Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 85, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
REGULARITY CRITERIA FOR THE WAVE MAP AND
RELATED SYSTEMS
JISHAN FAN, YONG ZHOU
Abstract. We obtain some regularity criteria for the wave map, a liquid
crystals model, and the Hall-MHD with ion-slip effect.
1. Introduction
First, we consider the nD wave maps d : R1+n → Sm ⊂ R1+m which obey the
nonlinear wave equation
∂t2 d − ∆d = d(|∇d|2 − |∂t d|2 )
(1.1)
with the initial conditions
(d, ∂t d)(·, 0) = (d0 , d1 ), d0 ∈ Sm ,
d0 · d1 = 0.
(1.2)
Wave maps have wide applications in physics from the harmonic gauge in general
relativity to the nonlinear σ-models in particle physics.
Local well-posedness of (1.1) (1.2) has been proved by Tao [20]. Shatah [19]
showed that solutions to the Cauchy problem for wave maps may blow up in finite
time. However, some smallness assumption on the initial data or integrability
condition on the solution itself are sufficient to guarantee the regularity. Fan and
Ozawa [11] obtained the regularity criterion
0
∇d, ∂t d ∈ L1 (0, T ; Ḃ∞,∞
(Rn ))
(1.3)
when n = 2.
The first aim of this article is to prove a following regularity criterion when
n ≥ 3.
Theorem 1.1. Let n ≥ 3 and (∇d0 , d1 ) ∈ H 1+s (Rn ) with s > n2 , |d0 | = 1, d0 · d1 =
0 and d be a smooth solution of (1.1), (1.2). If (1.3) and ∂t d ∈ L∞ (0, T ; Ln (Rn ))
hold true with 0 < T < ∞, then the solution d can be extended beyond T > 0.
Next, we consider the liquid crystals model [2, 3, 4, 16]:
∂t u + u · ∇u + ∇π − ∆u = −∇ · (∇d ∇d),
2
∂t d + u · ∇d − ∆d = d|∇d| ,
|d| = 1,
div u = 0,
2010 Mathematics Subject Classification. 35K55, 35Q35, 70S15.
Key words and phrases. Regularity criterion; wave map; liquid crystals; Hall-MHD.
c
2016
Texas State University.
Submitted February 18, 2016. Published March 29, 2016.
1
(1.4)
(1.5)
(1.6)
2
J. FAN, Y. ZHOU
(u, d)(·, 0) = (u0 , d0 )
EJDE-2016/85
in Rn , |d0 | = 1.
(1.7)
Here
P u is the velocity, π is the pressure, d is the direction vector, and (∇d∇d)i,j :=
k ∂i dk ∂j dk , and hence
X
1
∇ · (∇d ∇d) =
∆dk ∇dk + ∇|∇d|2 .
2
k
If u = 0, then (1.5) is the harmonic heat flow.
Fan-Gao-Guo [9] proved the blow-up criterion
0
u, ∇d ∈ L2 (0, T ; Ḃ∞,∞
)
(1.8)
when n = 3. One can find other related results in [8, 24] and references therein.
We will prove the following theorem.
Theorem 1.2. Let n ≥ 3 and s > n2 be an integer. Let u0 and d0 satisfy u0 , ∇d0 ∈
H s , div u0 = 0, and |d0 | = 1 in Rn . Let (u, d) be a local strong solution to the
problem (1.4)-(1.7). If ∇u and ∇2 d satisfy
2
−α
∇u, ∇2 d ∈ L 2−α (0, T ; Ḃ∞,∞
(Rn ))
(1.9)
with 0 < α < 1 and 0 < T < ∞, then the solution (u, d) can be extended beyond
T > 0.
Also we consider the incompressible MHD with the Hall or ion-slip system
1
∂t u + u · ∇u + ∇ π + |b|2 − ∆u = b · ∇b,
(1.10)
2
∂t b + u · ∇b − b · ∇u + h rot(rot b × b) − γ rot[(rot b × b) × b] = ∆b,
(1.11)
div u = div b = 0,
(u, b)(·, 0) = (u0 , b0 )
(1.12)
3
in R .
