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A gradient flow model in the space of signed measures
Edoardo Mainini
` Dipar timento di Matematica F. Casorati, Universita degli Studi di Pavia

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

1 / 22


Evolution model for ginzburg-landau vor tices
Mean field model for the evolution of the vor tex densities in a superconductor, derived by W. E (1994), Lin and Zhang (2000): d µ(t ) - div( h dt -hµ
(t ) µ(t )

µ(t )) = 0

in R2 ,

= µ(t ) in R2 .

Model proposed by Chapman, Rubinstein and Schatzman (1996) d µ(t ) - div( h dt
µ(t )

|µ(t )|) = 0,

in

(CRS)

-hµ + hµ = µ in hµ = 1 on . The model involves signed measures.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 2 / 22


Evolution model for ginzburg-landau vor tices
Mean field model for the evolution of the vor tex densities in a superconductor, derived by W. E (1994), Lin and Zhang (2000): d µ(t ) - div( h dt -hµ
(t ) µ(t )

µ(t )) = 0

in R2 ,

= µ(t ) in R2 .

Model proposed by Chapman, Rubinstein and Schatzman (1996) d µ(t ) - div( h dt
µ(t )

|µ(t )|) = 0,

in

(CRS)

-hµ + hµ = µ in hµ = 1 on . The model involves signed measures.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 2 / 22


Evolution model for ginzburg-landau vor tices
Mean field model for the evolution of the vor tex densities in a superconductor, derived by W. E (1994), Lin and Zhang (2000): d µ(t ) - div( h dt -hµ
(t ) µ(t )

µ(t )) = 0

in R2 ,

= µ(t ) in R2 .

Model proposed by Chapman, Rubinstein and Schatzman (1996) d µ(t ) - div( h dt
µ(t )

|µ(t )|) = 0,

in

(CRS)

-hµ + hµ = µ in hµ = 1 on . The model involves signed measures.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 2 / 22


Evolution model for ginzburg-landau vor tices
Mean field model for the evolution of the vor tex densities in a superconductor, derived by W. E (1994), Lin and Zhang (2000): d µ(t ) - div( h dt -hµ
(t ) µ(t )

µ(t )) = 0

in R2 ,

= µ(t ) in R2 .

Model proposed by Chapman, Rubinstein and Schatzman (1996) d µ(t ) - div( h dt
µ(t )

|µ(t )|) = 0,

in

(CRS)

-hµ + hµ = µ in hµ = 1 on . The model involves signed measures.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 2 / 22


Evolution model for ginzburg-landau vor tices

References
W. E : Dynamics of vor tex-liquids in Ginzburg-Landau theories with applications to superconductivity, Phys. Rev. B 50 (1994), no. 3, 1126-1135. J. S. C H A P M A N , J. RU B I N S T E I N , A N D M . S C H AT Z M A N : A mean-field model for superconducting vor tices, Eur. J. Appl. Math. 7 (1996), no. 2, 97­111. F. H . L I N A N D P. Z H A N G : On the hydrodynamic limit of Ginzburg-Landau vor tices. Discrete cont. dyn. systems 6 (2000), 121­142.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

3 / 22


The positive measure setting
Suppose first that the vor tex density µ is a positive measure. In this case we search for solutions in H -1 () P2 (), where P2 () is the space of probability measures over with finite second moment. The basic idea is to view a solution to (CRS) as a steepest descent curve in P2 () of the related energy: (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,


0.

Formally find a curve t µ(t ) such that µ(t ) = - (µ(t )).

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

4 / 22


The positive measure setting
Suppose first that the vor tex density µ is a positive measure. In this case we search for solutions in H -1 () P2 (), where P2 () is the space of probability measures over with finite second moment. The basic idea is to view a solution to (CRS) as a steepest descent curve in P2 () of the related energy: (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,


0.

Formally find a curve t µ(t ) such that µ(t ) = - (µ(t )).

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

4 / 22


The positive measure setting
Suppose first that the vor tex density µ is a positive measure. In this case we search for solutions in H -1 () P2 (), where P2 () is the space of probability measures over with finite second moment. The basic idea is to view a solution to (CRS) as a steepest descent curve in P2 () of the related energy: (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,


0.

Formally find a curve t µ(t ) such that µ(t ) = - (µ(t )).

