We obtain some regularity criteria for the wave map, a liquid crystals model, and the Hall-MHD with ion-slip effect

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 thenD wave maps d :R¹⁺ⁿ → S^m ⊂R^1+m which obey the nonlinear wave equation

∂²_td−∆d=d(|∇d|²− |∂td|²) (1.1) with the initial conditions

(d, ∂td)(·,0) = (d0, d1), d0∈S^m, d0·d1= 0. (1.2) Wave maps have wide applications in physics from the harmonic gauge in general relativity to the nonlinearσ-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

∇d, ∂_td∈L¹(0, T; ˙B⁰_∞,∞(Rⁿ)) (1.3) whenn= 2.

The first aim of this article is to prove a following regularity criterion when n≥3.

Theorem 1.1. Letn≥3and(∇d0, d1)∈H^1+s(Rⁿ)withs > ⁿ₂,|d0|= 1,d0·d1= 0 anddbe a smooth solution of (1.1),(1.2). If (1.3)and∂td∈L^∞(0, T;Lⁿ(Rⁿ)) hold true with0< T <∞, then the solution dcan be extended beyondT >0.

Next, we consider the liquid crystals model [2, 3, 4, 16]:

∂_tu+u· ∇u+∇π−∆u=−∇ ·(∇d ∇d), (1.4)

∂_td+u· ∇d−∆d=d|∇d|², |d|= 1, (1.5)

divu= 0, (1.6)

2010Mathematics 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

(u, d)(·,0) = (u0, d0) inRⁿ, |d0|= 1. (1.7) Hereuis the velocity,πis the pressure,dis the direction vector, and (∇d∇d)i,j:=

P

k∂_id_k∂_jd_k, and hence

∇ ·(∇d ∇d) =X

k

∆dk∇dk+1

2∇|∇d|². Ifu= 0, then (1.5) is the harmonic heat flow.

Fan-Gao-Guo [9] proved the blow-up criterion

u,∇d∈L²(0, T; ˙B⁰_∞,∞) (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. Letn≥3ands > ⁿ₂ be an integer. Letu0 andd0satisfyu0,∇d0∈ H^s,divu0 = 0, and |d0| = 1 in Rⁿ. Let (u, d) be a local strong solution to the problem (1.4)-(1.7). If∇uand∇²dsatisfy

∇u,∇²d∈L^2−α2 (0, T; ˙B_∞,∞^−α (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

∂tu+u· ∇u+∇ π+1

2|b|²

−∆u=b· ∇b, (1.10)

∂tb+u· ∇b−b· ∇u+hrot(rotb×b)−γrot[(rotb×b)×b] = ∆b, (1.11)

divu= divb= 0, (1.12)

(u, b)(·,0) = (u0, b0) inR³. (1.13) Here bis the magnetic field. his the Hall effect coefficient, andγ ≥0 the ion-slip effect coefficient, respectively.

Applications of the Hall-MHD system cover a very wide range of physical sub- jects, such as, magnetic reconnection in space plasmas, star formation, neutron stars, and geo-dynamos.

Very recently, Zhang [23] obtained the regularity criterion

u∈L^1−α2 (0, T; ˙B_∞,∞^−α ), ∇b∈L^1−β2 (0, T; ˙B^−β_∞,∞) (1.14) with−1< α <1 and 0< β <1 whenh= 1 andγ= 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

u∈L^q−32q (0, T;L^q), b∈L^∞(0, T;L^∞), ∇b∈L^p−32p (0, T;L^p) (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 ∈ H² with divu0 = divb0 = 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^∞)with0< T <∞, then the solution (u, b)can be extended beyond T >0.

