# Regularity Criteria for Hyperbolic Navier-Stokes and Related System

Authors: Jishan Fan; Tohru Ozawa
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

First, we consider the following hyperbolic Navier-Stokes equations [1]:
(1.1)τutt+ut-Δu+∇π+u⋅∇u+τut⋅∇u+τu⋅∇ut=0,(1.2)div⁡⁡u=0,(1.3)(u,ut)(x,0)=(u0,u1)(x),x∈Rn,n≥2.
Here u is the velocity, π is the pressure, and τ>0 is a small relaxation parameter. We will take τ=1 for simplicity.Whenτ=0, (1.1) and (1.2) reduce to the standard Navier-Stokes equations. Kozono et al. [2] proved the following regularity criterion:
(1.4)ω∶=curl⁡u∈L1(0,T;B˙∞,∞0).
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(u0,u1)∈Hs+1×Hs with s>n/2,n≥2 and divu0=divu1=0 in ℝn. Let (u,π) be a unique strong solution to the problem (1.1)–(1.3). If u satisfies
(1.5)u,∇u,ut∈L1(0,T;B˙∞,∞0),
then the solution u can be extended beyond T>0.In our proof, we will use the following logarithmic Sobolev inequality [2]:
(1.6)‖u‖L∞≤C(1+‖u‖B˙∞,∞0log⁡(e+‖u‖Hs))
and the following bilinear product and commutator estimates according to Kato and Ponce [3]:
(1.7)‖Λs(fg)‖Lp≤C(‖f‖Lp1‖Λsg‖Lq1+‖Λsf‖Lp2‖g‖Lq2),(1.8)‖Λs(fg)-fΛsg‖Lp≤C(‖∇f‖Lp1‖Λs-1g‖Lq1+‖Λsf‖Lp2‖g‖Lq2),
with s>0,  Λ:=(-Δ)1/2 and 1/p=(1/p1)+(1/q1)=(1/p2)+(1/q2).Next, we consider the fractional Landau-Lifshitz equation:(1.9)∂tϕ=ϕ×Λ2βϕ,(1.10)ϕ(x,0)=ϕ0(x)∈S2,x∈Rn,
where ϕ∈𝕊2 is a three-dimensional vector representing the magnetization and β is a positive constant.Whenβ=1, using the standard stereographic projection 𝕊2→ℂ∪{∞}, (1.9) can be rewritten as the derivative Schrödinger equation for w∈ℂ,
(1.11)iwt+Δw+4(∇w)21+|w|2w¯=0.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:
(1.12)∇ϕ∈L2(0,T;L∞(Rn))
with n≥2.When0<β≤1/2, Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion
(1.13)Λ2βϕ∈L1(0,T;L∞(Rn))
with n≤3.We will refine (1.13) as follows.Theorem 1.2.
Let0<β≤1/2. Let m be an integer such that 2m>(n+1)/2 for any n≥1. Let Λβϕ0∈H2m and ϕ0∈𝕊2 and ϕ be a local smooth solution to the problem (1.9) and (1.10). If ϕ satisfies
(1.14)Λ2βϕ∈L1(0,T;B˙∞,∞0(Rn))
for some finite T>0, then the solution ϕ can be extended beyond T>0.