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

4 / 22


The positive measure setting
Suppose first that the vor tex density µ is a positive measure. In this case we search for solutions in H -1 () P2 (), where P2 () is the space of probability measures over with finite second moment. The basic idea is to view a solution to (CRS) as a steepest descent curve in P2 () of the related energy: (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,


0.

Formally find a curve t µ(t ) such that µ(t ) = - (µ(t )).

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

4 / 22


Optimal transpor tation distance
Kantorovich optimal transpor tation problem inf
X âX 1 2 Transpor t plans: (µ, ) (i.e. P (X â X ), # = µ, # = ).

|x - y |2 d (x , y ) : (µ, ) .

Optimal plans set: 0 (µ, ). Plan induced by a map: = (I, t)# µ. Optimal transpor t distance (Wasserstein distance) W2 (µ, ) := inf
X âX

|x - y | d (x , y ) : (µ, )

2

1 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

5 / 22


Optimal transpor tation distance
Kantorovich optimal transpor tation problem inf
X âX 1 2 Transpor t plans: (µ, ) (i.e. P (X â X ), # = µ, # = ).

|x - y |2 d (x , y ) : (µ, ) .

Optimal plans set: 0 (µ, ). Plan induced by a map: = (I, t)# µ. Optimal transpor t distance (Wasserstein distance) W2 (µ, ) := inf
X âX

|x - y | d (x , y ) : (µ, )

2

1 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

5 / 22


Optimal transpor tation distance
Kantorovich optimal transpor tation problem inf
X âX 1 2 Transpor t plans: (µ, ) (i.e. P (X â X ), # = µ, # = ).

|x - y |2 d (x , y ) : (µ, ) .

Optimal plans set: 0 (µ, ). Plan induced by a map: = (I, t)# µ. Optimal transpor t distance (Wasserstein distance) W2 (µ, ) := inf
X âX

|x - y | d (x , y ) : (µ, )

2

1 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

5 / 22


Optimal transpor tation distance
Kantorovich optimal transpor tation problem inf
X âX 1 2 Transpor t plans: (µ, ) (i.e. P (X â X ), # = µ, # = ).

|x - y |2 d (x , y ) : (µ, ) .

Optimal plans set: 0 (µ, ). Plan induced by a map: = (I, t)# µ. Optimal transpor t distance (Wasserstein distance) W2 (µ, ) := inf
X âX

|x - y | d (x , y ) : (µ, )

2

1 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

5 / 22


Optimal transpor tation distance
Kantorovich optimal transpor tation problem inf
X âX 1 2 Transpor t plans: (µ, ) (i.e. P (X â X ), # = µ, # = ).

|x - y |2 d (x , y ) : (µ, ) .

Optimal plans set: 0 (µ, ). Plan induced by a map: = (I, t)# µ. Optimal transpor t distance (Wasserstein distance) W2 (µ, ) := inf
X âX

|x - y | d (x , y ) : (µ, )

2

1 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

5 / 22


Gradient flow in the space (P2 (X ), W2 )
Jordan-Kinderlehrer-Otto (1998) framework: Consider a functional : P2 (X ) R and a PDE of the form t µt + div (vt µt ) = 0. Given µ0 P2 (X ) and a time step > 0, find recursively µk among solutions of min ( ) + 12 W (, µk 2 2
-1

P2 (X )

),

µ0 = µ0 .

Construct a curve t [0, T ] µ(t ) P2 (X ) interpolating the discrete values and passing to the limit as 0. Show that the obtained limit curve satisfies the continuity equation.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

6 / 22


Gradient flow in the space (P2 (X ), W2 )
Jordan-Kinderlehrer-Otto (1998) framework: Consider a functional : P2 (X ) R and a PDE of the form t µt + div (vt µt ) = 0. Given µ0 P2 (X ) and a time step > 0, find recursively µk among solutions of min ( ) + 12 W (, µk 2 2
-1

P2 (X )

),

µ0 = µ0 .

Construct a curve t [0, T ] µ(t ) P2 (X ) interpolating the discrete values and passing to the limit as 0. Show that the obtained limit curve satisfies the continuity equation.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

6 / 22


Gradient flow in the space (P2 (X ), W2 )
Jordan-Kinderlehrer-Otto (1998) framework: Consider a functional : P2 (X ) R and a PDE of the form t µt + div (vt µt ) = 0. Given µ0 P2 (X ) and a time step > 0, find recursively µk among solutions of min ( ) + 12 W (, µk 2 2
-1

P2 (X )

),

µ0 = µ0 .

