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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 796368, 7 pages
doi:10.5402/2012/796368
Research Article

Regularity Criteria for Hyperbolic Navier-Stokes and Related System

Jishan Fan1 and Tohru Ozawa2

1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 10 July 2012; Accepted 2 August 2012

Academic Editors: S. Cingolani, P. Mironescu, L. Sanchez, and T. Tran

Copyright © 2012 Jishan Fan and Tohru Ozawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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]: 𝜏 𝑢 𝑡 𝑡 + 𝑢 𝑡 − Δ 𝑢 + ∇ 𝜋 + 𝑢 ⋅ ∇ 𝑢 + 𝜏 𝑢 𝑡 ⋅ ∇ 𝑢 + 𝜏 𝑢 ⋅ ∇ 𝑢 𝑡 = 0 , ( 1 . 1 ) d i v 𝑢 = 0 , ( 1 . 2 ) 𝑢 , 𝑢 𝑡 𝑢 ( 𝑥 , 0 ) = 0 , 𝑢 1 ( 𝑥 ) , 𝑥 ∈ ℝ 𝑛 , 𝑛 ≥ 2 . ( 1 . 3 ) Here 𝑢 is the velocity, 𝜋 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: 𝜔 ∶ = c u r l 𝑢 ∈ 𝐿 1 ̇ 𝐵 0 , 𝑇 ; 0 ∞ , ∞ . ( 1 . 4 ) Here ̇ 𝐵 0 ∞ , ∞ 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 ( 𝑢 0 , 𝑢 1 ) ∈ 𝐻 𝑠 + 1 × 𝐻 𝑠 with 𝑠 > 𝑛 / 2 , 𝑛 ≥ 2 and d i v 𝑢 0 = d i v 𝑢 1 = 0 in ℝ 𝑛 . Let ( 𝑢 , 𝜋 ) be a unique strong solution to the problem (1.1)–(1.3). If 𝑢 satisfies 𝑢 , ∇ 𝑢 , 𝑢 𝑡 ∈ 𝐿 1 ̇ 𝐵 0 , 𝑇 ; 0 ∞ , ∞ , ( 1 . 5 ) then the solution 𝑢 can be extended beyond 𝑇 > 0 .

In our proof, we will use the following logarithmic Sobolev inequality [2]: ‖ 𝑢 ‖ 𝐿 ∞ ≤ 𝐶 1 + ‖ 𝑢 ‖ ̇ 𝐵 0 ∞ , ∞ l o g 𝑒 + ‖ 𝑢 ‖ 𝐻 𝑠 ( 1 . 6 ) and the following bilinear product and commutator estimates according to Kato and Ponce [3]: ‖ Λ 𝑠 ‖ ( 𝑓 𝑔 ) 𝐿 𝑝 ≤ 𝐶 ‖ 𝑓 ‖ 𝐿 𝑝 1 ‖ Λ 𝑠 𝑔 ‖ 𝐿 𝑞 1 + ‖ Λ 𝑠 𝑓 ‖ 𝐿 𝑝 2 ‖ 𝑔 ‖ 𝐿 𝑞 2 , ( 1 . 7 ) ‖ Λ 𝑠 ( 𝑓 𝑔 ) − 𝑓 Λ 𝑠 𝑔 ‖ 𝐿 𝑝 ≤ 𝐶 ‖ ∇ 𝑓 ‖ 𝐿 𝑝 1 ‖ ‖ Λ 𝑠 − 1 𝑔 ‖ ‖ 𝐿 𝑞 1 + ‖ Λ 𝑠 𝑓 ‖ 𝐿 𝑝 2 ‖ 𝑔 ‖ 𝐿 𝑞 2 , ( 1 . 8 ) with 𝑠 > 0 ,   Λ ∶ = ( − Δ ) 1 / 2 and 1 / 𝑝 = ( 1 / 𝑝 1 ) + ( 1 / 𝑞 1 ) = ( 1 / 𝑝 2 ) + ( 1 / 𝑞 2 ) .

Next, we consider the fractional Landau-Lifshitz equation: 𝜕 𝑡 𝜙 = 𝜙 × Λ 2 𝛽 𝜙 𝜙 , ( 1 . 9 ) ( 𝑥 , 0 ) = 𝜙 0 ( 𝑥 ) ∈ 𝕊 2 , 𝑥 ∈ ℝ 𝑛 , ( 1 . 1 0 ) where 𝜙 ∈ 𝕊 2 is a three-dimensional vector representing the magnetization and 𝛽 is a positive constant.

