# Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping

Authors: Weihua Wang; Guopeng Zhou
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper is concerned with the regularity criterion of weak solutions to the three-dimensional Navier-Stokes equations with nonlinear damping in critical weakLq spaces. It is proved that if the weak solution satisfies ∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3/1+ln⁡e+∇uL22ds<∞,  q>3/2, then the weak solution of Navier-Stokes equations with nonlinear damping is regular on (0,T].


---

## Body

## 1. Introduction

In this study we consider the Cauchy problem of the three-dimensional Navier-Stokes equations with the nonlinear damping(1)∂tu+u·∇u+∇π+ur-2u=Δu,∇·u=0,together with the initial data(2)ux,0=u0,where u=(u1(x,t),u2(x,t),u3(x,t)) and π(x,t) denote the unknown velocity fields and the unknown pressure of the fluid. |u|r-2u, r>2 is the nonlinear damping. Moreover, (3)∇=∂1,∂2,∂3represents the gradient operator, (4)Δ=∇·∇=∑i=13∂2ux,t∂xi2denotes the Laplacian operator, and(5)u·∇u=∑i=13ui∂ux,t∂xi,∇·u=∑i=13∂uix,t∂xi.The mathematical model (1) is from the resistance to the motion of the flows. It describes various physical situations such as drag or friction effects, porous media flow, and some dissipative mechanisms [1, 2]. When the nonlinear damping term |u|r-2u in (1) disappears, the system reduces the classic Navier-Stokes equations [3, 4](6)∂tu+u·∇u+∇π=Δu,∇·u=0.In the mathematical viewpoint, therefore, Navier-Stokes equations with the nonlinear damping are a modification of the classic Navier-Stokes equations. There is a large literature on the well-posedness and large time behavior for solutions of Navier-Stokes equations with the nonlinear damping (see [1, 5, 6]).However it is not known whether the weak solution of Navier-Stokes equations with the nonlinear damping (1) is regular or smooth for a given smooth and compactly supported initial velocity u0. Fortunately, the regularity of weak solutions for Navier-Stokes equations with the nonlinear damping (1) can be derived when certain growth conditions are satisfied. This is known as a regularity criterion problem. Recently, Zhou [7] studied the regularity criterion for weak solutions for Navier-Stokes equations with the nonlinear damping (1) in critical Lebesgue spaces. That is, if a weak solution u of Navier-Stokes equations with the nonlinear damping (1) satisfies(7)∇u∈Lp0,T;LqR3for2p+3q=2,3<p≤∞,then the weak solution is smooth on (0,T].Since Navier-Stokes equations with the nonlinear damping (1) are a modification of the classic Navier-Stokes equations, it is necessary to mention some regularity criteria of weak solutions for Navier-Stokes equations and related fluid models [8, 9]. As for this direction, the first result of Navier-Stokes equations is studied by He [10] and improved by Dong and Zhang [11], Pokorý [12, 13], and Zhou [14]. One may also refer to some interesting regularity criteria on related fluid models (see [15] and the references therein).The main purpose of this paper is to investigate the regularity criteria of weak solutions with the aid of two components of velocity fields in critical weakLq space. To do so, we recall the definition of the weak solution of Navier-Stokes equations with the nonlinear damping (1).Definition 1.
A measurable functionu(x,t) is called a weak solution of Navier-Stokes equations with the nonlinear damping (1) on (0,T) if u satisfies the following properties:(i)
u∈L∞(0,T;L2(R3)) and ∇u∈L2(0,T;L2(R3));(ii)
for anyϕ∈C0∞(R3×[0,T)) with ∇·ϕ=0(8)∫0T∫R3u·∂tϕ-∇u·∇ϕ+∇ϕ:u⊗u+ur-2uϕdxdt=-∫R3u0ϕ0dx;(iii)
∇·u=0 in the distribution space D′((0,T)×R3);(iv)
u satisfies the energy inequality for 0≤t≤T(9)utL22+2∫0t∫R3∇ux,s2dxds+2∫0t∫R3ux,spdxds≤u0L22.The main result on the regularity criteria of the weak solutions of Navier-Stokes equations with the nonlinear damping (1) is now read.Theorem 2.
Supposeu0∈L2(R3)∩H˙1(R3) and u is a weak solution of Navier-Stokes equations with the nonlinear damping (1) in (0,T). If any two components of velocity fields satisfy(10)∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL22ds<∞,q>32,then u is smooth on (0,T].This result improves the earlier regularity criterion involving (7). Furthermore, Theorem 2 also implies the following regularity criterion for weak solutions of Navier-Stokes equations with the nonlinear damping (1).Theorem 3.
Supposeu0∈L2(R3)∩H˙1(R3) and u is a weak solution of Navier-Stokes equations with the nonlinear damping (1) in (0,T). If any two components of velocity fields satisfy(11)∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3ds<∞,q>32,then u is smooth on (0,T].Remark 4.
The main idea in the proof of Theorem2 is borrowing from the argument of previous results on classic Navier-Stokes equations [16] and together with energy methods.