Construct a curve t [0, T ] µ(t ) P2 (X ) interpolating the discrete values and passing to the limit as 0. Show that the obtained limit curve satisfies the continuity equation.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

6 / 22


Gradient flow in the space (P2 (X ), W2 )
Jordan-Kinderlehrer-Otto (1998) framework: Consider a functional : P2 (X ) R and a PDE of the form t µt + div (vt µt ) = 0. Given µ0 P2 (X ) and a time step > 0, find recursively µk among solutions of min ( ) + 12 W (, µk 2 2
-1

P2 (X )

),

µ0 = µ0 .

Construct a curve t [0, T ] µ(t ) P2 (X ) interpolating the discrete values and passing to the limit as 0. Show that the obtained limit curve satisfies the continuity equation.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

6 / 22


Gradient flow in the space (P2 (X ), W2 )
Jordan-Kinderlehrer-Otto (1998) framework: Consider a functional : P2 (X ) R and a PDE of the form t µt + div (vt µt ) = 0. Given µ0 P2 (X ) and a time step > 0, find recursively µk among solutions of min ( ) + 12 W (, µk 2 2
-1

P2 (X )

),

µ0 = µ0 .

Construct a curve t [0, T ] µ(t ) P2 (X ) interpolating the discrete values and passing to the limit as 0. Show that the obtained limit curve satisfies the continuity equation.

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

6 / 22


Gradient flow in the space (P2 (X ), W2 )

References:
R . J O R DA N , D. K I N D E R L E H R E R , F. OT TO, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), 1­17. F. OT TO, Dynamics of Labyrinthine Pattern Formation in Magnetic Fluids: A MeanField Theory, Arch.Rational Mech. Anal. 141 (1998), 63­103. ´ L . A M B R O S I O, N . G I G L I , G . S AVA R E , Gradient flows in metric spaces and in the ¨ spaces of probability measures, Lectures in Mathematics ETH Zurich, Birkhauser ¨ Verlag, Basel (2005).

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

7 / 22


Existence and regularity result
In the P2 () framework, we have the following Theorem (L. Ambrosio, S. Serfaty, 2008) Let µ0 H -1 () P2 (). Then there exists a curve t µ(t ) H -1 () P2 () such that: (i) µ(0) = µ0 and µ(t ) is solution to the (CRS) model; (ii) The above solution is the Wasserstein gradient flow of the energy (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,
p

0;

(iii) If moreover µ0 Lp (), then µ(t )

C;

Task The actual (CRS) model involves signed measures. Can we extend the above framework and results to the signed case?
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 8 / 22


Existence and regularity result
In the P2 () framework, we have the following Theorem (L. Ambrosio, S. Serfaty, 2008) Let µ0 H -1 () P2 (). Then there exists a curve t µ(t ) H -1 () P2 () such that: (i) µ(0) = µ0 and µ(t ) is solution to the (CRS) model; (ii) The above solution is the Wasserstein gradient flow of the energy (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,
p

0;

(iii) If moreover µ0 Lp (), then µ(t )

C;

Task The actual (CRS) model involves signed measures. Can we extend the above framework and results to the signed case?
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 8 / 22


Existence and regularity result
In the P2 () framework, we have the following Theorem (L. Ambrosio, S. Serfaty, 2008) Let µ0 H -1 () P2 (). Then there exists a curve t µ(t ) H -1 () P2 () such that: (i) µ(0) = µ0 and µ(t ) is solution to the (CRS) model; (ii) The above solution is the Wasserstein gradient flow of the energy (µ) := 1 |µ|() + 2 2 | hµ |2 + |hµ - 1|2 ,
p

0;

(iii) If moreover µ0 Lp (), then µ(t )

C;

Task The actual (CRS) model involves signed measures. Can we extend the above framework and results to the signed case?
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 8 / 22


Transpor t cost for signed measures
Extension of the 2-Wasserstein distance to the space M, M () := {µ M() : µ() = , |µ|() M }, R, M 0.
-

First attempt: given µ, M, M (), let µ = µ+ - µ- , = + - (Hahn decomposition), and let
2 W2 (µ, ) := W2 (µ+ + - , + + µ- ).

