Abstract

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

1. Introduction

The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1–4]. However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system: 𝑒𝑑𝑑+π‘’β‹…βˆ‡π‘’+βˆ‡πœ‹βˆ’Ξ”π‘’=βˆ’βˆ‡β‹…(βˆ‡π‘‘βŠ™βˆ‡π‘‘),(1.1)𝑑𝑒+π‘’β‹…βˆ‡π‘‘=Ξ”π‘‘βˆ’π‘“(𝑑),(1.2)div𝑒=0,(1.3)(𝑒,𝑑)(π‘₯,0)=0,𝑑0ξ€Έ(π‘₯)inℝ3,(1.4) which include the velocity vector π‘’βˆΆ=(𝑒1,𝑒2,𝑒3)𝑑, the scalar pressure πœ‹ being and the direction vector π‘‘βˆΆ=(𝑑1,𝑑2,𝑑3)𝑑. 𝑓(𝑑)∢=1/πœ‚(|𝑑|2βˆ’1)𝑑 with πœ‚, a positive constant. (βˆ‡π‘‘βŠ™βˆ‡π‘‘)𝑖,π‘—βˆ‘βˆΆ=π‘˜πœ•π‘–π‘‘π‘˜πœ•π‘—π‘‘π‘˜, and hence βˆ‘βˆ‡β‹…(βˆ‡π‘‘βŠ™βˆ‡π‘‘)=π‘˜Ξ”π‘‘π‘˜βˆ‡π‘‘π‘˜+(1/2)βˆ‡|βˆ‡π‘‘|2.

Lin-Liu [5] proved that the system (1.1)–(1.4) has a unique smooth solution globally in 2 space dimensions and locally in 3 dimensions. They also proved the global existence of weak solutions. However, the regularity of solutions to the system is still open. Fan-Guo [6] and Fan-Ozawa [7] showed the following regularity criteria: π‘’βˆˆπΏπ‘Ÿξ€·0,𝑇;𝐿𝑠ℝ32ξ€Έξ€Έforπ‘Ÿ+3𝑠=1,3<π‘ β‰€βˆž,βˆ‡π‘’βˆˆπΏπ‘Ÿξ€·0,𝑇;𝐿𝑠ℝ32ξ€Έξ€Έforπ‘Ÿ+3𝑠3=2,2<π‘ β‰€βˆž,π‘’βˆˆπΏ2̇𝐡0,𝑇;0∞,βˆžξ€Έ,πœ”βˆΆ=curlπ‘’βˆˆπΏ1̇𝐡0,𝑇;0∞,βˆžξ€Έ,(1.5) where ̇𝐡0∞,∞ denotes the homogeneous Besov space.

The first aim of this paper is to prove a new regularity criterion as follows.

Theorem 1.1. Let 𝑒0∈𝐻3,𝑑0∈𝐻4 with div𝑒0=0 in ℝ3. Let (𝑒,𝑑) be a smooth solution to the problem (1.1)–(1.4) on [0,𝑇). If 𝑒 satisfies ξ€œπ‘‡0‖𝑒(β‹…,𝑑)‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,βˆžξ‚€1+log𝑒+‖𝑒(β‹…,𝑑)β€–Μ‡π΅βˆ’π‘ βˆž,βˆžξ‚π‘‘π‘‘<∞(1.6) for some 𝑠 with 0<𝑠<1, then the solution (𝑒,𝑑) can be extended beyond 𝑇>0.

When the penalization parameter πœ‚β†’0, (1.1)–(1.4) reduce to 𝑒𝑑𝑑+π‘’β‹…βˆ‡π‘’+βˆ‡πœ‹βˆ’Ξ”π‘’=βˆ’βˆ‡β‹…(βˆ‡π‘‘βŠ™βˆ‡π‘‘),(1.7)div𝑒=0,(1.8)𝑑||||+π‘’β‹…βˆ‡π‘‘=Δ𝑑+βˆ‡π‘‘2||𝑑||𝑒𝑑,=1,(1.9)(𝑒,𝑑)(π‘₯,0)=0,𝑑0ξ€Έ(π‘₯)inℝ3,||𝑑0||=1.(1.10)

When 𝑒=0, then (1.9) is the well-known harmonic heat flow equation onto a sphere.