(1.13)
Here b is the magnetic field. h is the Hall effect coefficient, and γ ≥ 0 the ion-slip
effect coefficient, respectively.
Applications of the Hall-MHD system cover a very wide range of physical subjects, such as, magnetic reconnection in space plasmas, star formation, neutron
stars, and geo-dynamos.
Very recently, Zhang [23] obtained the regularity criterion
2
−α
u ∈ L 1−α (0, T ; Ḃ∞,∞
),
2
−β
∇b ∈ L 1−β (0, T ; Ḃ∞,∞
)
(1.14)
with −1 < α < 1 and 0 < β < 1 when h = 1 and γ = 0.
Local well-posedness of strong solutions to (1.10)-(1.13) has been proved by Fan,
Jia, Nakamura and Zhou [10], they also obtained the regularity criterion
2q
u ∈ L q−3 (0, T ; Lq ),
b ∈ L∞ (0, T ; L∞ ),
2p
∇b ∈ L p−3 (0, T ; Lp )
(1.15)
with 3 < p, q ≤ ∞. For standard Hall-MHD system we refer to [1, 5, 6, 7, 13, 21, 22]
and references therein.
By the method in [23], we will refine (1.15) as follows.
Theorem 1.3. Let u0 , b0 ∈ H 2 with div u0 = div b0 = 0 in R3 . Let (u, b) be a
local strong solution to the problem (1.10)-(1.13). If u and b satisfy (1.14) and
b ∈ L∞ (0, T ; L∞ ) with 0 < T < ∞, then the solution (u, b) can be extended beyond
T > 0.
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WAVE MAP AND RELATED SYSTEMS
3
In the following proofs, we use the logarithmic Sobolev inequality [15]:
k∇dkL∞ ≤ C(1 + k∇dkḂ 0
log(e + k∇dkH 1+s )),
(1.16)
k∂t dkL∞ ≤ C(1 + k∂t dkḂ 0
log(e + k∂t dkH 1+s ))
(1.17)
∞,∞
∞,∞
for s > n2 − 1, and the bilinear product and commutator estimates due to KatoPonce [14]:
kΛs (f g)kLp ≤ C(kΛs f kLp1 kgkLq1 + kf kLp2 kΛs gkLq2 ),
s
s
s−1
kΛ (f g) − f Λ gkLp ≤ C(k∇f kLp1 kΛ
s
gkLq1 + kΛ f kLp2 kgkLq2 ),
(1.18)
(1.19)
1
with s > 0, Λ := (−∆) 2 and p1 = p11 + q11 = p12 + q12 .
We also use the Gagliardo-Nirenberg inequalities
k∇dk2L2p ≤ CkdkL∞ k∇2 dkLp ,
(1.20)
2+s
k∇ dkLp ≤ Ck∇dk1−θ
dkθL2 ,
L∞ kΛ
kΛ1+s dk 2p ≤ Ck∇dkθL∞ kΛ2+s dk1−θ
L2
L p−2
(1.21)
2
(1.22)
with p := 2s + 2 and θ := 1/(1 + s).