## 2. Preliminaries

To start with, let us recall the definitions of some functional spaces.Lq(R3) with 1≤q≤∞ is a Lebesgue space under the norm (12)gLq=∫R3gxqdx1/q,1≤q<∞,esssupx∈R3gx,q=∞,and H˙m(R3) the Hilbert space (13)g∈L2R3;∇mgL2<∞.To define the Lorenz spaceLp,q(R3) with 1≤p, q≤∞, g∈Lp,q(R3) if and only if(14)gLp,q=∫0∞tqmg,tq/pdtt1/q<∞for1≤q<∞,gLp,∞=supt≥0tmg,t1/p<∞forq=∞,where (15)mf,t≔mx∈R3:gx>tis Lebesgue measure of the set {x∈R3:|g(x)|>t}.Actually Lorentz spaceLp,q(R3) may be alternatively defined by real interpolation (see Triebel [17])(16)Lp,qR3=Lp1R3,Lp2R3θ,qwith (17)1p=1-θp1+θp2,1≤p1<p<p2≤∞,0<θ<1.In particular, gLq,∞ is equivalently to the norm(18)sup0<S<∞⁡S1/q-1∫Sgxdx.Furthermore, the definition implies the continuous relationship(19)LqR3↪Lq,∞R3,1<q<∞.In fact it is easy to check and thus it is readily seen that (20)x-3/q∈¯LqR3,butx-3/q∈Lq,∞R3.We also recall the Hölder inequality in Lorentz space which plays an important role in the next section.Lemma 5 (O’Neil [18]).
Letf∈Lp1,q1(R3) and g∈Lp2,q2(R3) with (21)1≤p1,p2≤∞,1≤q1,q2≤∞.Then fg∈Lp,q(R3) satisfies the Hölder inequality of Lorentz spaces(22)fgLp,q≤CfLp1,q1gLp2,q2,where (23)1p=1p1+1p2,1q≤1q1+1q2.