Fan-Gao-Guo [8] proved the following blow-up criteria:𝑒,βˆ‡π‘‘βˆˆπΏ2̇𝐡0,𝑇;0∞,βˆžξ€Έ,πœ”,Ξ”π‘‘βˆˆπΏ1̇𝐡0,𝑇;0∞,βˆžξ€Έ.(1.11)

We will prove the folowing theorem

Theorem 1.2. Let 𝑒0,βˆ‡π‘‘0∈𝐻3(ℝ3) with div𝑒0=0,|𝑑0|=1 in ℝ3. Let (𝑒,𝑑) be a smooth solution to the problem (1.7)–(1.10) on [0,𝑇). If the following condition is satisfied: ξ€œπ‘‡0‖𝑒(β‹…,𝑑)‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘(β‹…,𝑑)β€–2̇𝐡0∞,βˆžξ‚€1+log𝑒+‖𝑒(β‹…,𝑑)β€–Μ‡π΅βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘(β‹…,𝑑)‖̇𝐡0∞,βˆžξ‚π‘‘π‘‘<∞,(1.12) for some 𝑠 with 0<𝑠<1, then the solution (𝑒,𝑑) can be extended beyond 𝑇>0.

2. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Since it is well-known that there are 𝑇0>0 and a unique smooth solution (𝑒,𝑑) to the problem (1.1)–(1.4) in (0,𝑇0], we only need to show a priori estimates.

Testing (1.1) by 𝑒 and using (1.3), we see that 12π‘‘ξ€œπ‘’π‘‘π‘‘2ξ€œ||||𝑑π‘₯+βˆ‡π‘’2ξ€œ(𝑑π‘₯=βˆ’π‘’β‹…βˆ‡)𝑑⋅Δ𝑑𝑑π‘₯.(2.1)

Testing (1.2) by Ξ”π‘‘βˆ’π‘“(𝑑) and using (1.3), we find that π‘‘ξ€œ1𝑑𝑑2||||βˆ‡π‘‘2+1ξ‚€||𝑑||4πœ‚2ξ‚βˆ’12ξ€œ||||𝑑π‘₯+Ξ”π‘‘βˆ’π‘“(𝑑)2ξ€œ(𝑑π‘₯=π‘’β‹…βˆ‡)𝑑⋅Δ𝑑𝑑π‘₯.(2.2)

Summing up (2.1) and (2.2), we infer that ξ€œ12𝑒2+12||||βˆ‡π‘‘2+1ξ‚€||𝑑||4πœ‚2ξ‚βˆ’12ξ€œπ‘‘π‘₯+𝑇0ξ€œ||||βˆ‡π‘’2+||||Ξ”π‘‘βˆ’π‘“(𝑑)2β‰€ξ€œ1𝑑π‘₯2𝑒20+12||βˆ‡π‘‘0||2+1ξ‚€||𝑑4πœ‚0||2ξ‚βˆ’12𝑑π‘₯.(2.3)

Testing (1.2) by 𝑑 and using (1.3), we deduce that 12π‘‘ξ€œπ‘‘π‘‘π‘‘2ξ€œ||||𝑑π‘₯+βˆ‡π‘‘21𝑑π‘₯+πœ‚ξ€œ||𝑑||41𝑑π‘₯=πœ‚ξ€œπ‘‘2𝑑π‘₯,(2.4) which yields β€–π‘‘β€–πΏβˆž(0,𝑇;𝐿2)+‖𝑑‖𝐿2(0,𝑇;𝐻1)≀𝐢.(2.5)

Next, we prove the following estimate: β€–π‘‘β€–πΏβˆž(0,𝑇;𝐿∞)‖‖𝑑≀max0β€–β€–πΏβˆžξ€Έ,1.(2.6)

Without loss of generality, we assume that 1≀‖𝑑0β€–πΏβˆž. Multiplying (1.2) by 𝑑, we get πœ™π‘‘||𝑑||+π‘’β‹…βˆ‡πœ™βˆ’Ξ”πœ™+222πœ™=βˆ’πœ‚ξ‚€β€–β€–π‘‘0β€–β€–2πΏβˆžξ‚||𝑑||βˆ’12||||βˆ’2βˆ‡π‘‘2≀0(2.7) with πœ™=|𝑑|2βˆ’β€–π‘‘0β€–2𝐿∞ and πœ™|𝑑=0=|𝑑0|2βˆ’β€–π‘‘0β€–2πΏβˆžβ‰€0. Then (2.6) follows immediately from πœ™β‰€0 by the maximum principle.