When 𝛽 = 1 , using the standard stereographic projection 𝕊 2 → ℂ ∪ { ∞ } , (1.9) can be rewritten as the derivative Schrödinger equation for 𝑤 ∈ ℂ , 𝑖 𝑤 𝑡 + Δ 𝑤 + 4 ( ∇ 𝑤 ) 2 1 + | 𝑤 | 2 𝑤 = 0 . ( 1 . 1 1 )

Equation (1.9) is also called the Schrödinger map and has been studied by many authors [4–31]. Guo and Han [32] proved the following regularity criterion: ∇ 𝜙 ∈ 𝐿 2 ( 0 , 𝑇 ; 𝐿 ∞ ( ℝ 𝑛 ) ) ( 1 . 1 2 ) with 𝑛 ≥ 2 .

When 0 < 𝛽 ≤ 1 / 2 , Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion Λ 2 𝛽 𝜙 ∈ 𝐿 1 ( 0 , 𝑇 ; 𝐿 ∞ ( ℝ 𝑛 ) ) ( 1 . 1 3 ) with 𝑛 ≤ 3 .

We will refine (1.13) as follows.

Theorem 1.2. Let 0 < 𝛽 ≤ 1 / 2 . Let 𝑚 be an integer such that 2 𝑚 > ( 𝑛 + 1 ) / 2 for any 𝑛 ≥ 1 . Let Λ 𝛽 𝜙 0 ∈ 𝐻 2 𝑚 and 𝜙 0 ∈ 𝕊 2 and 𝜙 be a local smooth solution to the problem (1.9) and (1.10). If 𝜙 satisfies Λ 2 𝛽 𝜙 ∈ 𝐿 1 ̇ 𝐵 0 , 𝑇 ; 0 ∞ , ∞ ( ℝ 𝑛 ) ( 1 . 1 4 ) for some finite 𝑇 > 0 , then the solution 𝜙 can be extended beyond 𝑇 > 0 .

2. Proof of Theorem 1.1

Since ( 𝑢 , 𝜋 ) is a local smooth solution, we only need to prove a priori estimates.

First, testing (1.1) by 𝑢 and using (1.2), we see that 𝑑 1 𝑑 𝑡 2 𝑢 2 + 𝑢 𝑢 𝑡 | | | | 𝑑 𝑥 + ∇ 𝑢 2 = 𝑢 𝑑 𝑥 2 𝑡 𝑑 𝑥 + 𝑢 ⋅ ∇ 𝑢 ⋅ 𝑢 𝑡 ≤ u 𝑑 𝑥 2 𝑡 1 𝑑 𝑥 + 2 ‖ ∇ 𝑢 ‖ 𝐿 ∞ ‖ 𝑢 ‖ 2 𝐿 2 + ‖ ‖ 𝑢 𝑡 ‖ ‖ 2 𝐿 2 . ( 2 . 1 )

Testing (1.1) by 4 𝑢 𝑡 and using (1.2), we find that 𝑑 𝑑 𝑡 2 𝑢 2 𝑡 | | | | + 2 ∇ 𝑢 2 𝑢 𝑑 𝑥 + 4 2 𝑡 𝑑 𝑥 = − 4 𝑢 ⋅ ∇ 𝑢 + 𝑢 𝑡 𝑢 ⋅ ∇ 𝑢 𝑡 𝑑 𝑥 ≤ 𝐶 ‖ ∇ 𝑢 ‖ 𝐿 ∞ ‖ 𝑢 ‖ 2 𝐿 2 + ‖ ‖ 𝑢 𝑡 ‖ ‖ 2 𝐿 2 . ( 2 . 2 )