In the following proofs, we use the logarithmic Sobolev inequality [15]:

k∇dk_L^∞ ≤C(1 +k∇dkB˙⁰_∞,∞log(e+k∇dk_H1+s)), (1.16) k∂tdkL^∞ ≤C(1 +k∂tdkB˙_∞,∞⁰ log(e+k∂tdkH^1+s)) (1.17) for s > ⁿ₂ −1, and the bilinear product and commutator estimates due to Kato- Ponce [14]:

kΛ^s(f g)kL^p≤C(kΛ^sfkL^p1kgkL^q1 +kfkL^p2kΛ^sgkL^q2), (1.18) kΛ^s(f g)−fΛ^sgkL^p≤C(k∇fkL^p1kΛ^s−1gkL^q1 +kΛ^sfkL^p2kgkL^q2), (1.19) withs >0, Λ := (−∆)¹² and ¹_p = _p¹

1 +_q¹

1 =_p¹

2 +_q¹

2. We also use the Gagliardo-Nirenberg inequalities

k∇dk²_L2p≤CkdkL^∞k∇²dkL^p, (1.20) k∇²dkL^p≤Ck∇dk^1−θ_L∞kΛ^2+sdk^θ_L2, (1.21) kΛ^1+sdk

L

2p

p−2 ≤Ck∇dk^θ_L∞kΛ^2+sdk^1−θ_L2 (1.22) withp:= 2s+ 2 and θ:= 1/(1 +s).

We also use the improved Gagliardo-Nirenberg inequalities [12, 17, 18]:

k∇ukL^q1 ≤Ck∇uk^1−θ_˙ ¹

B∞,∞^−α kuk^θ_˙¹

H^s+α, (1.23)

kΛ^suk

L

2q1

q1−2 ≤Ck∇uk^θ_˙¹

B^−α∞,∞kuk^1−θ_˙ ¹

H^s+α, (1.24)

withq1:=^2(s−1+2α)_α andθ1:= 2/q1, and k∇dkL^q2 ≤Ck∇dk^1−θ_˙ ²

B^−α_∞,∞k∇dk^θ_˙²

H^s+α, (1.25)

kΛ^s∇dk

L

2q2

q2−2 ≤Ck∇dk^θ_˙²

B_∞,∞^−α k∇dk^1−θ_˙ ²

H^s+α, (1.26)

withq2:=^2(s+2α)_α andθ2:= _q²

2,

k∇dkB˙^−α∞,∞≤Ckdk

1 2−α

L^∞ k∇²dk

1−α 2−α

B˙^−α_∞,∞, (1.27) and

kD^kukL^pk ≤Ck∇uk¹⁻_˙ ^θ˜k

B^−α∞,∞kuk¹⁻_˙ ^θ˜k

H^s+α, (1.28)

kD^s+2−kdk

L

2pk pk−2

≤Ck∇²dk^θ˜_˙^k

B^−α∞,∞

k∇dk¹⁻_˙ ^θ˜k

H^s+α, (1.29)

withpk:= _˜²

θ_k and ˜θk:= _s+2α−1^k+α−1, and k∇uk³_L3 ≤CkukB˙^−α_∞,∞kuk²

H˙^3+α2

with −1< α <1, (1.30) kuk_˙

H^3+α2 ≤Ck∇uk

1−α 2

L² k∆uk_L^1+α2² with −1< α <1, (1.31) and

k∇bk²_L4 ≤Ck∇bkB˙∞,∞^−β kbkH˙^1+β with 0< β <1, (1.32) kbkH˙^1+β ≤Ck∇bk^1−β_L2 k∆bk^β_L2 with 0< β <1. (1.33)

2. Proof of Theorem 1.1

Testing (1.1) by∂tdand using|d|= 1 andd·∂td= 0, we easily get the conser- vation of the energy:

d dt

Z

(|∂td|²+|∇d|²)dx= 0. (2.1) Applying the operator Λ^1+s to equation (1.1), testing by Λ^1+s∂td, using (1.18), (1.16), (1.17), (1.20), (1.21) and (1.22), we reach