Testing (1.1) by βˆ’Ξ”π‘’ and using (1.3), we see that 12π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘’2ξ€œ||||𝑑π‘₯+Δ𝑒2ξ€œ(βˆ’ξ“π‘‘π‘₯=π‘’β‹…βˆ‡)𝑒⋅Δ𝑒𝑑π‘₯𝑖,π‘˜ξ€œΞ”π‘‘π‘˜β‹…πœ•π‘–βˆ‡π‘‘π‘˜β‹…βˆ‡π‘’π‘–ξ“π‘‘π‘₯βˆ’π‘–,π‘˜ξ€œπœ•π‘–π‘‘π‘˜β‹…βˆ‡Ξ”π‘‘π‘˜β‹…βˆ‡π‘’π‘–π‘‘π‘₯.(2.8)

Applying Ξ” to (1.2), testing by Δ𝑑, and using (1.3), we find that 12π‘‘ξ€œ||||𝑑𝑑Δ𝑑2ξ€œ||||𝑑π‘₯+βˆ‡Ξ”π‘‘2𝑑π‘₯=𝑖,π‘˜ξ€œπœ•π‘–π‘‘π‘˜β‹…βˆ‡Ξ”π‘‘π‘˜β‹…βˆ‡π‘’π‘–βˆ’ξ“π‘‘π‘₯𝑖,π‘˜ξ€œπœ•π‘–πœ•π‘—π‘‘π‘˜β‹…πœ•π‘—βˆ‡π‘‘π‘˜β‹…βˆ‡π‘’π‘–ξ€œπ‘‘π‘₯βˆ’Ξ”π‘“(𝑑)⋅Δ𝑑𝑑π‘₯.(2.9)

Summing up (2.8) and (2.9), we get 12π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘’2+||||Δ𝑑2ξ€œ||||𝑑π‘₯+Δ𝑒2+||||βˆ‡Ξ”π‘‘2=ξ€œξ“π‘‘π‘₯(π‘’β‹…βˆ‡)𝑒⋅Δ𝑒𝑑π‘₯βˆ’π‘–,π‘˜ξ€œΞ”π‘‘π‘˜β‹…πœ•π‘–βˆ‡π‘‘π‘˜β‹…βˆ‡π‘’π‘–βˆ’ξ“π‘‘π‘₯𝑖,π‘˜ξ€œπœ•π‘–πœ•π‘—π‘‘π‘˜β‹…πœ•π‘—βˆ‡π‘‘π‘˜β‹…βˆ‡π‘’π‘–ξ€œπ‘‘π‘₯βˆ’Ξ”π‘“(𝑑)Δ𝑑𝑑π‘₯=∢𝐼1+𝐼2+𝐼3+𝐼4.(2.10)

By using (2.6), 𝐼4 is simply bounded as 𝐼4≀𝐢‖Δ𝑑‖2𝐿2.(2.11)

By using the inequalities [9] β€–π‘’β‹…βˆ‡π‘’β€–πΏ2β‰€πΆβ€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžβ€–βˆ‡π‘’β€–Μ‡π΅π‘ 2,2,β€–βˆ‡π‘’β€–Μ‡π΅π‘ 2,2β‰€πΆβ€–βˆ‡π‘’β€–πΏ1βˆ’π‘ 2‖Δ𝑒‖𝑠𝐿2.(2.12)

𝐼1 can be bounded as follows: 𝐼1β‰€β€–π‘’β‹…βˆ‡π‘’β€–πΏ2‖Δ𝑒‖𝐿2β‰€πΆβ€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžβ€–π‘’β€–Μ‡π΅1+𝑠2,2‖Δ𝑒‖𝐿2β‰€πΆβ€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžβ€–βˆ‡π‘’β€–πΏ1βˆ’π‘ 2‖Δ𝑒‖𝐿1+𝑠2≀12‖Δ𝑒‖2𝐿2+𝐢‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,βˆžβ€–βˆ‡π‘’β€–2𝐿2.(2.13)

