Ding, S. and C. Wang. (2007) “Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation,” International Mathematics Research Notices, Vol. 2007, Article ID rnm012, 25 pages. doi:10.1093/imrn/rnm012 Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation Shijin Ding 1 and Changyou Wang 2 1 School of Mathematical Sciences, South China Normal University , Guangzhou 510631, P. R. China and 2 Department of Mathematics, University of Kentucky , Lexington, KY 40506, USA Correspondence to be sent to: Changyou Wang, Department of Mathematics, University of Kentucky , Lexington, KY 40506, USA. e-mail: [email protected]We prove that in dimensions three or four , for suitably chosen initial data, the short time smooth solution to the Landau-Lifshitz-Gilbert equation blows up at finite time. 1 Introduction The Landau-Lifshitz-Gilbert equation is the fundamental evolution equation for spin fields in the continuum theory of ferromagnetism, first proposed by Landau and Lifshitz [1] in 1935. In the simplest case, where the energy of spin interactions is modeled by E(u) = 1 2 Ω |∇u| 2 for magnetic moment u : Ω ⊂ R n → S 2 , the Landau-Lifshitz-Gilbert equation for u : Ω × (0, +∞) → S 2 is given by αu t + βu ∧ u t = ∆u + |∇u| 2 u, (1.1) where α ≥ 0, β ∈ R, α 2 + β 2 = 1, and ∧ is the vector product in R 3 . Note that (1.1) reduces to the heat flow of harmonic maps to S 2 for α = 1, β = 0, and to the Schr ¨ odinger flow of harmonic maps to S 2 for α = 0, β = 1. We assume throughout this article that 0 < α < 1, Received October 30, 2006; Revised January 24, 2007; Accepted January 28, 2007 Communicated by Michael Struwe See http://www.oxfordjournals.org/our journals/imrn/ for proper citation instructions. c The Author 2007. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected].
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Ding, S. and C. Wang. (2007) “Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation,”International Mathematics Research Notices, Vol. 2007, Article ID rnm012, 25 pages.doi:10.1093/imrn/rnm012
Finite Time Singularity of the Landau-Lifshitz-GilbertEquation
Shijin Ding1 and Changyou Wang2
1School of Mathematical Sciences, South China Normal University,Guangzhou 510631, P. R. China and 2Department of Mathematics,University of Kentucky, Lexington, KY 40506, USA
Correspondence to be sent to: Changyou Wang, Department of Mathematics, University ofKentucky, Lexington, KY 40506, USA. e-mail: [email protected]
We prove that in dimensions three or four, for suitably chosen initial data, the short time
smooth solution to the Landau-Lifshitz-Gilbert equation blows up at finite time.
1 Introduction
The Landau-Lifshitz-Gilbert equation is the fundamental evolution equation for spin
fields in the continuum theory of ferromagnetism, first proposed by Landau and Lifshitz
[1] in 1935. In the simplest case, where the energy of spin interactions is modeled by
E(u) =12
∫Ω
|∇u|2 for magnetic moment u : Ω ⊂ Rn→ S2, the Landau-Lifshitz-Gilbert
equation for u : Ω × (0,+∞) → S2 is given by
αut + βu ∧ ut = ∆u + |∇u|2u, (1.1)
where α ≥ 0,β ∈ R, α2 + β2 = 1, and ∧ is the vector product in R3. Note that (1.1) reduces
to the heat flow of harmonic maps to S2 for α = 1,β = 0, and to the Schrodinger flow of
harmonic maps to S2 for α = 0,β = 1. We assume throughout this article that 0 < α < 1,
Received October 30, 2006; Revised January 24, 2007; Accepted January 28, 2007
Communicated by Michael Struwe
See http://www.oxfordjournals.org/our journals/imrn/ for proper citation instructions.
In particular, if M = Bn,n ≥ 3, then we can find u0 ∈ C1(Bn,S2) such that u0|∂Bn =
constant, u0 is not homotopic to a constant relative to ∂Bn, and E(u0) is arbitrarily small.
For the Neumann boundary value problem, we have the following.
