0 ≤ t0 < t1 ≤ T , µ uniquely minimizes

Eµ∫ t1

t0

12|dξt

dt|2dt ,

among all generalized incompressible flows (GIF) µ̃with same end-points: µ̃t0,t1 = µt0,t1, as (t1 − t0)2r < π2.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 23 / 31

The generalized least action principle (31 AC)Let D be a convex body and (µ,p) be a generalizedsolution to the Euler model.Assume D2

xp(t , x) ≤ rId uniformly for some finite r .

Then, for all 0 ≤ t0 < t1 ≤ T , µ uniquely minimizes

Eµ∫ t1

t0

12|dξt

dt|2dt ,

among all generalized incompressible flows (GIF) µ̃with same end-points: µ̃t0,t1 = µt0,t1, as (t1 − t0)2r < π2.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 23 / 31

The generalized least action principle (31 AC)Let D be a convex body and (µ,p) be a generalizedsolution to the Euler model.Assume D2

xp(t , x) ≤ rId uniformly for some finite r .

Then, for all 0 ≤ t0 < t1 ≤ T , µ uniquely minimizes

Eµ∫ t1

t0

12|dξt

dt|2dt ,

among all generalized incompressible flows (GIF) µ̃with same end-points: µ̃t0,t1 = µt0,t1, as (t1 − t0)2r < π2.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 23 / 31

The converse part of the action principle (35 AC)

Let µ0,T be a probability measure on D × D such thatµ0 = µT = LD. Then, µ0,T is always achieved by ageneralized incompressible flow µ of minimal energyand there is a unique distribution ∇p(t , x) such that

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A >, ∀A ∈ D(int([0,T ]× D)).

NB. There is no such result in the classical framework (Shnirelman1985).

Proof: treat the problem as a continuous multi-marginal Monge-Kantorovich problem.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 24 / 31

The converse part of the action principle (35 AC)Let µ0,T be a probability measure on D × D such thatµ0 = µT = LD.

Then, µ0,T is always achieved by ageneralized incompressible flow µ of minimal energyand there is a unique distribution ∇p(t , x) such that

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A >, ∀A ∈ D(int([0,T ]× D)).

NB. There is no such result in the classical framework (Shnirelman1985).

Proof: treat the problem as a continuous multi-marginal Monge-Kantorovich problem.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 24 / 31

The converse part of the action principle (35 AC)Let µ0,T be a probability measure on D × D such thatµ0 = µT = LD. Then, µ0,T is always achieved by ageneralized incompressible flow µ of minimal energy

and there is a unique distribution ∇p(t , x) such that

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A >, ∀A ∈ D(int([0,T ]× D)).

NB. There is no such result in the classical framework (Shnirelman1985).

Proof: treat the problem as a continuous multi-marginal Monge-Kantorovich problem.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 24 / 31

The converse part of the action principle (35 AC)Let µ0,T be a probability measure on D × D such thatµ0 = µT = LD. Then, µ0,T is always achieved by ageneralized incompressible flow µ of minimal energyand there is a unique distribution ∇p(t , x) such that

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A >, ∀A ∈ D(int([0,T ]× D)).

NB. There is no such result in the classical framework (Shnirelman1985).

Proof: treat the problem as a continuous multi-marginal Monge-Kantorovich problem.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 24 / 31

Euler, Monge, Kantorovich

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 25 / 31

Regularity of the pressure gradient (41-54 AC)

In the proof, the existence of a unique ∇p requiresspecial efforts (with respect to more standard optimaltransport or MFG problems).Main open question: is p semi-concave? So far, weonly know (*) ∇p ∈ L2

loc(Mloc) and we have examplewhere p is not better than locally semi-concave.(*) after Y.B. CPAM 1999, square integrability in time due to Ambrosio-Figalli 2008.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 26 / 31

Regularity of the pressure gradient (41-54 AC)In the proof, the existence of a unique ∇p requiresspecial efforts (with respect to more standard optimaltransport or MFG problems).Main open question: is p semi-concave? So far, weonly know (*) ∇p ∈ L2

loc(Mloc) and we have examplewhere p is not better than locally semi-concave.(*) after Y.B. CPAM 1999, square integrability in time due to Ambrosio-Figalli 2008.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 26 / 31

SOME REFERENCES...1) Euler’s equations of incompressible fluidsL. Euler, opera omnia, seria secunda 12, p. 274,Arnold, Ann. Fourier 1966, Ebin-Marsden, Ann. Maths 1970,Shnirelman Math USSR Sb. 1987. Books:Marchioro-Pulvirenti 1994, Lions 1996, Arnold-Khesin 1998...