Applying Λ 𝑠 to (1.1), testing by Λ 𝑠 𝑢 𝑡 and using (1.2), (1.7), (1.8), and (1.6), we have 1 2 𝑑 | | Λ 𝑑 𝑡 𝑠 + 1 𝑢 | | 2 + | | Λ 𝑠 𝑢 𝑡 | | 2 | | Λ 𝑑 𝑥 + 𝑠 𝑢 𝑡 | | 2 𝑑 𝑥 = − 𝑖 Λ 𝑠 𝜕 𝑖 𝑢 𝑖 𝑢 ⋅ Λ 𝑠 𝑢 𝑡 Λ 𝑑 𝑥 − 𝑠 𝑢 𝑡 ⋅ ∇ 𝑢 ⋅ Λ 𝑠 𝑢 𝑡 − 𝑑 𝑥 𝑖 Λ 𝑠 𝜕 𝑖 𝑢 𝑖 𝑢 𝑡 − 𝑢 𝑖 𝜕 𝑖 Λ 𝑠 𝑢 𝑡 Λ 𝑠 𝑢 𝑡 𝑑 𝑥 ≤ 𝐶 ‖ 𝑢 ‖ 𝐿 ∞ ‖ ‖ Λ 𝑠 + 1 𝑢 ‖ ‖ 𝐿 2 ‖ ‖ Λ 𝑠 𝑢 𝑡 ‖ ‖ 𝐿 2 ‖ ‖ 𝑢 + 𝐶 𝑡 ‖ ‖ 𝐿 ∞ ‖ ‖ Λ 𝑠 + 1 𝑢 ‖ ‖ 𝐿 2 + ‖ ∇ 𝑢 ‖ 𝐿 ∞ ‖ ‖ Λ 𝑠 𝑢 𝑡 ‖ ‖ 𝐿 2 ‖ ‖ Λ 𝑠 𝑢 𝑡 ‖ ‖ 𝐿 2 ≤ 𝐶 ‖ 𝑢 ‖ 𝐿 ∞ + ‖ ∇ 𝑢 ‖ 𝐿 ∞ + ‖ ‖ 𝑢 𝑡 ‖ ‖ 𝐿 ∞ ‖ ‖ Λ 𝑠 + 1 𝑢 ‖ ‖ 2 𝐿 2 + ‖ ‖ Λ 𝑠 𝑢 𝑡 ‖ ‖ 2 𝐿 2 ≤ 𝐶 1 + ‖ 𝑢 ‖ ̇ 𝐵 0 ∞ , ∞ + ‖ ∇ 𝑢 ‖ ̇ 𝐵 0 ∞ , ∞ + ‖ ‖ 𝑢 𝑡 ‖ ‖ ̇ 𝐵 0 ∞ , ∞ l o g 𝑒 + ‖ 𝑢 ‖ 2 𝐻 𝑠 + 1 + ‖ ‖ 𝑢 𝑡 ‖ ‖ 2 𝐻 𝑠 ⋅ ‖ ‖ Λ 𝑠 𝑢 𝑡 ‖ ‖ 2 𝐿 2 + ‖ ‖ Λ 𝑠 + 1 𝑢 ‖ ‖ 2 𝐿 2 . ( 2 . 3 )

Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that ‖ 𝑢 ‖ 𝐿 ∞ ( 0 , 𝑇 ; 𝐻 𝑠 + 1 ) + ‖ ‖ 𝑢 𝑡 ‖ ‖ 𝐿 ∞ ( 0 , 𝑇 ; 𝐻 𝑠 ) ≤ 𝐶 . ( 2 . 4 )

This completes the proof.

3. Proof of Theorem 1.2

Since 𝜙 is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by ( ⋅ , ⋅ ) the standard 𝐿 2 scalar product.

First, testing (1.9) by Λ 2 𝛽 𝜙 and using ( 𝑎 × 𝑏 ) ⋅ 𝑎 = 0 , we see that 1 2 𝑑 | | Λ 𝑑 𝑡 𝛽 𝜙 | | 2 𝑑 𝑥 = 0 . ( 3 . 1 )