We also use the improved Gagliardo-Nirenberg inequalities [12, 17, 18]:
1
kukθḢ1s+α ,
k∇ukLq1 ≤ Ck∇uk1−θ
Ḃ −α
(1.23)
∞,∞
kΛs uk
with q1 :=
2(s−1+2α)
α
2q1
L q1 −2
1
,
≤ Ck∇ukθḂ1−α kuk1−θ
Ḣ s+α
(1.24)
∞,∞
and θ1 := 2/q1 , and
2
k∇dkθḢ2s+α ,
k∇dkLq2 ≤ Ck∇dk1−θ
Ḃ −α
(1.25)
∞,∞
kΛs ∇dk
2q2
L q2 −2
with q2 :=
2(s+2α)
α
and θ2 :=
2
,
≤ Ck∇dkθḂ2−α k∇dk1−θ
Ḣ s+α
(1.26)
∞,∞
2
q2 ,
1−α
1
2−α
2
−α
≤ CkdkL2−α
k∇dkḂ∞,∞
∞ k∇ dk −α ,
Ḃ
(1.27)
θ̃k
θ̃k
kDk ukLpk ≤ Ck∇uk1−
kuk1−
,
Ḣ s+α
Ḃ −α
(1.28)
∞,∞
and
∞,∞
kDs+2−k dk
2pk
L pk −2
with pk :=
2
θ̃k
and θ̃k :=
k+α−1
s+2α−1 ,
θ̃k
≤ Ck∇2 dkθ̃Ḃk−α k∇dk1−
,
Ḣ s+α
and
2
−α kuk
k∇uk3L3 ≤ CkukḂ∞,∞
Ḣ
kuk
Ḣ
3+α
2
(1.29)
∞,∞
1−α
2
L2
≤ Ck∇uk
3+α
2
k∆uk
1+α
2
L2
with − 1 < α < 1,
(1.30)
with − 1 < α < 1,
(1.31)
with 0 < β < 1,
(1.32)
and
k∇bk2L4 ≤ Ck∇bkḂ∞,∞
−β kbk 1+β
Ḣ
β
kbkḢ 1+β ≤ Ck∇bk1−β
L2 k∆bkL2
with 0 < β < 1.
(1.33)
4
J. FAN, Y. ZHOU
EJDE-2016/85
2. Proof of Theorem 1.1
Testing (1.1) by ∂t d and using |d| = 1 and d · ∂t d = 0, we easily get the conservation of the energy:
Z
d
(|∂t d|2 + |∇d|2 )dx = 0.
(2.1)
dt
Applying the operator Λ1+s to equation (1.1), testing by Λ1+s ∂t d, using (1.18),
(1.16), (1.17), (1.20), (1.21) and (1.22), we reach
Z
1 d
(|Λ1+s ∂t d|2 + |Λ2+s d|2 )dx
2 dt
Z
= Λ1+s (d|∇d|2 − d|∂t d|2 )Λ1+s ∂t ddx
≤ (kΛ1+s (d|∇d|2 )kL2 + kΛ1+s (d|∂t d|2 )kL2 )kΛ1+s ∂t dkL2
≤ C(kdkL∞ kΛ1+s (|∇d|2 )kL2 + k∇dk2L2p kΛ1+s dk
)kΛ1+s ∂t dkL2
2p
L p−2
+ C(kdkL∞ kΛ1+s (|∂t d|2 )kL2 + k∂t dk2L2n kΛ1+s dk
≤ C(k∇dkL∞ kΛ2+s dkL2 + k∇2 dkLp kΛ1+s dk
2p
L p−2
2n
)kΛ1+s ∂t dkL2
L n−2
1+s
)kΛ
∂t dkL2
+ C(k∂t dkL∞ kΛ1+s ∂t dkL2 + k∂t dkLn k∂t dkL∞ kΛ2+s dkL2 )kΛ1+s ∂t dkL2
≤ Ck∇dkL∞ kΛ2+s dkL2 kΛ1+s ∂t dkL2
+ Ck∂t dkL∞ kΛ1+s ∂t dk2L2 + Ck∂t dkL∞ kΛ2+s dkL2 kΛ1+s ∂t dkL2
≤ C(k∇dkL∞ + k∂t dkL∞ )(kΛ2+s dk2L2 + kΛ1+s ∂t dk2L2 )
≤ C(1 + k∇dkḂ 0
∞,∞
+ k∂t dkḂ 0
∞,∞
) log(e + y 2 )y 2 ,
with y 2 := kΛ1+s ∂t dk2L2 + kΛ2+s dk2L2 , which gives
sup (kΛ1+s ∂t dk2L2 + kΛ2+s dk2L2 ) ≤ C.