1 2

d dt

Z

(|Λ^1+s∂_td|²+|Λ^2+sd|²)dx

= Z

Λ^1+s(d|∇d|²−d|∂td|²)Λ^1+s∂_tddx

≤(kΛ^1+s(d|∇d|²)k_L2+kΛ^1+s(d|∂_td|²)k_L2)kΛ^1+s∂_tdk_L2

≤C(kdk_L^∞kΛ^1+s(|∇d|²)k_L2+k∇dk²_L2pkΛ^1+sdk

L

2p

p−2)kΛ^1+s∂_tdk_L2

+C(kdk_L^∞kΛ^1+s(|∂_td|²)k_L2+k∂_tdk²_L2nkΛ^1+sdk

Lⁿ⁻²²ⁿ )kΛ^1+s∂_tdk_L2

≤C(k∇dkL^∞kΛ^2+sdk_L2+k∇²dkL^pkΛ^1+sdk

L

2p

p−2)kΛ^1+s∂tdk_L2

+C(k∂tdkL^∞kΛ^1+s∂tdk_L2+k∂tdkLⁿk∂tdkL^∞kΛ^2+sdk_L2)kΛ^1+s∂tdk_L2

≤Ck∇dkL^∞kΛ^2+sdk_L2kΛ^1+s∂tdk_L2

+Ck∂tdkL^∞kΛ^1+s∂tdk²_L2+Ck∂tdkL^∞kΛ^2+sdk_L2kΛ^1+s∂tdk_L2

≤C(k∇dkL^∞+k∂tdkL^∞)(kΛ^2+sdk²_L2+kΛ^1+s∂tdk²_L2)

≤C(1 +k∇dkB˙⁰_∞,∞+k∂tdkB˙_∞,∞⁰ ) log(e+y²)y², withy²:=kΛ^1+s∂tdk²_L2+kΛ^2+sdk²_L2, which gives

sup

0≤t≤T

(kΛ^1+s∂tdk²_L2+kΛ^2+sdk²_L2)≤C.

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) byuand using (1.6), we see that 1

2 d dt

Z

|u|²dx+ Z

|∇u|²dx=− Z

(u· ∇)d·∆ddx. (3.1) Testing (1.5) by−∆d, usingd∆d=−|∇d|²and|d|= 1, we find that

1 2

d dt

Z

|∇d|²dx+ Z

|∆d|²dx= Z

(u· ∇)d·∆ddx+ Z

(d∆d)²dx

≤ Z

(u· ∇)d·∆ddx+ Z

|∆d|²dx.

(3.2)

Summing up (3.1) and (3.2), we have Z

(|u|²+|∇d|²)dx≤ Z

(|u0|²+|∇d0|²)dx. (3.3)

ApplyingD^sto (1.4), testing byD^su, using (1.6), (1.18), (1.19), (1.23), (1.24), (1.25), (1.26) and (1.27), we obtain

1 2

d dt

Z

|D^su|²dx+ Z

|D^1+su|²dx

=− Z

(D^s(u· ∇u)−u∇D^su)D^sudx+ Z

D^s(∇d ∇d) :∇D^sudx

≤Ck∇uk_L^q1kD^suk

L

2q1

q1−2kD^suk_L2+Ck∇dk_L^q2kD^s∇dk

L

2q2

q2−2k∇D^suk_L2

≤Ck∇ukB˙^−α_∞,∞kD^s+αukL²kD^sukL²+Ck∇dkB˙_∞,∞^−α kD^s+α∇dkL²kDΛ^sukL²

≤Ck∇ukB˙^−α∞,∞kD^suk^2−α_L2 kD^1+suk^α_L2

+Ck∇²dk

1−α 2−α

B˙∞,∞^−α kD^s+1dk^1−α_L2 kD^s+2dk^α_L2kD^1+suk_L2

≤ 1

8kD^1+suk²_L2+1

8kD^s+2dk²_L2+Ck∇uk

2 2−α

B˙∞,∞^−α kD^suk²_L2

+Ck∇²dk

2 2−α

B˙∞,∞^−α kD^s+1dk²_L2.

(3.4)

ApplyingD^s+1 to (1.5), testing byD^s+1dand using (1.6), we obtain 1

2 d dt

Z

|D^s+1d|²dx+ Z

|D^s+2d|²dx

= Z

D^s+1(d|∇d|²)D^s+1ddx

− Z

(D^s+1(u· ∇d)−u∇D^s+1d)D^s+1d dx=:I1+I2.

