|∂tQε|2dx+
(∇ ·ξ)∇qdεF(Qε):∂tQεdx +
(∇ ·ξ)HI· ∇ψεdx+
Hε·HI|∇Qε|dx
−
∇HI :(ξ−nε)⊗2|∇ψε|dx +Jε1+
(∇ ·HI)ε
2|∇Qε|2+1
εFε(Qε)− |∇ψε| dx +
(∇ ·HI) (1−ξ·nε)|∇ψε|dx+Jε2.
Now we complete squares for the first four terms on the RHS of (4.33): Re-ordering terms, we have
−ε|∂tQε|2+(∇ ·ξ)∇qdεF(Qε):∂tQε+(∇ ·ξ)HI · ∇ψε+Hε·HI|∇Qε|
= − 1 2ε
|ε∂tQε|2−2(∇ ·ξ)∇qdεF(Qε):ε∂tQε+(∇ ·ξ)2|∇qdεF(Qε)|2
− 1
2ε|ε∂tQε|2+ 1
2ε(∇ ·ξ)2|∇qdεF(Qε)|2+(∇ ·ξ)HI · ∇ψε
− 1 2ε
|Hε|2−2Hε·ε|∇Qε|HI +ε2|∇Qε|2|HI|2
+ 1 2ε
|Hε|2+ε2|∇Qε|2|HI|2
= − 1
2εε∂tQε−(∇ ·ξ)∇qdεF(Qε)2− 1
2εHε−ε|∇Qε|HI2
− 1
2ε|ε∂tQε|2+ 1 2ε|Hε|2 + 1
2ε
(∇ ·ξ)2|∇qdεF(Qε)|2+2ε(∇ ·ξ)∇ψε·HI+ |εQε∇Qε|2|HI|2
+ε 2
|∇Qε|2− |Qε∇Qε|2
|HI|2. (4.33)
Using the definition (4.8b) of the normal nεand the chain rule in form of (4.11a), the terms in (4.33) form the last missing square. Integrating over the domainand
substituting into (4.33) we arrive at (4.19).
5. Convergence to the Harmonic Map Heat Flow
This section is devoted to the proof of Theorem2.1. We start with a lemma about uniform estimates ofQε.
Lemma 5.1.There exists a universal constant C =C(I0)such that Proof. We first establish a priori estimates of the solutionsQεwhich are indepen-dent ofε. It follows from (4.18) and the assumption (2.12) that
ess supt∈[0,T]1 On the other hand, using the orthogonal projection (4.10), we obtain
ε∂tQε− ∇qdεF(Qε)(∇ ·ξ)2=ε∂tQε−εQε∂tQε2
+εQε∂tQε− ∇qdεF(Qε)(∇ ·ξ)2. This, together with (5.4), yields
1 The above two estimates together with (4.13b) implies (5.1). Moreover, (5.2) fol-lows from (5.4) and (4.13e). Now we turn to the time derivative. It folfol-lows from (5.4) that
By (1.13a) and (4.8a) we have Hε = −ε∂tQε : |∇∇QQεε|. Using this, we can expand the integrand in the above estimate and apply the Cauchy-Schwarz inequality to obtain
ε2|∂tQε|2− |Hε|2+ |Hε−εHI|∇Qε||2
=ε2|∂tQε|2+ε2|HI|2|∇Qε|2+2ε2(HI· ∇)Qε:∂tQε ε2|∂tQε|2+ε2|(HI· ∇)Qε|2+2ε2(HI· ∇)Qε:∂tQε
=ε2|∂tQε+(HI· ∇)Qε|2. This implies
T 0
|∂tQε+(HI · ∇)Qε|2dxdt e(1+T)C(I0), (5.7)
so combining (5.2) with (5.7) leads us to (5.3).