0≤t≤T
This completes the proof.
3. Proof of Theorem 1.2
Since it is easy to prove that there are T0 > 0 and a unique strong solution
(u, π, d) to the problem (1.4)-(1.7) in [0, T0 ], we only need to prove a priori estimates.
Testing (1.4) by u and using (1.6), we see that
Z
Z
Z
1 d
|u|2 dx + |∇u|2 dx = − (u · ∇)d · ∆ddx.
(3.1)
2 dt
Testing (1.5) by −∆d, using d∆d = −|∇d|2 and |d| = 1, we find that
Z
Z
Z
Z
1 d
2
2
|∇d| dx + |∆d| dx = (u · ∇)d · ∆ddx + (d∆d)2 dx
2 dt
Z
Z
≤ (u · ∇)d · ∆ddx + |∆d|2 dx.
Summing up (3.1) and (3.2), we have
Z
Z
2
2
(|u| + |∇d| )dx ≤ (|u0 |2 + |∇d0 |2 )dx.
(3.2)
(3.3)
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WAVE MAP AND RELATED SYSTEMS
5
Applying Ds to (1.4), testing by Ds u, using (1.6), (1.18), (1.19), (1.23), (1.24),
(1.25), (1.26) and (1.27), we obtain
1 d
2 dt
Z
|Ds u|2 dx +
Z
Z
|D1+s u|2 dx
s
=−
s
Z
s
(D (u · ∇u) − u∇D u)D udx +
Ds (∇d ∇d) : ∇Ds udx
≤ Ck∇ukLq1 kDs uk
≤
s
s
s
2q1 kD ukL2 + Ck∇dkLq2 kD ∇dk
2q2 k∇D ukL2
L q1 −2
L q2 −2
s+α
s+α
−α kD
−α kD
Ck∇ukḂ∞,∞
ukL2 kDs ukL2 + Ck∇dkḂ∞,∞
∇dkL2 kDΛs ukL2
2−α
s
1+s
−α kD uk 2 kD
≤ Ck∇ukḂ∞,∞
ukα
L2
L
(3.4)
1−α
1−α
1+s
s+1
s+2
ukL2
dkL
dkα
+ Ck∇2 dkḂ2−α
2 kD
−α kD
L2 kD
∞,∞
2
1
1
s
2
≤ kD1+s uk2L2 + kDs+2 dk2L2 + Ck∇ukḂ2−α
−α kD ukL2
∞,∞
8
8
2
s+1
dk2L2 .
+ Ck∇2 dkḂ2−α
−α kD
∞,∞
Applying Ds+1 to (1.5), testing by Ds+1 d and using (1.6), we obtain
Z
Z
1 d
s+1 2
|D d| dx + |Ds+2 d|2 dx
2 dt
Z
= Ds+1 (d|∇d|2 )Ds+1 ddx
Z
− (Ds+1 (u · ∇d) − u∇Ds+1 d)Ds+1 d dx =: I1 + I2 .
(3.5)
Using (1.18), |d| = 1, (1.25), (1.26), and (1.27), we bound I1 as follows.
I1 ≤ kDs+1 (d|∇d|2 )k
≤ C(kdkL∞ kD
2q2
kDs+1 dk
2q2
L q2 −2
L q2 +2
s+1
2
(|∇d| )k
2q2
L q2 +2
+ k∇dk2Lq2 kDs+1 dk
≤ C(k∇dkLq2 kDs+2 dkL2 + k∇dk2Lq2 kDs+1 dk
2q2
L q2 −2
+
≤ Ck∇dk2Ḃ −α k∇dk2Ḣ s+α +
1
kDs+2 dk2L2
16
∞,∞
≤
)kD
)kDs+1 dk
2q2
L q2 −2
dk
2q2
L q2 −2
1
kDs+2 dk2L2
16
≤ Ck∇dk2Lq2 kDs+1 dk2
2q2
L q2 −2
2q2
L q2 −2
s+1
2
1 s+2 2
s+1
kD dkL2 + Ck∇2 dkḂ2−α
dk2L2 .