Theorem 1.5. Let M = Ω = x ∈ R4 : 1 ≤ |x| ≤ 2, g = g0 be the Euclidean metric on R4,
and u0(x) = (H Ψλ)( x|x| ) : M → S2, where H Ψλ is given by Remark 1.2 (2). Then for any
T > 0, there exists λ = λ(T) > 0 such that the short time smooth solution u to (1.2)–(1.3)
and
∂u∂ν
(x, t) = 0, x ∈ ∂Ω, t > 0 (1.8)
must blow up before time t = T.
Remark 1.6.
(1) It is unknown whether theorem 1.5 holds in dimension three. Namely in
dimension three, we are unable to construct a map u0 ∈ C∞(M,S2)
such that E(u0) can be arbitrarily small, and it can’t be deformed into
a constant map through families of maps H ∈ C1(M × [0, 1],S2) with∂H∂ν (x, t) = 0 for (x, t) ∈ ∂M × [0, 1]. In fact, for M = B3 = x ∈ R3 : |x| ≤ 1,it is not difficult to show that for any map φ ∈ C∞(B3,S2), with ∂φ
∂ν = 0,
there exists Φ ∈ C1(B3 × [0, 1],S2) such that Φ(·, 0) = φ, Φ(·, 1) = constant,
and ∂Φ∂ν = 0 on ∂B3 × [0, 1].
(2) It is a very important open problem whether the Landau-Lifshitz-Gilbert
equation has finite time singularity in dimension two. It is well-known
(cf. Chang-Ding-Ye [22]) that there exists finite time singularity for the
heat flow of harmonic maps in two dimensions.
Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation 5
The article is organized as follows. In §2, we establish a priori estimates for
smooth solutions of (1.2) under a small energy condition and prove Theorem 1.1. In §3,
we establish boundary a priori estimates for smooth solutions of (1.1). In §4, we prove
both Theorem 1.3 and 1.5.
2 Holder continuity estimate and Proof of Theorem 1.1
In this section, we first establish a priori continuity estimate of smooth solutions to (1.2)
under small energy condition, and then give a proof of Theorem 1.1.
Lemma 2.1 (Energy inequality). For any n ≥ 1, T > 0, and u0 ∈ C∞(M,S2), let u ∈C∞(M × [0,T),S2) solve (1.2)–(1.3). For t ∈ (0,T), denote u(t) = u(·, t). Then we have
α
∫ t
0
∫M
|ut|2
+ E(u(t)) = E(u0), (2.1)
and, for any 0 ≤ s < t < T and φ ∈ C∞0 (M),
α
∫ t
s
∫M
|ut|2φ2
+
∫M
|∇u(t)|2φ2 ≤∫
M|∇u(s)|2φ2
+4α
∫ t
s
∫M
|∇u|2|∇φ|2. (2.2)
Proof. Since u ∧ ut · ut = 0, (2.1) follows from multiplying (1.2) by ut and integrating the
resulting equation over M × [0, t). To see (2.2), we multiply (1.2) by utφ2 and integrate the
resulting equation over M × [s, t] to get
α
∫ t
s
∫M
|ut|2φ2
+12
∫M
|∇u(t)|2φ2=
12
∫M
|∇u(s)|2φ2− 2
∫ t
s
∫M
ut · ∇uφ∇φ.
By the Holder inequality, we have
|2∫ t
s
∫M
ut · ∇uφ∇φ| ≤ α
2
∫ t
s
∫M
|ut|2φ2
+2α
∫ t
s
∫M
|∇u|2|∇φ|2.
Hence (2.2) follows.
Let iM > 0 be the injectivity radius of M. For x ∈ M, t > 0, and 0 < r <
miniM ,√
t, let Br(x) ⊂ M be the ball with center x and radius r, and Pr(x, t) = Br(x) ×(t − r2, t) ⊂ M × (0,+∞) be the parabolic ball with center (x, t) and radius r. Now we have
the localized energy inequality.