## 2. Proof of Theorem1.1

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)ddt∫‍(12u2+uut)dx+∫‍|∇u|2dx=∫‍ut2dx+∫‍u⋅∇u⋅utdx≤∫‍ut2dx+12‖∇u‖L∞(‖u‖L22+‖ut‖L22).Testing (1.1) by 4ut and using (1.2), we find that
(2.2)ddt∫‍(2ut2+2|∇u|2)dx+4∫‍ut2dx=-4∫‍(u⋅∇u+ut⋅∇u)utdx≤C‖∇u‖L∞(‖u‖L22+‖ut‖L22).ApplyingΛs to (1.1), testing by Λsut and using (1.2), (1.7), (1.8), and (1.6), we have
(2.3)12ddt∫‍(|Λs+1u|2+|Λsut|2)dx+∫‍|Λsut|2dx=-∑i‍∫‍Λs∂i(uiu)⋅Λsutdx-∫‍Λs(ut⋅∇u)⋅Λsutdx-∑i‍∫‍[Λs∂i(uiut)-ui∂iΛsut]Λsutdx≤C‖u‖L∞‖Λs+1u‖L2‖Λsut‖L2+C(‖ut‖L∞‖Λs+1u‖L2+‖∇u‖L∞‖Λsut‖L2)‖Λsut‖L2≤C(‖u‖L∞+‖∇u‖L∞+‖ut‖L∞)(‖Λs+1u‖L22+‖Λsut‖L22)≤C(1+(‖u‖B˙∞,∞0+‖∇u‖B˙∞,∞0+‖ut‖B˙∞,∞0)log⁡(e+‖u‖Hs+12+‖ut‖Hs2))⋅(‖Λsut‖L22+‖Λs+1u‖L22).Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that
(2.4)‖u‖L∞(0,T;Hs+1)+‖ut‖L∞(0,T;Hs)≤C.This completes the proof.


## 3. Proof of Theorem1.2

Sinceϕ is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by (·,·) the standard L2 scalar product.First, testing (1.9) by Λ2βϕ and using (a×b)·a=0, we see that
(3.1)12ddt∫‍|Λβϕ|2dx=0.Testing (1.9) by Δ2mΛ2βϕ and using (a×b)·a=0, (1.6) and (1.7), we obtain, with (1/p)+(1/q)=(1/pα)+(1/qα)=(1/p~α)+(1/q~α)=1/2,
(3.2)12ddt∫|ΔmΛβϕ|2dx=(ϕ×Λ2βϕ,Δ2mΛ2βϕ)=(Δm(ϕ×Λ2βϕ),ΔmΛ2βϕ)=(Δmϕ×Λ2βϕ+∑α=12m-1‍CαD2m-αϕ×Λ2βDαϕ,ΔmΛ2βϕ)=(Λβ(Δmϕ×Λ2βϕ+∑α=12m-1‍CαD2m-αϕ×Λ2βDαϕ),ΔmΛβϕ)≤C‖Λ2βϕ‖L∞‖ΔmΛβϕ‖L22+C‖Δmϕ‖Lp‖Λ3βϕ‖Lq‖ΔmΛβϕ‖L2+∑α=12m-2‍Cα(‖D2m-αϕ‖Lpα‖Λ3βDαϕ‖Lqα+‖D2m-αΛβϕ‖Lp~α‖Λ2βDαϕ‖Lq~α)‖ΔmΛβϕ‖L2≤C‖Λ2βϕ‖L∞‖ΔmΛβϕ‖L22≤C(1+‖Λ2βϕ‖B˙∞,∞0log⁡(e+‖ΔmΛβϕ‖L2))‖ΔmΛβϕ‖L22,
which yields
(3.3)‖Λβϕ(t)‖H2m≤C.
Here we have used the following interesting Gagliardo-Nirenberg inequalities:
(3.4)‖Δmϕ‖Lp≤C‖Λ2βϕ‖L∞1-θ0‖ΔmΛβϕ‖L2θ0withp=2m-βm-β,θ0=2m-2β2m-β,‖Λ3βϕ‖Lq≤C‖Λ2βϕ‖L∞θ0‖ΔmΛβϕ‖L21-θ0withq=2(2m-β)β,‖D2m-αϕ‖Lpα≤C‖Λ2βϕ‖L∞1-θα‖ΔmΛβϕ‖L2θαwithθα=2m-α-2β2m-β,pα=4m-2β2m-α-2β,‖Λ3βDαϕ‖Lqα≤C‖Λ2βϕ‖L∞θα‖ΔmΛβϕ‖L21-θαwithqα=4m-2βα+β,‖D2m-αΛβϕ‖Lp~α≤C‖Λ2βϕ‖L∞1-θ~α‖ΔmΛβϕ‖L2θ~αwithθ~α=2m-α-β2m-β,p~α=4m-2β2m-α-β,‖Λ2βDαϕ‖Lq~α≤C‖Λ2βϕ‖L∞θ~α‖ΔmΛβϕ‖L21-θ~αwithq~α=4m-2βα.
This completes the proof.



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*Source: 796368-2012-09-06.xml*