(3.5)

Using (1.18),|d|= 1, (1.25), (1.26), and (1.27), we boundI₁as follows.

I1≤ kD^s+1(d|∇d|²)k

L

2q2 q2 +2

kD^s+1dk

L

2q2 q2−2

≤C(kdkL^∞kD^s+1(|∇d|²)k

L

2q2 q2 +2

+k∇dk²_Lq2kD^s+1dk

L

2q2

q2−2)kD^s+1dk

L

2q2 q2−2

≤C(k∇dk_L^q2kD^s+2dk_L2+k∇dk²_Lq2kD^s+1dk

L

2q2

q2−2)kD^s+1dk

L

2q2 q2−2

≤Ck∇dk²_Lq2kD^s+1dk²

L

2q2 q2−2

+ 1

16kD^s+2dk²_L2

≤Ck∇dk²_B˙−α

∞,∞k∇dk²_H˙s+α+ 1

16kD^s+2dk²_L2

≤ 1

8kD^s+2dk²_L2+Ck∇²dk

2 2−α

B˙^−α_∞,∞kD^s+1dk²_L2. Using the Leibniz rule, we writeI2 as follows.

I2=− Z

(C1DuD^s+1d+

s

X

k=2

CkD^kuD^s+2−kd+Cs+1D^s+1u· ∇d)D^s+1ddx

=:I₂¹+

s

X

k=2

I₂^k+I₂^s+1.

(3.6)

By the same calculations as that ofI1, we have I₂^s+1≤Ck∇dkL^q2kD^s+1dk

L

2q2

q2−2kD^s+1ukL²

≤ 1

16kD^s+1uk²_L2+Ck∇²dk

2 2−α

B˙∞,∞^−α kD^s+1dk²_L2.

(3.7)

Using (1.23) and (1.24), we boundI₂¹as follows.

I₂¹≤Ck∇ukL^q1kD^s+1dk

L

2q1

q1−2kD^s+1dk_L2

≤Ck∇uk^1−θ_˙ ¹

B_∞,∞^−α kuk^θ_˙¹

H^s+α· k∇²dk^θ_˙¹

B_∞,∞^−α k∇dk^1−θ_˙ ¹

H^s+αkD^s+1dk_L2

≤C(k∇ukB˙^−α∞,∞+k∇²dkB˙∞,∞^−α )(kukH˙^s+α+k∇dkH˙^s+α)kD^s+1dkL²

≤ 1

16kD^s+1uk²_L2+ 1

16kD^s+2dk²_L2

+C(k∇uk

2 2−α

B˙∞,∞^−α +k∇²dk

2 2−α

B˙^−α∞,∞)(kD^suk²_L2+kD^s+1dk²_L2).

(3.8)

Using (1.28) and (1.29), we boundPs

k=2I₂^k as follows.

s

X

k=2

I₂^k ≤Ck∇uk¹⁻_˙ ^θ˜k

B∞,∞^−α kuk^θ˜_˙^k

H^s+αk∇²dk^θ˜_˙^k

B^−α∞,∞k∇dk¹⁻_˙ ^θ˜k

H^s+αkD^s+1dkL²

≤C(k∇ukB˙∞,∞^−α +k∇²dkB˙^−α∞,∞)(kukH˙^s+α+k∇dkH˙^s+α)kD^s+1dk_L2

≤C(k∇ukB˙∞,∞^−α +k∇²dkB˙^−α∞,∞)(kukH˙^s+α+k∇dkH˙^s+α)

×(kD^sukL²+kD^s+1dkL²)

≤ 1

16kD^s+1uk²_L2+ 1

16kD^s+2dk²_L2

+C(k∇uk

2 2−α

B˙^−α_∞,∞+k∇²dk

2 2−α

B˙^−α_∞,∞)(kD^suk²_L2+kD^s+1dk²_L2).