With the above uniform estimates, we can prove the following convergence result:
Proposition 5.2.There exists a subsequence ofεk>0such that ∂tQεk,Qεk
=(
∂tQεk−Qεk∂tQεk,Qεk )
k→∞
−−−→ ¯S0(x,t)weakly in L2(0,T;L2()), (5.8a) ∂iQεk,Qεk
=(
∂iQεk−Qεk∂iQεk,Qεk )
k→∞
−−−→ ¯Si(x,t)weakly-star in L∞(0,T;L2()) (5.8b) for1id. Moreover,
∂tQεk −−−→k→∞ ∂tQ, weakly in L2(0,T;L2loc(±(t))), (5.9a)
∇Qεk −−−→ ∇k→∞ Q, weakly in L∞(0,T;L2loc(±(t))), (5.9b) Qεk −−−→k→∞ Q, strongly in C([0,T];L2loc(±(t))), (5.9c) where Q=Q(x,t)is represented as
Q(x,t)=s±
u(x,t)⊗u(x,t)−1 3I3
a.e.(x,t)∈±T (5.10) for some unit vector field
u∈ L∞(0,T;H1(+(t);S2))∩ H1(0,T;L2(+(t);S2))
∩C([0,T];L2(+(t);S2)). (5.11)
Proof. We first deduce from (1.6) and (2.10) thatdF(Q)is an isotropic function, which only depends on the eigenvalue of Q ∈ Q. So by (2.8b), the mollified distance functiondεF(Q)is isotropic and smooth inQ. By [4] there exists a smooth symmetric function g(λ1, λ2, λ3)such thatdεF(Q) = g(λ1(Q), λ2(Q), λ3(Q)). Let Q0 ∈ Qbe a matrix having distinct eigenvalues, thenλi(Q)as well as the eigenvectors ni(Q)are real-analytic functions of Qnear Q0, and then by chain rule
∂dεF(Q)
∂Q = 3 k=1
∂g
∂λk
∂λk
∂Q = 3 k=1
∂g
∂λk
nk(Q)⊗nk(Q),
in a neighborhood ofQ0. (5.12)
In a neighborhood of Q0, we also haveQ= 3k=1λk(Q)nk(Q)⊗nk(Q). So we
have (
∇qdεF(Q),Q
)=0, (5.13)
holds in a neighborhood of Q0 having distinct eigenvalues, and thus for every Q∈Qby continuity. Now in view of (4.10), we have
[Qε∂tQε(x,t),Qε(x,t)] =0,
[Qε∂iQε(x,t),Qε(x,t)] =0a.e. (x,t)∈T (5.14) for 1i d. This together with (3.29) and (5.1) implies
[∂tQε,Qε]L2(0,T;L2())+ [∇Qε,Qε]L∞(0,T;L2())
=
∂tQε−Qε∂tQε,Qε
L2(0,T;L2())
+
∇Qε−Qε∇Qε,Qε
L∞(0,T;L2())C (5.15) for some C independent ofε. Combining this estimate with weak compactness implies (5.8).
It follows from (5.2), (5.3), (3.29), and the Aubin-Lions lemma that, for any δ >0, there exists a subsequenceεk =εk(δ) >0 such that
∂tQεk −−−→k→∞ ∂tQ¯δ, weakly inL2(0,T;L2(±(t)\It(δ))), (5.16a)
∇Qεk −−−→ ∇ ¯k→∞ Qδ, weakly-star inL∞(0,T;L2(±(t)\It(δ))), (5.16b) Qεk −−−→ ¯k→∞ Qδ, weakly-star inL∞(×(0,T)), (5.16c) Qεk −−−→ ¯k→∞ Qδ, strongly inC([0,T];L2(±(t)\It(δ))). (5.16d) By a diagonal argument, we infer there exists
Q∈L2(0,T;Hloc1 (±(t)))∩L∞(±),with∂tQ∈ L2(0,T;L2loc(±(t))) (5.17) such that the convergence (5.9) as well as
Q(x,t)= ¯Qδ(x,t) in L∞(0,T;H1(±(t)\It(δ))
hold for everyδ >0 and everyt ∈ [0,T]. Moreover, by (5.17), the interpolation theory and (5.16c), we have
Q∈C([0,T];L2(±(t))∩L∞(×(0,T)).
To prove (5.10), we first deduce thatF(Q)has the same regularity asQin (5.17), and thus by interpolation theory we obtain
F(Q)∈C([0,T];L2(±(t)).
We use (5.9c), (5.2), and Fatou’s lemma to deduce that
F(Q(x,t))=0, ∀t∈ [0,T] and a.e. inx∈±(t). (5.18) This, together with (1.10), implies
|Q|(x,t)∈ {0,s+ 2
3}, ∀t∈ [0,T] and a.e. inx∈±(t). (5.19) By taking theL2-norm, we obtain two continuous functions:
f±(t):=Q(·,t)L2(±(t))∈C([0,T]; {0,s+
2
3|±(t)|}).
On the other hand, by the choice of the initial condition (2.21) and the convergence (5.16d), we deduce that
Q(x,0)=1+(0)s+
ui n(x)⊗ui n(x)−1 3I3
, a.e. in±(0)\I0(δ)
for anyδ >0 and thus forδ=0. This implies f+(0)=s+ 2
3|+(0)|, f−(0)=0 and thus
f+(t)=s+
2
3|+(t)|, f−(t)=0, ∀t ∈ [0,T].