−α kD
∞,∞
8
Using the Leibniz rule, we write I2 as follows.
Z
I2 = −
(C1 DuDs+1 d +
s
X
k=2
=:
I21
+
s
X
k=2
I2k
+
I2s+1 .
Ck Dk uDs+2−k d + Cs+1 Ds+1 u · ∇d)Ds+1 ddx
(3.6)
6
J. FAN, Y. ZHOU
EJDE-2016/85
By the same calculations as that of I1 , we have
I2s+1 ≤ Ck∇dkLq2 kDs+1 dk
2q2
L q2 −2
kDs+1 ukL2
2
1
s+1
kDs+1 uk2L2 + Ck∇2 dkḂ2−α
dk2L2 .
−α kD
∞,∞
16
≤
(3.7)
Using (1.23) and (1.24), we bound I21 as follows.
I21 ≤ Ck∇ukLq1 kDs+1 dk
2q1
L q1 −2
kDs+1 dkL2
θ1
1
1
≤ Ck∇uk1−θ
· k∇2 dkθḂ1−α k∇dk1−θ
kDs+1 dkL2
kukḢ
s+α
Ḣ s+α
Ḃ −α
∞,∞
∞,∞
s+1
−α
−α )(kuk s+α + k∇dk s+α )kD
≤ C(k∇ukḂ∞,∞
+ k∇2 dkḂ∞,∞
dkL2
Ḣ
Ḣ
(3.8)
1
1
kDs+1 uk2L2 + kDs+2 dk2L2
≤
16
16
2
2
s
2
s+1
+ k∇2 dkḂ2−α
dk2L2 ).
+ C(k∇ukḂ2−α
−α
−α )(kD ukL2 + kD
∞,∞
∞,∞
Using (1.28) and (1.29), we bound
s
X
Ps
k
k=2 I2
as follows.
θ̃k
θ̃k
1−θ̃k
1−θ̃k
2
s+1
I2k ≤ Ck∇ukḂ
dkL2
−α kuk s+α k∇ dk −α k∇dk s+α kD
Ḣ
Ḣ
Ḃ
∞,∞
∞,∞
k=2
s+1
−α )(kuk s+α + k∇dk s+α )kD
−α
dkL2
+ k∇2 dkḂ∞,∞
≤ C(k∇ukḂ∞,∞
Ḣ
Ḣ
−α )(kuk s+α + k∇dk s+α )
−α
+ k∇2 dkḂ∞,∞
≤ C(k∇ukḂ∞,∞
Ḣ
Ḣ
× (kDs ukL2 + kDs+1 dkL2 )
1
1
≤
kDs+1 uk2L2 + kDs+2 dk2L2
16
16
2
2
∞,∞
∞,∞
(3.9)
s
2
s+1
+ k∇2 dkḂ2−α
dk2L2 ).
+ C(k∇ukḂ2−α
−α
−α )(kD ukL2 + kD
Inserting the above estimates into (3.5) and combining with (3.4) and using the
Gronwall inequality, we arrive at
kDs ukL∞ (0,T ;L2 ) + kDs+1 dkL∞ (0,T ;L2 ) ≤ C.
This completes the proof.
4. Proof of Theorem 1.3
We only need to show a priori estimates. For simplicity, we will take h = γ = 1.
First, testing (1.10) by u and using (1.12), we see that
Z
Z
Z
1 d
|u|2 dx + |∇u|2 dx = (b · ∇)b · udx.
(4.1)
2 dt
Testing (1.11) by b and using (1.12), we find that
Z
Z
Z
Z
1 d
|b|2 dx + |∇b|2 dx + |b × rot b|2 dx = (b · ∇)u · bdx.
2 dt
Summing up (4.1) and (4.2), we obtain
Z
Z
1 d
2
2
(|u| + |b| )dx + (|∇u|2 + |∇b|2 + |b × rot b|2 )dx = 0.