W2 does not satisfy the triangle inequality: Let µ = 0 , = 4 . Let = 1 - 2 + 3 . We get W2 (µ, ) = 4 and W2 (µ, ) + W2 (, ) = 2 2.

µ W2 (·, µ) is not weakly l.s.c.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 9 / 22


Transpor t cost for signed measures
Extension of the 2-Wasserstein distance to the space M, M () := {µ M() : µ() = , |µ|() M }, R, M 0.
-

First attempt: given µ, M, M (), let µ = µ+ - µ- , = + - (Hahn decomposition), and let
2 W2 (µ, ) := W2 (µ+ + - , + + µ- ).

W2 does not satisfy the triangle inequality: Let µ = 0 , = 4 . Let = 1 - 2 + 3 . We get W2 (µ, ) = 4 and W2 (µ, ) + W2 (, ) = 2 2.

µ W2 (·, µ) is not weakly l.s.c.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 9 / 22


Transpor t cost for signed measures
Extension of the 2-Wasserstein distance to the space M, M () := {µ M() : µ() = , |µ|() M }, R, M 0.
-

First attempt: given µ, M, M (), let µ = µ+ - µ- , = + - (Hahn decomposition), and let
2 W2 (µ, ) := W2 (µ+ + - , + + µ- ).

W2 does not satisfy the triangle inequality: Let µ = 0 , = 4 . Let = 1 - 2 + 3 . We get W2 (µ, ) = 4 and W2 (µ, ) + W2 (, ) = 2 2.

µ W2 (·, µ) is not weakly l.s.c.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 9 / 22


Transpor t cost for signed measures
Extension of the 2-Wasserstein distance to the space M, M () := {µ M() : µ() = , |µ|() M }, R, M 0.
-

First attempt: given µ, M, M (), let µ = µ+ - µ- , = + - (Hahn decomposition), and let
2 W2 (µ, ) := W2 (µ+ + - , + + µ- ).

W2 does not satisfy the triangle inequality: Let µ = 0 , = 4 . Let = 1 - 2 + 3 . We get W2 (µ, ) = 4 and W2 (µ, ) + W2 (, ) = 2 2.

µ W2 (·, µ) is not weakly l.s.c.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 9 / 22


Transpor t cost for signed measures
Extension of the 2-Wasserstein distance to the space M, M () := {µ M() : µ() = , |µ|() M }, R, M 0.
-

First attempt: given µ, M, M (), let µ = µ+ - µ- , = + - (Hahn decomposition), and let
2 W2 (µ, ) := W2 (µ+ + - , + + µ- ).

W2 does not satisfy the triangle inequality: Let µ = 0 , = 4 . Let = 1 - 2 + 3 . We get W2 (µ, ) = 4 and W2 (µ, ) + W2 (, ) = 2 2.

µ W2 (·, µ) is not weakly l.s.c.
` Edoardo Mainini (Universita di Pavia) Gradient flow of a signed measures model 9 / 22


Transpor t cost for signed measures
At least by Holder inequality we have, if 0 (µ+ + - , + + µ- ),
1/2

|x - y |2 d
â



1 2M

|x - y | d
â

1 W1 (µ, ), 2M

where W1 (µ, ) := W1 (µ+ + - , + + µ- ) =
(µ+ + - , + +µ- ) â

inf

|x - y | d .

The new object W1 is a distance, as clearly seen from the duality formula W1 (µ+ + - , + + µ- ) = sup
Lip(),
Lip

d (µ - ).
1

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

10 / 22


Transpor t cost for signed measures
At least by Holder inequality we have, if 0 (µ+ + - , + + µ- ),
1/2

|x - y |2 d
â



1 2M

|x - y | d
â

1 W1 (µ, ), 2M

where W1 (µ, ) := W1 (µ+ + - , + + µ- ) =
(µ+ + - , + +µ- ) â

inf

|x - y | d .

