[PPT] - Gradient flows in the framework of (Cartesian) Currents Malte PowerPoint Presentation

SLIDE 1

Gradient flows in the framework of (Cartesian) Currents

Malte Kampschulte

Department for Mathematics I, RWTH Aachen University

Conference on Nonlinearity, Transport, Physics and Patterns Fields Institute 07.10.2014

SLIDE 2

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 3

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 4

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

◮ Enforces planar values m(x) ∈ S1 × {0} a.e.

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 5

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

◮ Enforces planar values m(x) ∈ S1 × {0} a.e. ◮ ⇒ Vortices form, can behave similar to point particles.

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 6

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

◮ Enforces planar values m(x) ∈ S1 × {0} a.e. ◮ ⇒ Vortices form, can behave similar to point particles. ◮ Information about orientation is lost!

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 7

Vortices as singularities

Consider functions m : Ω ⊂ R2 → S2 What happens if we penalize the third component?

◮ Enforces planar values m(x) ∈ S1 × {0} a.e. ◮ ⇒ Vortices form, can behave similar to point particles. ◮ Information about orientation is lost!

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 2 / 14

SLIDE 8

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 9

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 10

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 11

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 12

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 13

Bubbling and Vertical parts

Seen in the simpler case [a, b] → S1, information in the limit still exists in vertical parts:

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 3 / 14

SLIDE 14

Cartesian currents

Approach due to Giaquinta, Modica, Souˇ cek (’89):

◮ Consider graphs of (nice enough) functions Ω ⊂ Rn → M as

rectifiable n-current in Ω × M.

◮ Cartesian Currents ≈ closure the class of graphs in the topology of

currents

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 4 / 14

SLIDE 15

Cartesian currents

Approach due to Giaquinta, Modica, Souˇ cek (’89):

◮ Consider graphs of (nice enough) functions Ω ⊂ Rn → M as

rectifiable n-current in Ω × M.

◮ Cartesian Currents ≈ closure the class of graphs in the topology of

currents

Reminder (de Rham (’55), Federer & Fleming (’60)):

◮ k-Currents ≈ dual space of compactly supported smooth differential

k-forms (approach similar to distributions)

◮ Rectifiable k-currents ≈ countable unions of orientable manifolds with

integer multiplicity

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 4 / 14

SLIDE 16

Cartesian currents

Approach due to Giaquinta, Modica, Souˇ cek (’89):

◮ Consider graphs of (nice enough) functions Ω ⊂ Rn → M as

rectifiable n-current in Ω × M.

◮ Cartesian Currents ≈ closure the class of graphs in the topology of

currents

Some features:

◮ Cartesian Currents usually consist of flat “graph” part and “vertical”

singularities

◮ Cart. Currents are boundaryless (boundary in ∂(Ω × M) does not

count)

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 4 / 14

SLIDE 17

Why gradient flows

◮ Large class of similar problems ◮ Many have some sort of singularities ◮ Canonical example: Harmonic map heat flow ◮ Good abstract approach available (s. book by Ambrosio, Gigli,

Savar´ e)

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 5 / 14

SLIDE 18

Minimizing movements (de Giorgi)

Ingredients

◮ Set of admissible Currents A ◮ Metric d(., .) ◮ Energy E(.)

Implicit Euler iteration

S(h)

k+1 := arg min

1 2hd

S, S(h)

k

2 + E(S)

S ∈ A
.

Then for h → 0 the limit S(t) = limh→0 S(h)

k/h should converge to a solution

to the gradient flow ∂ ∂t S + ∇dE(S) = 0

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 6 / 14

SLIDE 19

Convergence theorem (K. 2014)

Assume we have some closed class of cartesian currents A for which i) d2 and E lower semi-continuous ii) E bounded from below iii) The mass of currents with bounded energy is bounded iv) ˙ F(S − T) ≤ c · d(S, T) for some c and all S, T of bounded energy Then the minimizing movements iteration is well defined and converges (up to a subsequence) on any time interval [0, τ] to an k + 1 space-time current A s.t. the approximations S(h)(r) converge to the slices A, t < r.

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 7 / 14

SLIDE 20

Convergence theorem (K. 2014)

Assume we have some closed class of cartesian currents A for which i) d2 and E lower semi-continuous ii) E bounded from below iii) The mass of currents with bounded energy is bounded iv) ˙ F(S − T) ≤ c · d(S, T) for some c and all S, T of bounded energy Then the minimizing movements iteration is well defined and converges (up to a subsequence) on any time interval [0, τ] to an k + 1 space-time current A s.t. the approximations S(h)(r) converge to the slices A, t < r.

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 7 / 14

SLIDE 21

Convergence theorem (K. 2014)

Assume we have some closed class of cartesian currents A for which i) d2 and E lower semi-continuous ii) E bounded from below iii) The mass of currents with bounded energy is bounded iv) ˙ F(S − T) ≤ c · d(S, T) for some c and all S, T of bounded energy Then the minimizing movements iteration is well defined and converges (up to a subsequence) on any time interval [0, τ] to an k + 1 space-time current A s.t. the approximations S(h)(r) converge to the slices A, t < r.

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 7 / 14

SLIDE 22

The boundary free case: a homogeneous Flat norm

Reminder: Flat norm

F(S − T) := sup {(S − T)(ω) : ω ≤ 1 ∧ dω ≤ 1} = inf {M(A) + M(B) : S − T = ∂A + B}

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 8 / 14

SLIDE 23

The boundary free case: a homogeneous Flat norm

Reminder: Flat norm

F(S − T) := sup {(S − T)(ω) : ω ≤ 1 ∧ dω ≤ 1} = inf {M(A) + M(B) : S − T = ∂A + B}

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 8 / 14

SLIDE 24

The boundary free case: a homogeneous Flat norm

Reminder: Flat norm

F(S − T) := sup {(S − T)(ω) : ω ≤ 1 ∧ dω ≤ 1} = inf {M(A) + M(B) : S − T = ∂A + B}

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 8 / 14

SLIDE 25

The boundary free case: a homogeneous Flat norm

Reminder: Flat norm

F(S − T) := sup {(S − T)(ω) : ω ≤ 1 ∧ dω ≤ 1} = inf {M(A) + M(B) : S − T = ∂A + B}

Variant: Homogeneous flat norm

˙ F(S − T) := sup {(S − T)(ω) : dω ≤ 1} = inf {M(A) : S − T = ∂A}

Malte Kampschulte (RWTH Aachen) Gradient flows in Cartesian Currents 07.10.2014 8 / 14

SLIDE 26

The boundary free case: a homogeneous Flat norm

Reminder: Flat norm

F(S − T) := sup {(S − T)(ω) : ω ≤ 1 ∧ dω ≤ 1} = inf {M(A) + M(B) : S − T = ∂A + B}

Variant: Homogeneous flat norm