Global Solutions of the Landau-Lifshitz Equation › SIAM11 › PDE › PDF › ...Nov 14, 2011 · S. Gustafson (UBC), Eva Koo (UBC) Global Solutions of the Landau-Lifshitz Equation

Global Solutions of the Landau-Lifshitz Equation

S. Gustafson (UBC), Eva Koo (UBC)

SIAM PDE, San Diego, Nov. 14, 2011

S. Gustafson (UBC), Eva Koo (UBC) Global Solutions of the Landau-Lifshitz Equation

Landau-Lifshitz Equations: Dynamics of Maps

Landau-Lifshitz (30s): ferromagnetism

“magnetization vector” ~u(x , t) ∈ R3

|~u(x , t)| ≡ const.~ut = ~u ×∆~u

Broader context:

maps: ~u(·, t) : R2 → S2 (~u ∈ R3, |~u| ≡ 1)energy: E(~u) = 12

∫R2 |∇~u|

2dx

heat-flow: ~ut = ∆~u + |∇~u|2~u (= −E ′(~u))wave maps: ~utt = ∆~u + (|∇~u|2 − |~ut |2)~uSchrödinger maps: ~ut = ~u ×∆~u (= JE ′(~u))

J = −~u× = complex structure

|~u(x , t)| ≡ const.~ut = ~u ×∆~u

Broader context:

∫R2 |∇~u|

2dx

|~u(x , t)| ≡ const.~ut = ~u ×∆~u

Broader context:

∫R2 |∇~u|

2dx

|~u(x , t)| ≡ const.~ut = ~u ×∆~u

Broader context:

∫R2 |∇~u|

2dx

|~u(x , t)| ≡ const.~ut = ~u ×∆~u

Broader context:

∫R2 |∇~u|

2dx

Regularity vs. Singularity: Energy Critical Problems

All these equations are energy critical in R2 (E(~u) scale invariant).

E =∫R2 |∇~u|

2 ≥ 4π|degree(~u)| with = iff ~u a harmonic map.

Emin := “ground state energy”= lowest energy of a non-trivial harmonic map ( = 4π).

We expect a dichotomy:

global existence below the ground state: E < Eminblow-up examples above the ground state: E > Emin

heat-flow:

E < 4π =⇒ global smooth solution [Struwe 85]E > 4π =⇒ singularities may form: [Chang-Ding-Ye 92]

wave map:

E < 4π =⇒ global solution [Sterbenz-Tataru 09]E > 4π =⇒ singularities may form [Krieger-Schlag-Tataru 08,Rodnianski-Sterbenz 09, Raphael-Rodnianski 09]

E =∫R2 |∇~u|

heat-flow:

wave map:

E =∫R2 |∇~u|

heat-flow:

wave map:

Some Schrödinger Map Results

For the Schrödinger map ~ut = ~u ×∆~u on R2:E small =⇒ global solutions:[Chang-Shatah-Uhlenbeck 00, Bejenaru-Ionescu-Kenig-Tataru 08]

for equivariant maps of degree m ∈ Z (Emin = 4π|m|):I |m| ≥ 3: global existence for 0 ≤ E − Emin � 1 [G-Kang-Tsai 08,

G-Nakanishi-Tsai 10]

I |m| = 2: ??I |m| = 1: construction of singular solutions for E > 4π

[Merle-Raphael-Rodnianski 11]

I m = 0 (radially-symmetric): solutions are global [G-Koo 11]

open: global existence for all E ≤ Emin without symmetryrestrictions

Global Existence of Radial Schrödinger Maps

Consider radially-symmetric Schrödinger maps: ~u(r , t), r = |x |:{~ut = ~u ×∆~u

~u(r , 0) = ~u0(r), ~u0 − k̂ ∈ H2(R2)(SM)rad

There are no radially symmetric harmonic maps, so Emin =∞ .The generalized Hasimoto transform [Chang-Shatah-Uhlenbeck 00]

T~uS2 3 ~ur = q1ê + q2Jê, D~ur ê ≡ 0

transforms solutions of (SM)rad to solutions q(r , t) = q1 + iq2 of{iqt = −∆q + 1r2 q + (

∫∞r |q(ρ, t)|

2 dρρ −

12 |q|

2)q

q(r , 0) = q0(r)(nlNLS)

Thm:[G-Koo 11]: (nlNLS) is globally well-posed for q0(r) ∈ L2(R2).

Corollary: (SM)rad has a unique global solution ~u(t)− k̂ ∈ H2(R2).S. Gustafson (UBC), Eva Koo (UBC) Global Solutions of the Landau-Lifshitz Equation

L2-Critical NLS: Local vs. Non-local

Compare our non-local NLS

iqt = −∆q +1

r2q +

(∫ ∞r|q(ρ, t)|2 dρ

ρ− 1

2|q|2)q (nlNLS)

with the well-studied local version{iqt = −∆q ± |q|2qq(r , 0) = q0(r)

(NLS±).

Common features:

conservation of L2 norm:∫R2 |q(x , t)|

2dx ≡ const.invariant under L2-critical scaling q(r , t) 7→ λq(λr , λ2t)local existence of a solution q = C ([0,T );Hs(R2)) with

I 0 < T = T (‖q0‖Hs ) for s > 0I 0 < T 6= T (‖q0‖L2) for s = 0

iqt = −∆q +1

r2q +

(∫ ∞r|q(ρ, t)|2 dρ

ρ− 1

2|q|2)q (nlNLS)

(NLS±).