The new object W1 is a distance, as clearly seen from the duality formula W1 (µ+ + - , + + µ- ) = sup
Lip(),
Lip

d (µ - ).
1

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

10 / 22


Transpor t cost for signed measures
At least by Holder inequality we have, if 0 (µ+ + - , + + µ- ),
1/2

|x - y |2 d
â



1 2M

|x - y | d
â

1 W1 (µ, ), 2M

where W1 (µ, ) := W1 (µ+ + - , + + µ- ) =
(µ+ + - , + +µ- ) â

inf

|x - y | d .

The new object W1 is a distance, as clearly seen from the duality formula W1 (µ+ + - , + + µ- ) = sup
Lip(),
Lip

d (µ - ).
1

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

10 / 22


Transpor t cost for signed measures
Relaxed l.s.c. version: for | |() |µ|(),
2 W2 (, µ) := + - - =

inf

2 2 W2 ( + , µ+ ) + W2 ( - , µ- ) .

We have W2 (, µ) 1 W1 (µ, ). 2M
, M

The discrete scheme: given µ0 M min
M, M (), | |()|µk |()

(), find µk

+1

by

( ) +

12 W (, µk ). 2 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

11 / 22


Transpor t cost for signed measures
Relaxed l.s.c. version: for | |() |µ|(),
2 W2 (, µ) := + - - =

inf

2 2 W2 ( + , µ+ ) + W2 ( - , µ- ) .

We have W2 (, µ) 1 W1 (µ, ). 2M
, M

The discrete scheme: given µ0 M min
M, M (), | |()|µk |()

(), find µk

+1

by

( ) +

12 W (, µk ). 2 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

11 / 22


Transpor t cost for signed measures
Relaxed l.s.c. version: for | |() |µ|(),
2 W2 (, µ) := + - - =

inf

2 2 W2 ( + , µ+ ) + W2 ( - , µ- ) .

We have W2 (, µ) 1 W1 (µ, ). 2M
, M

The discrete scheme: given µ0 M min
M, M (), | |()|µk |()

(), find µk

+1

by

( ) +

12 W (, µk ). 2 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

11 / 22


Transpor t cost for signed measures
Relaxed l.s.c. version: for | |() |µ|(),
2 W2 (, µ) := + - - =

inf

2 2 W2 ( + , µ+ ) + W2 ( - , µ- ) .

We have W2 (, µ) 1 W1 (µ, ). 2M
, M

The discrete scheme: given µ0 M min
M, M (), | |()|µk |()

(), find µk

+1

by

( ) +

12 W (, µk ). 2 2

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

11 / 22


Convergence of the discrete scheme
Existence of a limit curve in M, M ().
From (µk ) + we have
n 2 W2 (µk , µ k =1 N i =1 2 k -1

1 2 W2 (µk , µ 2

k -1

)=

min
M, M ()

( ) +
k -1

1 2 W2 (, µ 2

k -1

),

1 2 W2 (µk , µ 2 1 2

k -1

) + (µk ) (µ

) (µ0 ), (if 0)

) (µ0 ) - (µn ) (µ0 ). ai
-1

Hence, for n, m N, n > m, using
n

N
1/2

N i =1

ai2 , we get
/2

W2 (µk , µ
k =m + 1

k +1

)

1

n 2 W2 (µk , µk k =m +1

)

((n - m) )1



2 (µ0 )(n - m).

By W2 (, µ)

1 2M

W1 (µ, ) and triangle inequality, 1 2M
n n

1 W1 (µm , µn ) 2M
` Edoardo Mainini (Universita di Pavia)

W1 (µk , µk +1 )
k =m +1 k = m +1

W2 (µk , µk +1 ).
12 / 22

Gradient flow of a signed measures model


Convergence of the discrete scheme
Existence of a limit curve in M, M ().
From (µk ) + we have
n 2 W2 (µk , µ k =1 N i =1 2 k -1

1 2 W2 (µk , µ 2

k -1

)=

min
M, M ()

( ) +
k -1

1 2 W2 (, µ 2

k -1

),

1 2 W2 (µk , µ 2 1 2

k -1

) + (µk ) (µ

) (µ0 ), (if 0)

) (µ0 ) - (µn ) (µ0 ). ai
-1

Hence, for n, m N, n > m, using
n

N
1/2

N i =1

ai2 , we get
/2

W2 (µk , µ
k =m + 1

k +1

)

1

n 2 W2 (µk , µk k =m +1

)

((n - m) )1



2 (µ0 )(n - m).