We bound 𝐼2 and 𝐼3 as follows: 𝐼2,𝐼3β‰€πΆβ€–βˆ‡π‘’β€–πΏ2‖Δ𝑑‖2𝐿4β‰€πΆβ€–βˆ‡π‘’β€–πΏ2β€–βˆ‡Ξ”π‘‘β€–πΏ2≀14β€–βˆ‡Ξ”π‘‘β€–2𝐿2+πΆβ€–βˆ‡π‘’β€–2𝐿2.(2.14)

Here we used the Gagliardo-Nirenberg inequality ‖Δ𝑑‖2𝐿4β‰€πΆβ€–π‘‘β€–πΏβˆžβ€–βˆ‡Ξ”π‘‘β€–πΏ2.(2.15) Inserting the above estimates into (2.10), we derive π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘’2+||||Δ𝑑2ξ€œ||||𝑑π‘₯+Δ𝑒2+||||βˆ‡Ξ”π‘‘2𝑑π‘₯≀𝐢‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,∞+1ξ‚ξ‚€β€–βˆ‡π‘’β€–2𝐿2+‖Δ𝑑‖2𝐿2≀𝐢1+‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,βˆžξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžξ‚ξ‚€ξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžξ‚ξ‚ξ‚€β€–βˆ‡π‘’β€–2𝐿2+‖Δ𝑑‖2𝐿2≀𝐢1+‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,βˆžξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžξ‚ξ‚€β€–(1+log(𝑒+𝑦))βˆ‡π‘’β€–2𝐿2+‖Δ𝑑‖2𝐿2.(2.16)

Due to (1.6), one concludes that for any small constant πœ–>0, there exists π‘‡βˆ—<𝑇 such that ξ€œπ‘‡π‘‡βˆ—β€–π‘’β€–Μ‡π΅2/(1βˆ’π‘ )βˆ’π‘ βˆž,βˆžξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,βˆžξ‚π‘‘π‘‘<πœ–.(2.17)

For any π‘‡βˆ—<𝑑≀𝑇, we set 𝑦(𝑑)∢=supπ‘‡βˆ—β‰€πœβ‰€π‘‘β€–β€–Ξ›3β€–β€–(𝑒,βˆ‡π‘‘)(β‹…,𝜏)𝐿2withΞ›βˆΆ=(βˆ’Ξ”)1/2.(2.18)

Applying Gronwall’s inequality to (2.16) in the interval [π‘‡βˆ—,𝑑], one has β€–β€–βˆ‡π‘’(β‹…,𝑑)2𝐿2β€–+‖Δ𝑑(β‹…,𝑑)2𝐿2≀𝐢(1+𝑦)𝐢0πœ–.(2.19)

Now, we derive a bound on 𝑦(𝑑) defined by (2.18). To this end, we will use the following commutator and product estimates due to Kato-Ponce [10]: ‖Λ𝛼(𝑓𝑔)βˆ’π‘“Ξ›π›Όπ‘”β€–πΏπ‘ξ€·β‰€πΆβ€–βˆ‡π‘“β€–πΏπ‘1β€–β€–Ξ›π›Όβˆ’1π‘”β€–β€–πΏπ‘ž1+‖Λ𝛼𝑓‖𝐿𝑝2β€–π‘”β€–πΏπ‘ž2ξ€Έ,(2.20)‖Λ𝛼‖(𝑓𝑔)𝐿𝑝≀𝐢‖𝑓‖𝐿𝑝1β€–Ξ›π›Όπ‘”β€–πΏπ‘ž1+‖Λ𝛼𝑓‖𝐿𝑝2β€–π‘”β€–πΏπ‘ž2ξ€Έ,(2.21) with 𝛼>0 and 1/𝑝=1/𝑝1+1/π‘ž1=1/𝑝2+1/π‘ž2.

