Abstract

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

1. Introduction

First, we consider the following hyperbolic Navier-Stokes equations [1]: πœπ‘’π‘‘π‘‘+π‘’π‘‘βˆ’Ξ”π‘’+βˆ‡πœ‹+π‘’β‹…βˆ‡π‘’+πœπ‘’π‘‘β‹…βˆ‡π‘’+πœπ‘’β‹…βˆ‡π‘’π‘‘ξ€·=0,(1.1)div𝑒=0,(1.2)𝑒,𝑒𝑑𝑒(π‘₯,0)=0,𝑒1ξ€Έ(π‘₯),π‘₯βˆˆβ„π‘›,𝑛β‰₯2.(1.3) Here 𝑒 is the velocity, πœ‹ is the pressure, and 𝜏>0 is a small relaxation parameter. We will take 𝜏=1 for simplicity.

When 𝜏=0, (1.1) and (1.2) reduce to the standard Navier-Stokes equations. Kozono et al. [2] proved the following regularity criterion: πœ”βˆΆ=curlπ‘’βˆˆπΏ1̇𝐡0,𝑇;0∞,βˆžξ€Έ.(1.4) Here ̇𝐡0∞,∞ is the homogeneous Besov space.

Rack and Saal [1] proved the local well posedness of the problem (1.1)–(1.3). The global regularity is still open. The first aim of this paper is to prove a regularity criterion. We will prove the following theorem.

Theorem 1.1. Let (𝑒0,𝑒1)βˆˆπ»π‘ +1×𝐻𝑠 with 𝑠>𝑛/2,𝑛β‰₯2 and div𝑒0=div𝑒1=0 in ℝ𝑛. Let (𝑒,πœ‹) be a unique strong solution to the problem (1.1)–(1.3). If 𝑒 satisfies 𝑒,βˆ‡π‘’,π‘’π‘‘βˆˆπΏ1̇𝐡0,𝑇;0∞,βˆžξ€Έ,(1.5) then the solution 𝑒 can be extended beyond 𝑇>0.

In our proof, we will use the following logarithmic Sobolev inequality [2]: β€–π‘’β€–πΏβˆžξ‚€β‰€πΆ1+‖𝑒‖̇𝐡0∞,βˆžξ€·log𝑒+‖𝑒‖𝐻𝑠(1.6) and the following bilinear product and commutator estimates according to Kato and Ponce [3]: ‖Λ𝑠‖(𝑓𝑔)𝐿𝑝≀𝐢‖𝑓‖𝐿𝑝1β€–Ξ›π‘ π‘”β€–πΏπ‘ž1+‖Λ𝑠𝑓‖𝐿𝑝2β€–π‘”β€–πΏπ‘ž2ξ€Έ,(1.7)‖Λ𝑠(𝑓𝑔)βˆ’π‘“Ξ›π‘ π‘”β€–πΏπ‘ξ€·β‰€πΆβ€–βˆ‡π‘“β€–πΏπ‘1β€–β€–Ξ›π‘ βˆ’1π‘”β€–β€–πΏπ‘ž1+‖Λ𝑠𝑓‖𝐿𝑝2β€–π‘”β€–πΏπ‘ž2ξ€Έ,(1.8) with 𝑠>0,β€‰β€‰Ξ›βˆΆ=(βˆ’Ξ”)1/2 and 1/𝑝=(1/𝑝1)+(1/π‘ž1)=(1/𝑝2)+(1/π‘ž2).

Next, we consider the fractional Landau-Lifshitz equation: πœ•π‘‘πœ™=πœ™Γ—Ξ›2π›½πœ™πœ™,(1.9)(π‘₯,0)=πœ™0(π‘₯)βˆˆπ•Š2,π‘₯βˆˆβ„π‘›,(1.10) where πœ™βˆˆπ•Š2 is a three-dimensional vector representing the magnetization and 𝛽 is a positive constant.