## 3. A Priori Estimates

In this section we will prove a priori estimates for smooth solutions of (1) described in the following.Theorem 6.
LetT>0, let u0∈L2(R3)∩H˙1(R3), and let u be a local smooth solution of the Navier-Stokes equations with the nonlinear damping (1). If u also satisfies (11), namely,(24)∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL22ds<∞,q>32,then the a priori estimate(25)sup0<t<TutL4≤Cholds true.Proof of Theorem6.
Multiplying both sides of the Navier-Stokes equations with the nonlinear damping (1) with Δu and integrating in R3, we have(26)12ddt∫R3∇u2dx+∫R3Δu2dx+r-1∫R3ur-2∇u2dx=-∫R3u·∇uΔudx,where we have used (27)∫R3∇pΔudx=-∫R3pΔdiv⁡udx=0.
For the right hand side of (26) we have (28)-∫R3u·∇uΔudx=-∑i,j,k=13∫R3ui∂iuj∂kkujdx=∑i,j,k=13∫R3∂kui∂iuj∂kujdx=∑i,j,k=13∫R3∂kui∂iuj∂kujdx+12∑i,j,k=13∫R3ui∂i∂kuj∂kujdx=∑i,j,k=13∫R3∂kui∂iuj∂kujdx=∑i=12∑j,k=13∫R3∂kui∂iuj∂kujdx+∑j=12∑k=13∫R3∂ku3∂3uj∂kujdx+∑k=13∫R3∂ku3∂3u3∂ku3dx=∑m=13Im,where we have used the fact that the divergence-free condition (29)∑k=13∂kuk=0.
ForI1, we have(30)I1≤∑i=12∑j,k=13∫R3∂kui∂iuj∂kujdx≤∫R3∇u1+∇u2∇u2dx.
ForI2, similarly we obtain(31)I2≤∑j=12∑k=13∫R3∂ku3∂3uj∂kujdx≤∫R3∇u1+∇u2∇u2dx.
Finally forI3, applying the fact (32)∂3u3=-∂2u2-∂1u1,I3≤∑k=13∫R3∂ku3∂3u3∂ku3dx≤∑k=13∫R3∂ku3-∂2u2-∂1u1∂ku3dx≤∫R3∇u1+∇u2∇u2dx.
Plugging the estimatesIi into the right hand side of (26), it follows that(33)12ddt∫R3∇u2dx+∫R3Δu2dx+r-1∫R3ur-2∇u2dx≤C∫R3∇u1+∇u2∇u∇udx.
Applying Hölder inequality and Young inequality, we have for the right hand side of (33)(34)∫R3∇u1+∇u2∇u∇udx≤C∇u1Lq,∞+∇u2Lq,∞∇uL2q/q-1,22.
Applying the Gagliardo-Nirenberg inequality in Lorentz spaces, that is,(35)∇uL2q/q-1,2≤C∇uL22q-3/2qΔuL23/2q,
thus we have from (33) (36)12ddt∫R3∇u2dx+∫R3Δu2dx+r-1∫R3ur-2∇u2dx≤C∇u1Lq,∞+∇u2Lq,∞∇uL22q-3/qΔuL23/q≤12ΔuL22+C∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3∇uL22which implies(37)ddt∫R3∇u2dx+∫R3Δu2dx≤C∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3∇uL22.In particular,(38)ddt∫R3∇u2dx+∫R3Δu2dx≤C∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL22·1+lne+∇uL22∇uL22.
Employing the Gronwall inequality, it follows that(39)∫R3∇u2dx≤∫R3∇u02dxexp⁡∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL221+lne+∇uL22dt.Hence we have(40)lne+∫R3∇u2dx≤lne+∫R3∇u02dx+∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL221+lne+∇uL22dt.
We take the Gronwall inequality into account again to get(41)lne+∫R3∇u2dx≤Cu0·exp∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL22ds<∞.
Hence we obtain a priori estimates of∇u:(42)esssup0<t<T∫R3∇u2dx<Cu0.


## 4. Proof of Theorem2

Under the a priori estimates in Theorem6, we now are in a position to complete the proof of Theorem 2. Since u0∈L2(R3)∩H1(R3) with ∇·u0=0, by the existence theorem of local strong solutions to the Navier-Stokes equations with nonlinear damping r>2, there exist a constant T∗>0 and a unique smooth solution u~ of (1) satisfying (refer to [19]) (43)u~∈BC0,T∗;H1,u~x,0=u0.Note that the weak solution satisfies the energy inequality (9). It follows from the weak-strong uniqueness criterion that (44)u~≡uon0,T∗.Thus it is sufficient to show that T∗=T. Suppose that T∗<T. Without loss of generality, we may assume that T∗ is the maximal existence time for u~. Since u~≡u on [0,T∗), by the assumptions (11), (45)∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-31+lne+∇uL22ds<∞,q>32.Therefore it follows from (25) that the existence time of u~ can be extended after t=T∗ which contradicts the maximality of t=T∗.This completes the proof of Theorem2.



---

*Source: 310934-2015-07-29.xml*