Applying Ξ›3 to (1.1), testing by Ξ›3𝑒, and using (1.3), (2.20), (2.21) and (2.19), we obtain 12π‘‘ξ€œ||Λ𝑑𝑑3𝑒||2ξ€œ||Λ𝑑π‘₯+4𝑒||2ξ€œξ€·Ξ›π‘‘π‘₯=βˆ’3(π‘’β‹…βˆ‡π‘’)βˆ’π‘’βˆ‡Ξ›3𝑒Λ3ξ€œΞ›π‘’π‘‘π‘₯+3(βˆ‡π‘‘βŠ™βˆ‡π‘‘)βˆΆΞ›3βˆ‡π‘’π‘‘π‘₯β‰€πΆβ€–βˆ‡π‘’β€–πΏ3β€–β€–Ξ›3𝑒‖‖2𝐿3+πΆβ€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›4𝑑‖‖𝐿2β€–β€–Ξ›4𝑒‖‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ3/42β€–β€–Ξ›3𝑒‖‖𝐿1/42β‹…β€–βˆ‡π‘’β€–πΏ1/32β€–β€–Ξ›4𝑒‖‖𝐿5/32+1β€–β€–Ξ›164𝑒‖‖2𝐿2+πΆβ€–βˆ‡π‘‘β€–2πΏβˆžβ€–β€–Ξ›4𝑑‖‖2𝐿2≀14β€–β€–Ξ›4𝑒‖‖2𝐿2+πΆβ€–βˆ‡π‘’β€–πΏ13/22β€–β€–Ξ›3𝑒‖‖𝐿3/22+𝐢‖Δ𝑑‖𝐿3/22β€–β€–Ξ›4𝑑‖‖𝐿1/22⋅‖Δ𝑑‖𝐿2/32β€–β€–Ξ›5𝑑‖‖𝐿4/32≀14β€–β€–Ξ›4𝑒‖‖2𝐿2+14β€–β€–Ξ›5𝑑‖‖2𝐿2+πΆβ€–βˆ‡π‘’β€–πΏ13/22β€–β€–Ξ›3𝑒‖‖𝐿3/22+𝐢‖Δ𝑑‖𝐿13/22β€–β€–Ξ›4𝑑‖‖𝐿3/22.(2.22) Here we have used the following Gagliardo-Nirenberg inequalities: β€–βˆ‡π‘’β€–πΏ3β‰€πΆβ€–βˆ‡π‘’β€–πΏ3/42β€–β€–Ξ›3𝑒‖‖𝐿1/42,β€–β€–Ξ›3𝑒‖‖𝐿3β‰€πΆβ€–βˆ‡π‘’β€–πΏ1/62β€–β€–Ξ›4𝑒‖‖𝐿5/62,β€–βˆ‡π‘‘β€–πΏβˆžβ‰€πΆβ€–Ξ”π‘‘β€–πΏ3/42β€–β€–Ξ›4𝑑‖‖𝐿1/42,β€–β€–Ξ›4𝑑‖‖𝐿2≀𝐢‖Δ𝑑‖𝐿1/32β€–β€–Ξ›5𝑑‖‖𝐿2/32.(2.23)

Taking Ξ›4 to (1.2), testing by Ξ›4𝑑, and using (1.3), (2.20), (2.23), and (2.6), we have 12π‘‘ξ€œ||Λ𝑑𝑑4𝑑||2ξ€œ||Λ𝑑π‘₯+5𝑑||2ξ€œξ€·Ξ›π‘‘π‘₯=βˆ’4(π‘’β‹…βˆ‡π‘‘)βˆ’π‘’βˆ‡Ξ›4𝑑Λ4ξ€œΞ›π‘‘π‘‘π‘₯βˆ’4𝑓(𝑑)β‹…Ξ›4𝑑𝑑π‘₯β‰€πΆβ€–βˆ‡π‘’β€–πΏ3β€–β€–Ξ›4𝑑‖‖𝐿6β€–β€–Ξ›4𝑑‖‖𝐿2+πΆβ€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›4𝑒‖‖𝐿2β€–β€–Ξ›4𝑑‖‖𝐿2β€–β€–Ξ›+𝐢4𝑑‖‖2𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ3/42β€–β€–Ξ›3𝑒‖‖𝐿1/42⋅‖Δ𝑑‖𝐿1/32β€–β€–Ξ›5𝑑‖‖𝐿5/32+14β€–β€–Ξ›4𝑒‖‖2𝐿2+πΆβ€–βˆ‡π‘‘β€–2πΏβˆžβ€–β€–Ξ›4𝑑‖‖2𝐿2β€–β€–Ξ›+𝐢4𝑑‖‖2𝐿2≀14β€–β€–Ξ›4𝑒‖‖2𝐿2+14β€–β€–Ξ›5𝑑‖‖2𝐿2+πΆβ€–βˆ‡π‘’β€–πΏ9/22‖Δ𝑑‖2𝐿2β€–β€–Ξ›3𝑒‖‖𝐿3/22+𝐢‖Δ𝑑‖𝐿13/22β€–β€–Ξ›4𝑑‖‖𝐿3/22β€–β€–Ξ›+𝐢4𝑑‖‖2𝐿2.(2.24)