When 𝛽=1, using the standard stereographic projection π•Š2β†’β„‚βˆͺ{∞}, (1.9) can be rewritten as the derivative SchrΓΆdinger equation for π‘€βˆˆβ„‚, 𝑖𝑀𝑑+Δ𝑀+4(βˆ‡π‘€)21+|𝑀|2𝑀=0.(1.11)

Equation (1.9) is also called the SchrΓΆdinger map and has been studied by many authors [4–31]. Guo and Han [32] proved the following regularity criterion: βˆ‡πœ™βˆˆπΏ2(0,𝑇;𝐿∞(ℝ𝑛))(1.12) with 𝑛β‰₯2.

When 0<𝛽≀1/2, Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion Ξ›2π›½πœ™βˆˆπΏ1(0,𝑇;𝐿∞(ℝ𝑛))(1.13) with 𝑛≀3.

We will refine (1.13) as follows.

Theorem 1.2. Let 0<𝛽≀1/2. Let π‘š be an integer such that 2π‘š>(𝑛+1)/2 for any 𝑛β‰₯1. Let Ξ›π›½πœ™0∈𝐻2π‘š and πœ™0βˆˆπ•Š2 and πœ™ be a local smooth solution to the problem (1.9) and (1.10). If πœ™ satisfies Ξ›2π›½πœ™βˆˆπΏ1̇𝐡0,𝑇;0∞,∞(ℝ𝑛)ξ€Έ(1.14) for some finite 𝑇>0, then the solution πœ™ can be extended beyond 𝑇>0.

2. Proof of Theorem 1.1

Since (𝑒,πœ‹) is a local smooth solution, we only need to prove a priori estimates.

First, testing (1.1) by 𝑒 and using (1.2), we see that π‘‘ξ€œξ‚€1𝑑𝑑2𝑒2+π‘’π‘’π‘‘ξ‚ξ€œ||||𝑑π‘₯+βˆ‡π‘’2=ξ€œπ‘’π‘‘π‘₯2π‘‘ξ€œπ‘‘π‘₯+π‘’β‹…βˆ‡π‘’β‹…π‘’π‘‘β‰€ξ€œu𝑑π‘₯2𝑑1𝑑π‘₯+2β€–βˆ‡π‘’β€–πΏβˆžξ‚€β€–π‘’β€–2𝐿2+‖‖𝑒𝑑‖‖2𝐿2.(2.1)

Testing (1.1) by 4𝑒𝑑 and using (1.2), we find that π‘‘ξ€œξ‚€π‘‘π‘‘2𝑒2𝑑||||+2βˆ‡π‘’2ξ‚ξ€œπ‘’π‘‘π‘₯+42π‘‘ξ€œξ€·π‘‘π‘₯=βˆ’4π‘’β‹…βˆ‡π‘’+π‘’π‘‘ξ€Έπ‘’β‹…βˆ‡π‘’π‘‘π‘‘π‘₯β‰€πΆβ€–βˆ‡π‘’β€–πΏβˆžξ‚€β€–π‘’β€–2𝐿2+‖‖𝑒𝑑‖‖2𝐿2.(2.2)