By W2 (, µ)

1 2M

W1 (µ, ) and triangle inequality, 1 2M
n n

1 W1 (µm , µn ) 2M
` Edoardo Mainini (Universita di Pavia)

W1 (µk , µk +1 )
k =m +1 k = m +1

W2 (µk , µk +1 ).
12 / 22

Gradient flow of a signed measures model


Convergence of the discrete scheme
Existence of a limit curve in M, M ().
From (µk ) + we have
n 2 W2 (µk , µ k =1 N i =1 2 k -1

1 2 W2 (µk , µ 2

k -1

)=

min
M, M ()

( ) +
k -1

1 2 W2 (, µ 2

k -1

),

1 2 W2 (µk , µ 2 1 2

k -1

) + (µk ) (µ

) (µ0 ), (if 0)

) (µ0 ) - (µn ) (µ0 ). ai
-1

Hence, for n, m N, n > m, using
n

N
1/2

N i =1

ai2 , we get
/2

W2 (µk , µ
k =m + 1

k +1

)

1

n 2 W2 (µk , µk k =m +1

)

((n - m) )1



2 (µ0 )(n - m).

By W2 (, µ)

1 2M

W1 (µ, ) and triangle inequality, 1 2M
n n

1 W1 (µm , µn ) 2M
` Edoardo Mainini (Universita di Pavia)

W1 (µk , µk +1 )
k =m +1 k = m +1

W2 (µk , µk +1 ).
12 / 22

Gradient flow of a signed measures model


Convergence of the discrete scheme
Existence of a limit curve in M, M ().
From (µk ) + we have
n 2 W2 (µk , µ k =1 N i =1 2 k -1

1 2 W2 (µk , µ 2

k -1

)=

min
M, M ()

( ) +
k -1

1 2 W2 (, µ 2

k -1

),

1 2 W2 (µk , µ 2 1 2

k -1

) + (µk ) (µ

) (µ0 ), (if 0)

) (µ0 ) - (µn ) (µ0 ). ai
-1

Hence, for n, m N, n > m, using
n

N
1/2

N i =1

ai2 , we get
/2

W2 (µk , µ
k =m + 1

k +1

)

1

n 2 W2 (µk , µk k =m +1

)

((n - m) )1



2 (µ0 )(n - m).

By W2 (, µ)

1 2M

W1 (µ, ) and triangle inequality, 1 2M
n n

1 W1 (µm , µn ) 2M
` Edoardo Mainini (Universita di Pavia)

W1 (µk , µk +1 )
k =m +1 k = m +1

W2 (µk , µk +1 ).
12 / 22

Gradient flow of a signed measures model


Convergence of the discrete scheme
Existence of a limit curve in M, M ().
From (µk ) + we have
n 2 W2 (µk , µ k =1 N i =1 2 k -1

1 2 W2 (µk , µ 2

k -1

)=

min
M, M ()

( ) +
k -1

1 2 W2 (, µ 2

k -1

),

1 2 W2 (µk , µ 2 1 2

k -1

) + (µk ) (µ

) (µ0 ), (if 0)

) (µ0 ) - (µn ) (µ0 ). ai
-1

Hence, for n, m N, n > m, using
n

N
1/2

N i =1

ai2 , we get
/2

W2 (µk , µ
k =m + 1

k +1

)

1

n 2 W2 (µk , µk k =m +1

)

((n - m) )1



2 (µ0 )(n - m).

By W2 (, µ)

1 2M

W1 (µ, ) and triangle inequality, 1 2M
n n

1 W1 (µm , µn ) 2M
` Edoardo Mainini (Universita di Pavia)

W1 (µk , µk +1 )
k =m +1 k = m +1

W2 (µk , µk +1 ).
12 / 22

Gradient flow of a signed measures model


Convergence of the discrete scheme

We are left with W1 (µm , µn ) Interpolation: for t > 0, let µ (t ) := µ We find the C
0, 1/2 k

4M (µ)0 (n - m) .

if t ((k - 1) , k ], k > 0.

estimate 2 (µ) |t - s| + s, t > 0.

W1 (µ (t ), µ (s)) A limit exists by compactness: µn (t )

µ(t ) in the sense of measures, for a.e. t .

