Regularity Criteria for Hyperbolic Navier-Stokes and Related System

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


Introduction
First, we consider the following hyperbolic Navier-Stokes equations 1 : 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: Here Ḃ0 ∞,∞ is the homogeneous Besov space.

ISRN Mathematical Analysis
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.
then the solution u can be extended beyond T > 0.
In our proof, we will use the following logarithmic Sobolev inequality 2 : and the following bilinear product and commutator estimates according to Kato and Ponce 3 : 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: 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 Schr ödinger equation for w ∈ C, Equation 1.9 is also called the Schr ödinger map and has been studied by many authors 4-31 .Guo and Han 32 proved the following regularity criterion: 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 with n ≤ 3. We will refine 1.13 as follows.

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

2.1
Testing 1.1 by 4u t and using 1.2 , we find that

2.3
Combining 2.1 , 2.2 , and 2.3 and using the Gronwall inequality, we conclude that This completes the proof.

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.

3.4
This completes the proof.