(3.9)

Inserting the above estimates into (3.5) and combining with (3.4) and using the Gronwall inequality, we arrive at

kD^suk_L^∞_(0,T;L2)+kD^s+1dk_L^∞_(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 takeh=γ= 1.

First, testing (1.10) byuand using (1.12), we see that 1

2 d dt

Z

|u|²dx+ Z

|∇u|²dx= Z

(b· ∇)b·udx. (4.1) Testing (1.11) byband using (1.12), we find that

1 2

d dt

Z

|b|²dx+ Z

|∇b|²dx+ Z

|b×rotb|²dx= Z

(b· ∇)u·bdx. (4.2) Summing up (4.1) and (4.2), we obtain

1 2

d dt

Z

(|u|²+|b|²)dx+ Z

(|∇u|²+|∇b|²+|b×rotb|²)dx= 0. (4.3)

Testing (1.10) by−∆u, using (1.12), (1.30) and (1.31), we infer that 1

2 d dt

Z

|∇u|²dx+ Z

|∆u|²dx

= Z

(u.∇)u·∆udx− Z

(b· ∇)b·∆udx

=−X

i,j

Z

∂jui∂iu∂judx− Z

(b· ∇)b·∆udx

≤Ck∇uk³_L3+kbkL^∞k∇bk_L2k∆uk_L2

≤CkukB˙∞,∞^−α kuk²

H˙ ^3+α2

+Ck∇bk_L2k∆uk_L2

≤CkukB˙∞,∞^−α k∇uk^1−α_L2 k∆uk^1+α_L2 +Ck∇bk_L2k∆uk_L2

≤ 1

8k∆uk²_L2+Ckuk^1−α_˙²

B∞,∞^−α

k∇uk²_L2+Ck∇bk²_L2.

(4.4)

Testing (1.11) by−∆band using (1.12), we deduce that 1

2 d dt

Z

|∇b|²dx+ Z

|∆b|²dx

= Z

(u· ∇)b·∆bdx− Z

(b· ∇)u·∆bdx +

Z

(rotb×b) rot ∆bdx− Z

[(rotb×b)×b] rot ∆bdx

=:`1+`2+`3+`4.

(4.5)

We bound`<sub>1</sub> and`₂ as follows.

`₁=X

i,j

Z

u_i∂_ib∂_j²bdx=−X

i,j

Z

∂_ju_i∂_ib∂_jbdx≤Ck∇ukL²k∇bk²_L4

≤Ck∇ukL²kbkL^∞k∆bkL²≤Ck∇ukL²k∆bkL²≤ 1

16k∆bk²_L2+Ck∇uk²_L2.

`₂≤ kbk_L^∞k∇uk_L2k∆bk_L2 ≤Ck∇uk_L2k∆bk_L2 ≤ 1

16k∆bk²_L2+Ck∇uk²_L2. Using (1.32) and (1.33), we bound`<sub>3</sub> and`₄ as follows.

`3=−X

i

Z

(rotb×∂ib)∂irotbdx≤Ck∇bk²_L4k∆bk_L2

≤Ck∇bk_B˙−β

∞,∞kbkH˙^1+βk∆bk_L2 ≤Ck∇bk_B˙−β

∞,∞k∇bk^1−β_L2 k∆bk^1+β_L2

≤ 1

16k∆bk²_L2+Ck∇bk

2 1−β

B˙^−β∞,∞

k∇bk²_L2.

`4=X

i

Z

∂i[(rotb×b)×b]∂irotbdx≤X

i

Z

[(rotb×∂ib)×b]∂irotbdx

+X

i

Z

[(rotb×b)×∂ib]∂irotbdx≤CkbkL^∞k∇bk²_L4k∆bk_L2

≤ 1

16k∆bk²_L2+Ck∇bk

2 1−β

B˙_∞,∞^−β k∇bk²_L2.

Inserting the above estimates into (4.5), and combining this with (4.4), and using the Gronwall inequality, we conclude that

k∇uk_L∞(0,T;L²)+k∇bk_L∞(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:[email protected]

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:[email protected]