6 S. Ding and C. Wang
Lemma 2.2. For any n ≥ 1 and T > 0, let u ∈ C∞(M × (0,T),S2) solve (1.2). Then for any
z0 = (x0, t0) ∈ M × (0,T), 0 < r < miniM ,√
t0, and t ∈ (t0 −r2
4 , t0), there exists Cα > 0
depending only on α,M such that
r2−n∫
B r2(x0)
|∇u(t)|2 + r2−n∫
P r2(z0)
|ut|2 ≤ Cαr−n
∫Pr(z0)
|∇u|2. (2.3)
Proof. For 0 < r < miniM ,√
t0, let φ ∈ C∞0 (Br(x0)) be such that 0 ≤ φ ≤ 1, φ ≡ 1 on
B r2(x0), and |∇φ| ≤ 4r−1. Let s0 ∈ (t0 − r2, t0 −
r2
4 ) be such that
∫Br(x0)
|∇u(s0)|2 ≤ 2r−2∫
Pr(z0)|∇u|2.
Putting φ into (2.2), we have
α
∫ t0
s0
∫B r
2(x0)
|ut|2 ≤
∫Br0 (x0)
|∇u(s0)|2 +64αr2
∫ t0
s0
∫Br0 (x0)
|∇u|2, (2.4)
and, for any t ∈ (t0 −r2
4 , t0),
∫B r
2(x0)
|∇u(t)|2 ≤∫
Br(x0)|∇u(s0)|2 +
64αr2
∫ t0
s0
∫Br(x0)
|∇u|2. (2.5)
It is clear that (2.4) and (2.5) imply (2.3).
Now we are ready to prove the following decay estimate.
Lemma 2.3. For 1 ≤ n ≤ 4, and any T > 0 and γ ∈ (0, 1), there exist ε0 > 0 and Cα > 0
depending only on M, g, γ,α such that if u ∈ C∞(M × (0,T),S2) solves (1.2) and satisfies,
for z0 = (x0, t0) ∈ M × (0,T) and 0 < r ≤ miniM ,√
t0,
r−n∫
Pr(z0)|∇u|2 ≤ ε2
0,
then u ∈ Cγ(P r2(z0),S2), and
[u]2Cγ (Ps(z0)) ≤ Cαr−(n+2γ)∫
Pr(z0)|∇u|2, ∀0 < s ≤ r
2. (2.6)
In order to prove Lemma 2.3, we first need to recall the following decay Lemma
that can be proved by a simple blowing up argument (see e.g. [10] Lemma 3.3 or [17]
Lemma 5.10).
Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation 7
Lemma 2.4. There exists a constant C0 > 0 such that for any θ ∈ (0, 14), there exists
ε1(θ) > 0 such that for any solution u ∈ C∞(M× [0,T),S2) of (1.2), z0 = (x0, t0) ∈ M×(0,T)
and 0 < r ≤ miniM ,√
t0, if u satisfies
r−n∫
Pr(z0)|∇u|2 ≤ ε2
1(θ),
then we have
(θr)−n−2∫
Pθr(z0)|u − uz0,θr|
2 ≤ C0θ2r−n
∫Pr(z0)
|∇u|2, (2.7)
where uz0,θr =1
|Pθr(z0)|
∫Pθr(z0) u.
2.1 Proof of Lemma 2.3
For simplicity, we may assume n = 4. In fact, if n ≤ 3, then we let M = M × S4−n,
g(x, y) = g(x)+h0(y) with h0 the standard metric on S4−n, and define u(x, y, t) = u(x, t) for
x ∈ M, y ∈ S4−n. One can easily check that u ∈ C∞(M× (0,T),S2) solves (1.2) and satisfies
r−4∫
Pr(z0)|∇u|2 ≤ ε2
0. Hence it suffices to prove (2.6) for u. Since it is a local result, we
may further assume that M = R4 and g is the Euclidean metric. One can modify without
difficulties the following argument to handle the general case, see [4] for example.
Now we have
Claim. For any δ ∈ (0, 1), there exist C(δ) > 0 and ε2(δ) > 0 such that if
r−4∫
Pr(z0)|∇u|2 ≤ ε2
2(δ),
then
( r8
)−4∫
P r8(z0)
|∇u|2 ≤ δr−4∫
Pr(z0)|∇u|2 +
C(δ)δ
r−6∫
Pr(z0)|u − uPr(z0)|
2. (2.8)
First, by considering ur(x, t) = u(z0+(rx, r2t)) : R4×(− t0r2 , 0) → S2,we may assume
r = 1, z0 = (0, 0), and u ∈ C∞(R4 × (−1, 0],S2) solves (1.2). Denote Br(0) by Br and Pr(0, 0)
by Pr. Now we divide the proof of the claim into three steps.