2) Generalized incompressible flowsY.B.: JAMS 1990, ARMA 1993, CPAM 1999, CVPDE 2012,Shnirelman: GAFA 1994, Ambrosio-Figalli: Arma 2008, CVPDE2008, Bernot-Figalli-Santambrogio, IHP NL 2009.3) Entropic (Schrödinger) regularizationArnaudon, Cruzeiro, Fang, Léonard, Zambrini...—>Ask Christian!4) NumericsY. B.: Comp. Mech. 1989, Physica D 2008 (combinatorics),Mérigot-Mirebeau arXiv-2015, Gallouët-Mérigot arXiv-2016(Monge-Ampère solver),Nenna’s PhD 2016 (entropic regularization).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 27 / 31

SOME REFERENCES...1) Euler’s equations of incompressible fluidsL. Euler, opera omnia, seria secunda 12, p. 274,Arnold, Ann. Fourier 1966, Ebin-Marsden, Ann. Maths 1970,Shnirelman Math USSR Sb. 1987. Books:Marchioro-Pulvirenti 1994, Lions 1996, Arnold-Khesin 1998...2) Generalized incompressible flowsY.B.: JAMS 1990, ARMA 1993, CPAM 1999, CVPDE 2012,Shnirelman: GAFA 1994, Ambrosio-Figalli: Arma 2008, CVPDE2008, Bernot-Figalli-Santambrogio, IHP NL 2009.

3) Entropic (Schrödinger) regularizationArnaudon, Cruzeiro, Fang, Léonard, Zambrini...—>Ask Christian!4) NumericsY. B.: Comp. Mech. 1989, Physica D 2008 (combinatorics),Mérigot-Mirebeau arXiv-2015, Gallouët-Mérigot arXiv-2016(Monge-Ampère solver),Nenna’s PhD 2016 (entropic regularization).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 27 / 31

SOME REFERENCES...1) Euler’s equations of incompressible fluidsL. Euler, opera omnia, seria secunda 12, p. 274,Arnold, Ann. Fourier 1966, Ebin-Marsden, Ann. Maths 1970,Shnirelman Math USSR Sb. 1987. Books:Marchioro-Pulvirenti 1994, Lions 1996, Arnold-Khesin 1998...2) Generalized incompressible flowsY.B.: JAMS 1990, ARMA 1993, CPAM 1999, CVPDE 2012,Shnirelman: GAFA 1994, Ambrosio-Figalli: Arma 2008, CVPDE2008, Bernot-Figalli-Santambrogio, IHP NL 2009.3) Entropic (Schrödinger) regularizationArnaudon, Cruzeiro, Fang, Léonard, Zambrini...—>Ask Christian!

4) NumericsY. B.: Comp. Mech. 1989, Physica D 2008 (combinatorics),Mérigot-Mirebeau arXiv-2015, Gallouët-Mérigot arXiv-2016(Monge-Ampère solver),Nenna’s PhD 2016 (entropic regularization).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 27 / 31

SOME REFERENCES...1) Euler’s equations of incompressible fluidsL. Euler, opera omnia, seria secunda 12, p. 274,Arnold, Ann. Fourier 1966, Ebin-Marsden, Ann. Maths 1970,Shnirelman Math USSR Sb. 1987. Books:Marchioro-Pulvirenti 1994, Lions 1996, Arnold-Khesin 1998...2) Generalized incompressible flowsY.B.: JAMS 1990, ARMA 1993, CPAM 1999, CVPDE 2012,Shnirelman: GAFA 1994, Ambrosio-Figalli: Arma 2008, CVPDE2008, Bernot-Figalli-Santambrogio, IHP NL 2009.3) Entropic (Schrödinger) regularizationArnaudon, Cruzeiro, Fang, Léonard, Zambrini...—>Ask Christian!4) NumericsY. B.: Comp. Mech. 1989, Physica D 2008 (combinatorics),Mérigot-Mirebeau arXiv-2015, Gallouët-Mérigot arXiv-2016(Monge-Ampère solver),Nenna’s PhD 2016 (entropic regularization).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 27 / 31

What about the initial value problem?A priori, convex minimization techniques are hopelessfor the initial value problem (IVP).For a generalized incompressible flow (GIF) µ with finite energy Eµ

∫ T0

12 |

dξtdt |

2dt < +∞,

it does not make sense to prescribe the initial velocity dξtdt |t=0 for µ−a.e. paths ξ.

However, we are going to see that the initial bulkvelocity makes sense for a GIF of minimal energy.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 28 / 31

What about the initial value problem?A priori, convex minimization techniques are hopelessfor the initial value problem (IVP).For a generalized incompressible flow (GIF) µ with finite energy Eµ

∫ T0

12 |

dξtdt |

2dt < +∞,

it does not make sense to prescribe the initial velocity dξtdt |t=0 for µ−a.e. paths ξ.

However, we are going to see that the initial bulkvelocity makes sense for a GIF of minimal energy.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 28 / 31

Initial bulk velocity of a GIF of minimal energyFor a GIF µ of minimal energy, we already know

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A > ∀A ∈ D(int([0,T ]× D)).

This suggests to define the initial bulk velocity V0 by

< V0,A(0, ·) >= Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt ,

where A are test fields chosen to be divergence-free(in order to kill the pressure term) without vanishing att = 0 (to be able to "feel" V0).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 29 / 31

Initial bulk velocity of a GIF of minimal energyFor a GIF µ of minimal energy, we already know

Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt =< ∇p,A > ∀A ∈ D(int([0,T ]× D)).