Testing (1.9) by Δ 2 𝑚 Λ 2 𝛽 𝜙 and using ( 𝑎 × 𝑏 ) ⋅ 𝑎 = 0 , (1.6) and (1.7), we obtain, with ( 1 / 𝑝 ) + ( 1 / 𝑞 ) = ( 1 / 𝑝 𝛼 ) + ( 1 / 𝑞 𝛼 ) = ( 1 / ̃ 𝑝 𝛼 ) + ( 1 / ̃ 𝑞 𝛼 ) = 1 / 2 , 1 2 𝑑 | | Δ 𝑑 𝑡 𝑚 Λ 𝛽 𝜙 | | 2 = 𝑑 𝑥 𝜙 × Λ 2 𝛽 𝜙 , Δ 2 𝑚 Λ 2 𝛽 𝜙 = Δ 𝑚 𝜙 × Λ 2 𝛽 𝜙 , Δ 𝑚 Λ 2 𝛽 𝜙 = Δ 𝑚 𝜙 × Λ 2 𝛽 𝜙 + 2 𝑚 − 1 𝛼 = 1 𝐶 𝛼 𝐷 2 𝑚 − 𝛼 𝜙 × Λ 2 𝛽 𝐷 𝛼 𝜙 , Δ 𝑚 Λ 2 𝛽 𝜙 = Λ 𝛽 Δ 𝑚 𝜙 × Λ 2 𝛽 𝜙 + 2 𝑚 − 1 𝛼 = 1 𝐶 𝛼 𝐷 2 𝑚 − 𝛼 𝜙 × Λ 2 𝛽 𝐷 𝛼 𝜙 , Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 2 𝐿 2 + 𝐶 ‖ Δ 𝑚 𝜙 ‖ 𝐿 𝑝 ‖ ‖ Λ 3 𝛽 𝜙 ‖ ‖ 𝐿 𝑞 ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 𝐿 2 + 2 𝑚 − 2 𝛼 = 1 𝐶 𝛼 ‖ ‖ 𝐷 2 𝑚 − 𝛼 𝜙 ‖ ‖ 𝐿 𝑝 𝛼 ‖ ‖ Λ 3 𝛽 𝐷 𝛼 𝜙 ‖ ‖ 𝐿 𝑞 𝛼 + ‖ ‖ 𝐷 2 𝑚 − 𝛼 Λ 𝛽 𝜙 ‖ ‖ 𝐿 ̃ 𝑝 𝛼 ‖ ‖ Λ 2 𝛽 𝐷 𝛼 𝜙 ‖ ‖ 𝐿 ̃ 𝑞 𝛼 ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 𝐿 2 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 2 𝐿 2 ‖ ‖ Λ ≤ 𝐶 1 + 2 𝛽 𝜙 ‖ ‖ ̇ 𝐵 0 ∞ , ∞ ‖ ‖ Δ l o g 𝑒 + 𝑚 Λ 𝛽 𝜙 ‖ ‖ 𝐿 2 ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 2 𝐿 2 , ( 3 . 2 ) which yields ‖ ‖ Λ 𝛽 ‖ ‖ 𝜙 ( 𝑡 ) 𝐻 2 𝑚 ≤ 𝐶 . ( 3 . 3 ) Here we have used the following interesting Gagliardo-Nirenberg inequalities: ‖ Δ 𝑚 𝜙 ‖ 𝐿 𝑝 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 1 − 𝜃 0 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 𝜃 0 𝐿 2 w i t h 𝑝 = 2 𝑚 − 𝛽 𝑚 − 𝛽 , 𝜃 0 = 2 𝑚 − 2 𝛽 , ‖ ‖ Λ 2 𝑚 − 𝛽 3 𝛽 𝜙 ‖ ‖ 𝐿 𝑞 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 𝜃 0 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 1 − 𝜃 0 𝐿 2 w i t h 𝑞 = 2 ( 2 𝑚 − 𝛽 ) 𝛽 , ‖ ‖ 𝐷 2 𝑚 − 𝛼 𝜙 ‖ ‖ 𝐿 𝑝 𝛼 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 1 − 𝜃 𝛼 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 𝜃 𝛼 𝐿 2 w i t h 𝜃 𝛼 = 2 𝑚 − 𝛼 − 2 𝛽 2 𝑚 − 𝛽 , 𝑝 𝛼 = 4 𝑚 − 2 𝛽 , ‖ ‖ Λ 2 𝑚 − 𝛼 − 2 𝛽 3 𝛽 𝐷 𝛼 𝜙 ‖ ‖ 𝐿 𝑞 𝛼 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 𝜃 𝛼 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 1 − 𝜃 𝛼 𝐿 2 w i t h 𝑞 𝛼 = 4 𝑚 − 2 𝛽 , ‖ ‖ 𝐷 𝛼 + 𝛽 2 𝑚 − 𝛼 Λ 𝛽 𝜙 ‖ ‖ 𝐿 ̃ 𝑝 𝛼 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ 1 − ̃ 𝜃 𝛼 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ ̃ 𝜃 𝛼 𝐿 2 ̃ 𝜃 w i t h 𝛼 = 2 𝑚 − 𝛼 − 𝛽 2 𝑚 − 𝛽 , ̃ 𝑝 𝛼 = 4 𝑚 − 2 𝛽 , ‖ ‖ Λ 2 𝑚 − 𝛼 − 𝛽 2 𝛽 𝐷 𝛼 𝜙 ‖ ‖ 𝐿 ̃ 𝑞 𝛼 ‖ ‖ Λ ≤ 𝐶 2 𝛽 𝜙 ‖ ‖ ̃ 𝜃 𝛼 𝐿 ∞ ‖ ‖ Δ 𝑚 Λ 𝛽 𝜙 ‖ ‖ 1 − ̃ 𝜃 𝛼 𝐿 2 w i t h ̃ 𝑞 𝛼 = 4 𝑚 − 2 𝛽 𝛼 . ( 3 . 4 ) This completes the proof.

Acknowledgment

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

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