2 dt
(4.2)
(4.3)
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WAVE MAP AND RELATED SYSTEMS
7
Testing (1.10) by −∆u, using (1.12), (1.30) and (1.31), we infer that
Z
Z
1 d
|∇u|2 dx + |∆u|2 dx
2 dt
Z
Z
= (u.∇)u · ∆udx − (b · ∇)b · ∆udx
Z
XZ
∂j ui ∂i u∂j udx − (b · ∇)b · ∆udx
=−
i,j
(4.4)
≤ Ck∇uk3L3 + kbkL∞ k∇bkL2 k∆ukL2
2
−α kuk
≤ CkukḂ∞,∞
Ḣ
3+α
2
+ Ck∇bkL2 k∆ukL2
1−α
1+α
−α k∇uk 2 k∆uk 2
≤ CkukḂ∞,∞
+ Ck∇bkL2 k∆ukL2
L
L
≤
2
1
2
2
k∆uk2L2 + CkukḂ1−α
−α k∇ukL2 + Ck∇bkL2 .
∞,∞
8
Testing (1.11) by −∆b and using (1.12), we deduce that
Z
Z
1 d
2
|∇b| dx + |∆b|2 dx
2 dt
Z
Z
= (u · ∇)b · ∆bdx − (b · ∇)u · ∆bdx
Z
Z
+ (rot b × b) rot ∆bdx − [(rot b × b) × b] rot ∆bdx
(4.5)
=: `1 + `2 + `3 + `4 .
We bound `1 and `2 as follows.
XZ
XZ
`1 =
ui ∂i b∂j2 bdx = −
∂j ui ∂i b∂j bdx ≤ Ck∇ukL2 k∇bk2L4
i,j
i,j
≤ Ck∇ukL2 kbkL∞ k∆bkL2 ≤ Ck∇ukL2 k∆bkL2 ≤
`2 ≤ kbkL∞ k∇ukL2 k∆bkL2 ≤ Ck∇ukL2 k∆bkL2 ≤
1
k∆bk2L2 + Ck∇uk2L2 .
16
1
k∆bk2L2 + Ck∇uk2L2 .
16
Using (1.32) and (1.33), we bound `3 and `4 as follows.
XZ
`3 = −
(rot b × ∂i b)∂i rot bdx ≤ Ck∇bk2L4 k∆bkL2
i
1−β
1+β
≤ Ck∇bkḂ∞,∞
−β kbk 1+β k∆bkL2 ≤ Ck∇bk −β k∇bk 2 k∆bk 2
Ḣ
L
L
Ḃ∞,∞
2
1
2
k∆bk2L2 + Ck∇bk 1−β
−β k∇bkL2 .
Ḃ∞,∞
16
XZ
XZ
`4 =
∂i [(rot b × b) × b]∂i rot bdx ≤
[(rot b × ∂i b) × b]∂i rot bdx
≤
i
+
i
XZ
[(rot b × b) × ∂i b]∂i rot bdx ≤ CkbkL∞ k∇bk2L4 k∆bkL2
i
2
1
2
≤
k∆bk2L2 + Ck∇bk 1−β
−β k∇bkL2 .
Ḃ
∞,∞
16
8
J. FAN, Y. ZHOU
EJDE-2016/85
Inserting the above estimates into (4.5), and combining this with (4.4), and using
the Gronwall inequality, we conclude that
k∇ukL∞ (0,T ;L2 ) + k∇bkL∞ (0,T ;L2 ) ≤ C.
(4.6)
This completes the proof by (1.15).
Acknowledgment. This work is partially supported by NSFC (No. 11171154).
The authors would like to thank the referees for their careful reading and helpful
suggestions.
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Jishan Fan
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037,
China
E-mail address: fanjishan@njfu.com.cn
Yong Zhou (corresponding author)
School of Mathematics, Shanghai University of Finance and Economics,
Shanghai 200433, China.
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
E-mail address: yzhou@sufe.edu.cn