This, together with (5.19), implies
Q(x,t)=0, ∀t∈ [0,T] and a.e. inx∈−(t), (5.20) Q(x,t)∈N, ∀t∈ [0,T] and a.e. inx∈+(t), (5.21) and thus (5.10) is proved.
By (5.10), (5.17), and the orientability theorem by Ball–Zarnescu [6, Section 3.2] implies thatQis uniaxial (5.10) for some
u∈L∞(0,T;Hloc1 (+(t);S2))with∂tu∈ L2(0,T;L2loc(+(t);S2)).(5.22) It remains to improve the integrability of∇x,tu. To this end, we choose a sequence
ψ(x,t)∈Cc∞(+T) such that ψ(x,t)−−−→→∞ 1+
T(x,t). (5.23)
Since |u| = 1 a.e., by (5.8a), (5.8b) and (5.9), we deduce that for almost every (x,t)∈+T, it holds that
ψS¯i =ψ[∂iQ,Q]=s+2ψ(∂iu⊗u−u⊗∂iu) , 0i d, (5.24) where∂0:=∂t. Note that for each fixedi ∈ {0, . . . ,3},
∂iu⊗u−u⊗∂iu=
⎛
⎝ 0 (∂iu∧u)3 −(∂iu∧u)2
−(∂iu∧u)3 0 (∂iu∧u)1
(∂iu∧u)2 −(∂iu∧u)1 0
⎞
⎠, (5.25)
where(∂iu∧u)kdenotes thek-th component of the 3-vector∂iu∧u. SinceS¯i are L2integrable inT, sending → ∞and applying the dominated convergence theorem to the above identity lead us to
∂tu∧u∈L∞(0,T;L2(+(t))), (5.26a)
∂iu∧u∈L2(0,T;L2(+(t))), fori ∈ {1, . . . ,d}. (5.26b) Retaining that u maps intoS2, we deduce
|∂tu|2= |∂tu∧u|2, |∂iu|2= |∂iu∧u|2a.e.in+T, 1i d,
so we improve (5.22) to (5.11).
Proof of Theorem2.1. In the course of the proof, we shall adopt the notationA : B=trATBfor anyA,B∈R3×3. We associate each testing vector fieldϕ(x,t)= (ϕ1, ϕ2, ϕ3)∈C1(T,R3)a matrix-valued function
(x,t)=
⎛
⎝ 0 ϕ3 −ϕ2
−ϕ3 0 ϕ1
ϕ2 −ϕ1 0
⎞
⎠ (5.27)
Since [∇qF(Qεk),Qεk] = 0, applying the anti-symmetric product [·,Qεk] to (1.13a) and integration by parts overT yields
T
∂tQεk,Qεk
:dxdt+
T
3 j=1
[∂jQεk,Qεk] :∂jdxdt =0. (5.28)
Note that no boundary integral will occur due to (1.13c). Recall that we denote It(δ)theδ−neighborhood ofIt. Equivalently, we can write the above equation by
±
T
0
±(t)\It(δ)
⎛
⎝∂tQεk,Qεk :+
3 j=1
[∂jQεk,Qεk] :∂j
⎞
⎠dxdt
+ T
0
It(δ)
⎛
⎝∂tQεk,Qεk :+
3 j=1
[∂jQεk,Qεk] :∂j
⎞
⎠dxdt =0. (5.29)
Using (5.9), (5.8) and (5.10), we can passk→ ∞and yield
By|u| =1 a.e., (5.10), (5.27) and (5.25), we obtain the following identities : [∂tQ,Q] :=s2+(∂tu⊗u−u⊗∂tu):=2s+2∂tu∧u·ϕ to the limitδ→0 in the above identity, which yields
T
This concludes the proof of Theorem2.1.
Acknowledgements. T. Laux is funded by the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) under GerGer-many’s Excellence Strategy – EXC-2047/1 – 390685813.
Y. Liu is partially supported by NSF of China under Grant 11971314.
Funding Open Access funding enabled and organized by Projekt DEAL.
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Tim Laux
Hausdorff Center for Mathematics, University of Bonn, Villa Maria, Endenicher Allee 62,
53115 Bonn Germany.
e-mail: tim.laux@hcm.uni-bonn.de and
Yuning Liu NYU Shanghai, 1555 Century Avenue,
Shanghai 200122 China.
e-mail: yl67@nyu.edu and
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North,
Shanghai 200062 China.
(Received October 19, 2020 / Accepted May 21, 2021) Published online June 28, 2021
© The Author(s)(2021)