Summing up (2.22) and (2.24) and taking πœ– small enough, we arrive at β€–π‘’β€–πΏβˆž(0,𝑇;𝐻3)+‖𝑒‖𝐿2(0,𝑇;𝐻4)≀𝐢,β€–π‘‘β€–πΏβˆž(0,𝑇;𝐻4)+‖𝑑‖𝐿2(0,𝑇;𝐻5)≀𝐢.(2.25)

This completes the proof.

3. Proof of Theorem 1.2

In this section, we will prove Theorem 1.2. Since it is easy to prove that there are 𝑇0>0 and a unique smooth solution (𝑒,𝑝,𝑑) to the problem (1.7)–(1.10) in [0,𝑇0], we only need to prove a priori estimates.

First, as in the previous section, we still have (2.1).

Testing (1.9) by βˆ’Ξ”π‘‘, using 𝑑Δ𝑑=βˆ’|βˆ‡π‘‘|2 and |𝑑|=1, we see that 12π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘‘2ξ€œ||||𝑑π‘₯+Δ𝑑2=ξ€œξ€œπ‘‘π‘₯(π‘’β‹…βˆ‡π‘‘)Δ𝑑𝑑π‘₯+(𝑑⋅Δ𝑑)2β‰€ξ€œξ€œ||||𝑑π‘₯(π‘’β‹…βˆ‡π‘‘)Δ𝑑𝑑π‘₯+Δ𝑑2𝑑π‘₯.(3.1)

Summing up (2.1) and (3.1), we find that ξ€œ12𝑒2+12||||βˆ‡π‘‘2ξ€œπ‘‘π‘₯+𝑇0ξ€œ||||βˆ‡π‘’2ξ€œ1𝑑π‘₯𝑑𝑑≀2𝑒20+||βˆ‡π‘‘0||2𝑑π‘₯.(3.2)

Similarly to (2.10), we have 12π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘’2+||||Δ𝑑2ξ€œ||||𝑑π‘₯+Δ𝑒2+||||βˆ‡Ξ”π‘‘2𝑑π‘₯=∢𝐼1+𝐼2+𝐼3+𝐼5.(3.3) Here 𝐼1,𝐼2, and 𝐼3 are the same as that in (2.10) and can be bounded as in the previous section. The corresponding last term 𝐼5 is written and bounded as 𝐼5ξ“βˆΆ=βˆ’π‘˜ξ€œπœ•π‘˜ξ‚€||||βˆ‡π‘‘2π‘‘ξ‚β‹…πœ•π‘˜ξ“Ξ”π‘‘π‘‘π‘₯=βˆ’π‘˜ξ€œπœ•π‘˜||||π‘‘β‹…βˆ‡π‘‘2β‹…πœ•π‘˜ξ“Ξ”π‘‘π‘‘π‘₯βˆ’π‘˜ξ€œπ‘‘πœ•π‘˜||||βˆ‡π‘‘2β‹…πœ•π‘˜=Δ𝑑𝑑π‘₯π‘˜ξ€œπœ•π‘˜ξ‚€πœ•π‘˜||||π‘‘β‹…βˆ‡π‘‘2⋅Δ𝑑𝑑π‘₯βˆ’π‘˜ξ€œπ‘‘πœ•π‘˜||||βˆ‡π‘‘2β‹…πœ•π‘˜Ξ”π‘‘π‘‘π‘₯β‰€πΆβ€–βˆ‡π‘‘β€–2𝐿4‖Δ𝑑‖2𝐿4+14β€–βˆ‡Ξ”π‘‘β€–2𝐿2≀𝐢‖Δ𝑑‖𝐿2β‹…β€–βˆ‡π‘‘β€–Μ‡π΅0∞,βˆžβ€–βˆ‡Ξ”π‘‘β€–πΏ2+14β€–βˆ‡Ξ”π‘‘β€–2𝐿2≀12β€–βˆ‡Ξ”π‘‘β€–2𝐿2+πΆβ€–βˆ‡π‘‘β€–2̇𝐡0∞,βˆžβ€–Ξ”π‘‘β€–2𝐿2.(3.4) Here we have used the following inequality [11, 12]: ‖Δ𝑑‖2𝐿4β‰€πΆβ€–βˆ‡π‘‘β€–Μ‡π΅0∞,βˆžβ€–βˆ‡Ξ”π‘‘β€–πΏ2(3.5) and the Gagliardo-Nirenberg inequality β€–βˆ‡π‘‘β€–2𝐿4β‰€πΆβ€–π‘‘β€–πΏβˆžβ€–Ξ”π‘‘β€–πΏ2.(3.6)