Common features:

iqt = −∆q +1

r2q +

(∫ ∞r|q(ρ, t)|2 dρ

ρ− 1

2|q|2)q (nlNLS)

(NLS±).

Common features:

iqt = −∆q +1

r2q +

(∫ ∞r|q(ρ, t)|2 dρ

ρ− 1

2|q|2)q (nlNLS)

(NLS±).

Common features:

Review: global existence for local NLS

Main difference: (NLS±) has a conserved energy

H = 12

∫R2|∇q|2 ± 1

4

∫R2.|q|4 ≡ const.

So global existence for q0 ∈ H1(R2) follows from:(NLS+): ‖q‖2H1 ≤ ‖q0‖

2L2 + 2H ≤ const.

(NLS−): by Sobolev interpolation ‖q‖4L4 ≤ C‖q‖2L2‖∇q‖

2L2 ,

‖∇q‖2L2 = 2H+12‖q‖

4L4 ≤ 2H+

12C‖q‖

2L2‖∇q‖

2L2 , so

‖q0‖L2 <√

2

C= ‖Q‖L2 =⇒ ‖∇q‖L2 ≤ const.

where Q = ground state [Weinstein 83].

If merely q0 ∈ L2(R2), the energy is unavailable, and globalexistence of radial solutions for (NLS+), and (NLS-) with‖q0‖L2 < ‖Q‖L2 , was proved by [Kilip-Tao-Visan 09] using much moresophisticated methods

Virial and Morawetz Identities

Back to (nlNLS) iqt = −∆q + 1r2 q +(∫∞

r |q(ρ, t)|2 dρρ −

12 |q|

2)q

It has no conserved energy. Is it “focusing” or “defocusing”?

Solutions of (nlNLS) formally obey:

virial-type identity:

d2

dt21

2

∫ ∞0

r2|q(r , t)|2rdr =∫ ∞0

{4|qr |2 + 4

|q|2

r2+ 2|q|4

}rdr > 0

Morawetz-type identity:

d2

dt2

∫ ∞0

r |q(r , t)|2rdr =∫ ∞0

{3|q|2

r3+

3

4

|q|4

r

}rdr > 0

So it looks defocusing!

However, to actually use identities like these, we must

cut off large (and small) r , and control errors

control derivative terms (no energy, and anyway q ∈ L2 only)S. Gustafson (UBC), Eva Koo (UBC) Global Solutions of the Landau-Lifshitz Equation

Reduction to a Minimal-Mass Blow-Up

[KTV]’s proof of L2 global well-posedness for radial (NLS+) (and(NLS−) below the ground state) follows Kenig-Merle’s framework:

if (GWP + scattering) fails, ∃ an L2-minimal “blow-up”solution q(r , t) with 1N(t)q(r/N(t), t) L

2-pre-compact: ∀� > 0,∫|x |>C�/N(t)

|q(x , t)|2dx < �,∫|ξ|>C�N(t)

|q̂(ξ, t)|2dξ < �.

3 possibilities:I N(t) ≡ 1 (“soliton”),I N(t) = 1/

√t (“self-similar”),

I lim inft→±∞ N(t) = 0 (“inverse cascade”)

regularity: q(t) ∈ Hs for any seach case ruled out by conservation of energy (plus virialidentity for “soliton”)

This “reduction to 3 enemies” + regularity extends to (nlNLS).But without conservation of energy, how do we rule them out?.

Eliminating Enemies: Soliton-Type

N(t) ≡ 1, q(r , t) compact, regular: ‖q(t)‖Hs . 1cut-off virial identity: consider IR(t) :=

∫ R0 r Im(qqr )rdr

virial identity, compactness, and regularity give

d

dtIR = 2

∫ R0{|qr |2 +

|q|2

r2+

1

2|q|4}rdr [1 + o(1)] & N2(t) ≡ 1

for fixed (sufficiently large) R

while regularity =⇒ |IR | . R‖q‖L2‖qr‖L2 . Ra contradiction.

Eliminating Enemies: Self-Similar-Type

N(t) = t−1/2, so take R ∼ 1N(T ) = T1/2. Then the virial

identity (as above) gives

d

dtIR & N

2(t) =1

t

implies IR(T ) & logT .

On the other hand, the [KTV] regularity theory gives

|IR | . R‖qr‖L2 .1

N(t)N(t) = O(1),

a contradiction for T large.

Eliminating Enemies: Inverse-Cascade-Type

N(t)→ 0 as t → ±∞introduce a modifier ψ(r) ∼ r〈r〉 into the Morawetz-typeidentity for: P(t) :=

∫∞0 Im(qqr )ψ(r)rdr ,

d

dtP(t) ∼

∫ ∞0

{|qr |2

〈r〉+|q|2

r2〈r〉+|q|4

〈r〉

}rdr > 0.

Now N(t)→ 0 as t → ±∞ implies

|P| . ‖q‖2‖qr‖2 → 0 t → ±∞,

a contradiction.

Conclusions

2D radial Schrödinger maps into the sphere are global

still open: GWP “below ground state” without symmetryrestriction

other issues: dimensions 3, more physical versions of theLandau-Lifshitz equations, etc.

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Global Solutions of the Landau-Lifshitz Equation › SIAM11 › PDE › PDF › ...Nov 14, 2011 · S. Gustafson (UBC), Eva Koo (UBC) Global Solutions of the Landau-Lifshitz Equation