This suggests to define the initial bulk velocity V0 by

< V0,A(0, ·) >= Eµ∫ T

0

(dξt

dt∂tA(t , ξt ) + (

dξt

dt⊗ dξt

dt) · ∇A(t , ξt )

)dt ,

where A are test fields chosen to be divergence-free(in order to kill the pressure term) without vanishing att = 0 (to be able to "feel" V0).

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 29 / 31

A minimization problem for the IVP (59 AC)Given a divergence-free velocity field V0, we look for aGIF µ of minimal energy with initial bulk velocity V0.

This reads as the saddle-point problem:

infµ≥0

supA,ϕ

< V0,A(0, ·) > +

Eµ∫ T

0(12|dξt

dt|2 − (

dξt

dt⊗ dξt

dt) · B(t , ξt)−

dξt

dt· E(t , ξt))dt

where E = ∂tA +∇ϕ, B = 12 (∇A +∇AT ), subject to ∇ · A = 0, A(T , ·) = 0 (to take

care of V0), while ϕ enforces the incompressibility condition.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 30 / 31

A minimization problem for the IVP (59 AC)Given a divergence-free velocity field V0, we look for aGIF µ of minimal energy with initial bulk velocity V0.This reads as the saddle-point problem:

infµ≥0

supA,ϕ

< V0,A(0, ·) > +

Eµ∫ T

0(12|dξt

dt|2 − (

dξt

dt⊗ dξt

dt) · B(t , ξt)−

dξt

dt· E(t , ξt))dt

where E = ∂tA +∇ϕ, B = 12 (∇A +∇AT ), subject to ∇ · A = 0, A(T , ·) = 0 (to take

care of V0), while ϕ enforces the incompressibility condition.

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 30 / 31

The dual (convex) minimization problem

inf(E ,B)

∫[0,T ]×D

E · (2Id + 4B)−1 · E + V0 · E

subject to B(t = T , ·) = 0, ∂tBij =12

(∂jEi + ∂iEj ) + ∂i∂j (−4)−1∂k Ek .

is always solvable, uniquely recovers smooth classicalsolutions to the Euler equations for short enough T ,and looks similar to the MK2 optimal transport problem

infρ,Q

∫[0,T ]×D

Q · ρ−1 ·Q, subject to ∂tρ+ ∂iQ i = 0, ρ prescribed at t = 0,T .

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 31 / 31

The dual (convex) minimization problem

inf(E ,B)

∫[0,T ]×D

E · (2Id + 4B)−1 · E + V0 · E

subject to B(t = T , ·) = 0, ∂tBij =12

(∂jEi + ∂iEj ) + ∂i∂j (−4)−1∂k Ek .

is always solvable

, uniquely recovers smooth classicalsolutions to the Euler equations for short enough T ,and looks similar to the MK2 optimal transport problem

infρ,Q

∫[0,T ]×D

Q · ρ−1 ·Q, subject to ∂tρ+ ∂iQ i = 0, ρ prescribed at t = 0,T .

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 31 / 31

The dual (convex) minimization problem

inf(E ,B)

∫[0,T ]×D

E · (2Id + 4B)−1 · E + V0 · E

subject to B(t = T , ·) = 0, ∂tBij =12

(∂jEi + ∂iEj ) + ∂i∂j (−4)−1∂k Ek .

is always solvable, uniquely recovers smooth classicalsolutions to the Euler equations for short enough T

,and looks similar to the MK2 optimal transport problem

infρ,Q

∫[0,T ]×D

Q · ρ−1 ·Q, subject to ∂tρ+ ∂iQ i = 0, ρ prescribed at t = 0,T .

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 31 / 31

The dual (convex) minimization problem

inf(E ,B)

∫[0,T ]×D

E · (2Id + 4B)−1 · E + V0 · E

subject to B(t = T , ·) = 0, ∂tBij =12

(∂jEi + ∂iEj ) + ∂i∂j (−4)−1∂k Ek .

is always solvable, uniquely recovers smooth classicalsolutions to the Euler equations for short enough T ,and looks similar to the MK2 optimal transport problem

infρ,Q

∫[0,T ]×D

Q · ρ−1 ·Q, subject to ∂tρ+ ∂iQ i = 0, ρ prescribed at t = 0,T .

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 31 / 31

The dual (convex) minimization problem

inf(E ,B)

∫[0,T ]×D

E · (2Id + 4B)−1 · E + V0 · E

subject to B(t = T , ·) = 0, ∂tBij =12

(∂jEi + ∂iEj ) + ∂i∂j (−4)−1∂k Ek .

is always solvable, uniquely recovers smooth classicalsolutions to the Euler equations for short enough T ,and looks similar to the MK2 optimal transport problem

infρ,Q

∫[0,T ]×D

Q · ρ−1 ·Q, subject to ∂tρ+ ∂iQ i = 0, ρ prescribed at t = 0,T .

Yann Brenier (CNRS) Generalized solutions to the Euler model TOULOUSE 6-9 June 2017 31 / 31