Step 1 (slice monotonicity inequality). For any t ∈ (−1, 0], x0 ∈ R4, 0 < r1 ≤ r2 < +∞, it
holds
r−21
∫Br1 (x0)
|∇u(t)|2 ≤ 2r−22
∫Br2 (x0)
|∇u(t)|2 + 2∫
Br2 (x0)|ut|
2. (2.9)
8 S. Ding and C. Wang
It is well-known that (2.9) follows from the standard Pohozaev type argument
(see [10, 16, 17] for more details). Here we sketch the proof. Assume x0 = 0, let R(u)(p) :=
αp + βu ∧ p : R3→ R3. Since u ∈ C∞(R4 × (−1, 0],S2), we multiply (1.2) by x · ∇u(t) and
integrate it over Br to get
∫Br
R(u)(ut)x · ∇u =
∫Br
∆ux · ∇u
= r∫
∂Br
∣∣∣∣∂u∂r
∣∣∣∣2
+
∫Br
|∇u|2 −r2
∫∂Br
|∇u|2. (2.10)
This, combined with |R(u)(ut)| = |ut|, yields
ddr
(r−2
∫Br
|∇u|2
2
)= r−2
∫∂Br
∣∣∣∣∂u∂r
∣∣∣∣2
− r−3∫
Br
R(u)(ut)x · ∇u
≥ r−2∫
∂Br
∣∣∣∣∂u∂r
∣∣∣∣2
+ddr
(r−1
∫Br
|ut|
∣∣∣∣∂u∂r
∣∣∣∣)
− r−1∫
∂Br
|ut|
∣∣∣∣∂u∂r
∣∣∣∣ .Integrating this inequality from r1 to r2, we have
r−22
∫Br2
|∇u|2 ≥ r−21
∫Br1
|∇u|2 + 2∫
Br2 \Br1
r−2
∣∣∣∣∂u∂r
∣∣∣∣2
−2r−11
∫Br1
|ut|
∣∣∣∣∂u∂r
∣∣∣∣ − 2∫ r2
r1
r−1∫
∂Br
|ut|
∣∣∣∣∂u∂r
∣∣∣∣ . (2.11)
By the Holder inequality, we have
2r−11
∫Br1
|ut|
∣∣∣∣∂u∂r
∣∣∣∣ ≤ 12
r−21
∫Br1
|∇u|2 + 2∫
Br2
|ut|2,
and
2∫ r2
r1
r−1∫
∂Br
|ut|
∣∣∣∣∂u∂r
∣∣∣∣ ≤∫
Br2 \Br1
r−2|∂u∂r
|2 +
∫Br2
|ut|2.
Putting these inequalities into (2.11), we obtain (2.9).
Step 2 (estimate on good Λ-slices). For any Λ ≥ 1, define the set of good Λ-slices by
GΛ=
t ∈
[−
14, 0
]|
∫B 1
2
|ut|2 ≤ Λ2
∫P 1
2
|ut|2
, (2.12)
Finite Time Singularity of the Landau-Lifshitz-Gilbert Equation 9
and the set of bad Λ-slices BΛ = [− 14 , 0] \ GΛ. By Fubini’s theorem, we have
|BΛ| ≤ 1Λ2
. (2.13)
For any t ∈ GΛ, by (2.3) and (2.12), we have
∫B 1
2
|∇u(t)|2 +
∫B 1
2
|ut(t)|2 ≤ CΛ2∫
P1
|∇u|2. (2.14)
This and (2.9) imply that for any t ∈ GΛ, we have
sup
s−2
∫Bs(x)
|∇u(t)|2 | x ∈ B 14, 0 < s ≤ 1
4
≤ C
∫B 1
2
(|∇u(t)|2 + |ut(t)|2)
≤ CΛ2∫
P1
|∇u|2. (2.15)
Let η ∈ C∞0 (B1) be such that 0 ≤ η ≤ 1, η = 1 on B 18, and |∇η| ≤ 16. For any t ∈ GΛ fixed,