Applying Λ𝑠 to (1.1), testing by Λ𝑠𝑒𝑑 and using (1.2), (1.7), (1.8), and (1.6), we have 12π‘‘ξ€œξ‚€||Λ𝑑𝑑𝑠+1𝑒||2+||Λ𝑠𝑒𝑑||2ξ‚ξ€œ||Λ𝑑π‘₯+𝑠𝑒𝑑||2𝑑π‘₯=βˆ’π‘–ξ€œΞ›π‘ πœ•π‘–ξ€·π‘’π‘–π‘’ξ€Έβ‹…Ξ›π‘ π‘’π‘‘ξ€œΞ›π‘‘π‘₯βˆ’π‘ ξ€·π‘’π‘‘ξ€Έβ‹…βˆ‡π‘’β‹…Ξ›π‘ π‘’π‘‘βˆ’ξ“π‘‘π‘₯π‘–ξ€œξ€ΊΞ›π‘ πœ•π‘–ξ€·π‘’π‘–π‘’π‘‘ξ€Έβˆ’π‘’π‘–πœ•π‘–Ξ›π‘ π‘’π‘‘ξ€»Ξ›π‘ π‘’π‘‘π‘‘π‘₯β‰€πΆβ€–π‘’β€–πΏβˆžβ€–β€–Ξ›π‘ +1𝑒‖‖𝐿2‖‖Λ𝑠𝑒𝑑‖‖𝐿2‖‖𝑒+πΆπ‘‘β€–β€–πΏβˆžβ€–β€–Ξ›π‘ +1𝑒‖‖𝐿2+β€–βˆ‡π‘’β€–πΏβˆžβ€–β€–Ξ›π‘ π‘’π‘‘β€–β€–πΏ2‖‖Λ𝑠𝑒𝑑‖‖𝐿2ξ€·β‰€πΆβ€–π‘’β€–πΏβˆž+β€–βˆ‡π‘’β€–πΏβˆž+β€–β€–π‘’π‘‘β€–β€–πΏβˆžξ€Έξ‚€β€–β€–Ξ›π‘ +1𝑒‖‖2𝐿2+‖‖Λ𝑠𝑒𝑑‖‖2𝐿2≀𝐢1+‖𝑒‖̇𝐡0∞,∞+β€–βˆ‡π‘’β€–Μ‡π΅0∞,∞+‖‖𝑒𝑑‖‖̇𝐡0∞,βˆžξ‚ξ‚€log𝑒+‖𝑒‖2𝐻𝑠+1+‖‖𝑒𝑑‖‖2𝐻𝑠⋅‖‖Λ𝑠𝑒𝑑‖‖2𝐿2+‖‖Λ𝑠+1𝑒‖‖2𝐿2.(2.3)

Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that β€–π‘’β€–πΏβˆž(0,𝑇;𝐻𝑠+1)+β€–β€–π‘’π‘‘β€–β€–πΏβˆž(0,𝑇;𝐻𝑠)≀𝐢.(2.4)

This completes the proof.

3. Proof of Theorem 1.2

Since πœ™ is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by (β‹…,β‹…) the standard 𝐿2 scalar product.

First, testing (1.9) by Ξ›2π›½πœ™ and using (π‘ŽΓ—π‘)β‹…π‘Ž=0, we see that 12π‘‘ξ€œ||Ξ›π‘‘π‘‘π›½πœ™||2𝑑π‘₯=0.(3.1)