Substituting the above estimates into (3.3), we obtain π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘’2+||||Δ𝑑2𝑑π‘₯≀𝐢‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘β€–2̇𝐡0∞,βˆžξ‚ξ‚€β€–βˆ‡π‘’β€–2𝐿2+‖Δ𝑑‖2𝐿2≀𝐢‖𝑒‖̇𝐡2/(1βˆ’π‘ )βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘β€–2̇𝐡0∞,βˆžξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘β€–Μ‡π΅0∞,βˆžξ‚ξ‚€(1+log(𝑒+𝑦))β€–βˆ‡π‘’β€–2𝐿2+‖Δ𝑑‖2𝐿2.(3.7)

Due to (1.12), one concludes that for any small constant πœ–>0, there exists π‘‡βˆ—<𝑇 such that ξ€œπ‘‡π‘‡βˆ—β€–π‘’β€–Μ‡π΅2/(1βˆ’π‘ )βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘β€–2̇𝐡0∞,βˆžξ‚€1+log𝑒+β€–π‘’β€–Μ‡π΅βˆ’π‘ βˆž,∞+β€–βˆ‡π‘‘β€–Μ‡π΅0∞,βˆžξ‚π‘‘π‘‘<πœ–.(3.8)

Applying Gronwall’s inequality to (3.7) in the interval [π‘‡βˆ—,𝑑], one has (2.19).

As in the previous section, we still have (2.22).

Similarly to (2.24), we obtain 12π‘‘ξ€œ||Λ𝑑𝑑4𝑑||2ξ€œ||Λ𝑑π‘₯+5𝑑||2ξ€œξ€·Ξ›π‘‘π‘₯=βˆ’4(π‘’β‹…βˆ‡π‘‘)βˆ’π‘’βˆ‡Ξ›4𝑑Λ4ξ€œΞ›π‘‘π‘‘π‘₯+4𝑑||||βˆ‡π‘‘2Λ4𝑑𝑑π‘₯=∢𝐽1+𝐽2.(3.9)𝐽1 is bounded as that in (2.24); 𝐽2β‰€πΆβ€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›4𝑑‖‖2𝐿2β€–β€–Ξ›+𝐢4ξ‚€||||βˆ‡π‘‘2‖‖𝐿2β€–β€–Ξ›4𝑑‖‖𝐿2,β‰€πΆβ€–βˆ‡π‘‘β€–2πΏβˆžβ€–β€–Ξ›4𝑑‖‖2𝐿2+πΆβ€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›5𝑑‖‖𝐿2β€–β€–Ξ›4𝑑‖‖𝐿2,≀14β€–β€–Ξ›5𝑑‖‖2𝐿2+πΆβ€–βˆ‡π‘‘β€–2πΏβˆžβ€–β€–Ξ›4𝑑‖‖2𝐿2,(3.10) then 𝐽2 can be bounded as that in (2.24).

Combining (2.22) and (3.9) and taking πœ– small enough, we conclude that β€–π‘’β€–πΏβˆž(0,𝑇;𝐻3)+‖𝑒‖𝐿2(0,𝑇;𝐻4)≀𝐢,β€–βˆ‡π‘‘β€–πΏβˆž(0,𝑇;𝐻3)+β€–βˆ‡π‘‘β€–πΏ2(0,𝑇;𝐻4)≀𝐢.(3.11) This completes the proof.

Acknowledgment

The paper is supported by NSFC (no. 11171154).