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

13 / 22


Convergence of the discrete scheme

We are left with W1 (µm , µn ) Interpolation: for t > 0, let µ (t ) := µ We find the C
0, 1/2 k

4M (µ)0 (n - m) .

if t ((k - 1) , k ], k > 0.

estimate 2 (µ) |t - s| + s, t > 0.

W1 (µ (t ), µ (s)) A limit exists by compactness: µn (t )

µ(t ) in the sense of measures, for a.e. t .

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

13 / 22


Convergence of the discrete scheme

We are left with W1 (µm , µn ) Interpolation: for t > 0, let µ (t ) := µ We find the C
0, 1/2 k

4M (µ)0 (n - m) .

if t ((k - 1) , k ], k > 0.

estimate 2 (µ) |t - s| + s, t > 0.

W1 (µ (t ), µ (s)) A limit exists by compactness: µn (t )

µ(t ) in the sense of measures, for a.e. t .

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

13 / 22


Convergence of the discrete scheme

We are left with W1 (µm , µn ) Interpolation: for t > 0, let µ (t ) := µ We find the C
0, 1/2 k

4M (µ)0 (n - m) .

if t ((k - 1) , k ], k > 0.

estimate 2 (µ) |t - s| + s, t > 0.

W1 (µ (t ), µ (s)) A limit exists by compactness: µn (t )

µ(t ) in the sense of measures, for a.e. t .

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

13 / 22


Convergence of the discrete scheme

We are left with W1 (µm , µn ) Interpolation: for t > 0, let µ (t ) := µ We find the C
0, 1/2 k

4M (µ)0 (n - m) .

if t ((k - 1) , k ], k > 0.

estimate 2 (µ) |t - s| + s, t > 0.

W1 (µ (t ), µ (s)) A limit exists by compactness: µn (t )

µ(t ) in the sense of measures, for a.e. t .

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

13 / 22


Variational result
For simplicity let = R2 . Theorem (L. Ambrosio, E. M., S. Serfaty, 2010) Consider a single step of the minimization problem above, star ting from µ Lp (R2 ), p 4. There exists a minimizer µ Lp (R2 ) such that µ (uniform in estimate) There holds - hµ µ = 11 11 + - ((x - y )0 ) + # ((x - y )0 ), #
p

µ p.

+ - where 0 0 (µ+ , µ+ ) and 0 0 (µ- , µ- ). 0 0

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

14 / 22


Variational result
For simplicity let = R2 . Theorem (L. Ambrosio, E. M., S. Serfaty, 2010) Consider a single step of the minimization problem above, star ting from µ Lp (R2 ), p 4. There exists a minimizer µ Lp (R2 ) such that µ (uniform in estimate) There holds - hµ µ = 11 11 + - ((x - y )0 ) + # ((x - y )0 ), #
p

µ p.

+ - where 0 0 (µ+ , µ+ ) and 0 0 (µ- , µ- ). 0 0

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

14 / 22


Variational result
For simplicity let = R2 . Theorem (L. Ambrosio, E. M., S. Serfaty, 2010) Consider a single step of the minimization problem above, star ting from µ Lp (R2 ), p 4. There exists a minimizer µ Lp (R2 ) such that µ (uniform in estimate) There holds - hµ µ = 11 11 + - ((x - y )0 ) + # ((x - y )0 ), #
p

µ p.

+ - where 0 0 (µ+ , µ+ ) and 0 0 (µ- , µ- ). 0 0

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

14 / 22


Variational result
For simplicity let = R2 . Theorem (L. Ambrosio, E. M., S. Serfaty, 2010) Consider a single step of the minimization problem above, star ting from µ Lp (R2 ), p 4. There exists a minimizer µ Lp (R2 ) such that µ (uniform in estimate) There holds - hµ µ = 11 11 + - ((x - y )0 ) + # ((x - y )0 ), #
p

µ p.

+ - where 0 0 (µ+ , µ+ ) and 0 0 (µ- , µ- ). 0 0

` Edoardo Mainini (Universita di Pavia)

Gradient flow of a signed measures model

14 / 22


Variational result
For simplicity let = R2 . Theorem (L. Ambrosio, E. M., S. Serfaty, 2010) Consider a single step of the minimization problem above, star ting from µ Lp (R2 ), p 4. There exists a minimizer µ Lp (R2 ) such that µ (uniform in estimate) There holds - hµ µ