Testing (1.9) by Ξ”2π‘šΞ›2π›½πœ™ and using (π‘ŽΓ—π‘)β‹…π‘Ž=0, (1.6) and (1.7), we obtain, with (1/𝑝)+(1/π‘ž)=(1/𝑝𝛼)+(1/π‘žπ›Ό)=(1/̃𝑝𝛼)+(1/Μƒπ‘žπ›Ό)=1/2, 12π‘‘ξ€œ||Ξ”π‘‘π‘‘π‘šΞ›π›½πœ™||2=𝑑π‘₯πœ™Γ—Ξ›2π›½πœ™,Ξ”2π‘šΞ›2π›½πœ™ξ€Έ=ξ€·Ξ”π‘šξ€·πœ™Γ—Ξ›2π›½πœ™ξ€Έ,Ξ”π‘šΞ›2π›½πœ™ξ€Έ=ξƒ©Ξ”π‘šπœ™Γ—Ξ›2π›½πœ™+2π‘šβˆ’1𝛼=1𝐢𝛼𝐷2π‘šβˆ’π›Όπœ™Γ—Ξ›2π›½π·π›Όπœ™,Ξ”π‘šΞ›2π›½πœ™ξƒͺ=ξƒ©Ξ›π›½ξƒ©Ξ”π‘šπœ™Γ—Ξ›2π›½πœ™+2π‘šβˆ’1𝛼=1𝐢𝛼𝐷2π‘šβˆ’π›Όπœ™Γ—Ξ›2π›½π·π›Όπœ™ξƒͺ,Ξ”π‘šΞ›π›½πœ™ξƒͺ‖‖Λ≀𝐢2π›½πœ™β€–β€–πΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–2𝐿2+πΆβ€–Ξ”π‘šπœ™β€–πΏπ‘β€–β€–Ξ›3π›½πœ™β€–β€–πΏπ‘žβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–πΏ2+2π‘šβˆ’2𝛼=1𝐢𝛼‖‖𝐷2π‘šβˆ’π›Όπœ™β€–β€–πΏπ‘π›Όβ€–β€–Ξ›3π›½π·π›Όπœ™β€–β€–πΏπ‘žπ›Ό+‖‖𝐷2π‘šβˆ’π›ΌΞ›π›½πœ™β€–β€–πΏΜƒπ‘π›Όβ€–β€–Ξ›2π›½π·π›Όπœ™β€–β€–πΏΜƒπ‘žπ›Όξ‚β€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–πΏ2‖‖Λ≀𝐢2π›½πœ™β€–β€–πΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–2𝐿2‖‖Λ≀𝐢1+2π›½πœ™β€–β€–Μ‡π΅0∞,βˆžξ€·β€–β€–Ξ”log𝑒+π‘šΞ›π›½πœ™β€–β€–πΏ2ξ€Έξ‚β€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–2𝐿2,(3.2) which yields β€–β€–Ξ›π›½β€–β€–πœ™(𝑑)𝐻2π‘šβ‰€πΆ.(3.3) Here we have used the following interesting Gagliardo-Nirenberg inequalities: β€–Ξ”π‘šπœ™β€–πΏπ‘β€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–1βˆ’πœƒ0πΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–πœƒ0𝐿2with𝑝=2π‘šβˆ’π›½π‘šβˆ’π›½,πœƒ0=2π‘šβˆ’2𝛽,β€–β€–Ξ›2π‘šβˆ’π›½3π›½πœ™β€–β€–πΏπ‘žβ€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–πœƒ0πΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–1βˆ’πœƒ0𝐿2withπ‘ž=2(2π‘šβˆ’π›½)𝛽,‖‖𝐷2π‘šβˆ’π›Όπœ™β€–β€–πΏπ‘π›Όβ€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–1βˆ’πœƒπ›ΌπΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–πœƒπ›ΌπΏ2withπœƒπ›Ό=2π‘šβˆ’π›Όβˆ’2𝛽2π‘šβˆ’π›½,𝑝𝛼=4π‘šβˆ’2𝛽,β€–β€–Ξ›2π‘šβˆ’π›Όβˆ’2𝛽3π›½π·π›Όπœ™β€–β€–πΏπ‘žπ›Όβ€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–πœƒπ›ΌπΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–1βˆ’πœƒπ›ΌπΏ2withπ‘žπ›Ό=4π‘šβˆ’2𝛽,‖‖𝐷𝛼+𝛽2π‘šβˆ’π›ΌΞ›π›½πœ™β€–β€–πΏΜƒπ‘π›Όβ€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–1βˆ’Μƒπœƒπ›ΌπΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–Μƒπœƒπ›ΌπΏ2Μƒπœƒwith𝛼=2π‘šβˆ’π›Όβˆ’π›½2π‘šβˆ’π›½,̃𝑝𝛼=4π‘šβˆ’2𝛽,β€–β€–Ξ›2π‘šβˆ’π›Όβˆ’π›½2π›½π·π›Όπœ™β€–β€–πΏΜƒπ‘žπ›Όβ€–β€–Ξ›β‰€πΆ2π›½πœ™β€–β€–Μƒπœƒπ›ΌπΏβˆžβ€–β€–Ξ”π‘šΞ›π›½πœ™β€–β€–1βˆ’Μƒπœƒπ›ΌπΏ2withΜƒπ‘žπ›Ό=4π‘šβˆ’2𝛽𝛼.(3.4) This completes the proof.

Acknowledgment

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