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