The regularity of Navier-Stokes equations in five-dimensional space

(Graduate School of China Academy of Engineering Physics, Beijing 100088, China)

The regularity of Navier-Stokes equations in five-dimensional space

MAXixia

(Graduate School of China Academy of Engineering Physics, Beijing 100088, China)

five dimensional space; Navier-Stokes; compact theorem; H?lder continuous

This paper is concerned with the partial regularity of weak solutions of incompressible Navier-Stokes equations in five dimensional space with unit viscosity and zero external force:

(1)

forx∈Ω?R5,t<0, and

(2)

The concepts of weak solutions of (1)-(2), and their regularity were already introduced in the fundamental paper of J.Leray. Pioneering works of J. Leray showed the existence of a functionuandpsuch that

(iii)usatisfies the Navier-Stokes equations in the distribution sense.

In the series of papers [1-2,4-5], when the spatial dimensiondis 3, Scheffer introduced the notions of suitable weak solutions and the generalized energy inequality. He also established various partial regularity results of such weak solutions. Scheffer’s results were further generalized and strengthened in the paper of Caffareli, Kohn and Nirenberg[2], ford=3.

Ford=4,V.Scheffer[6]provedthatthereexistsaweaksolutionuinR4×RsuchthatuiscontinuousoutsidealocallyclosedsetofR4×Rwhose3-DHausdorffmeasureisfinite.Ford=5,6,Struwe[2],DuandDong[3]obtainedthecorrespondingresultsinthesteadyNavier-Stokesequations.TianandXin[7]showedthepartialregularityforsmoothsolutionsandanyspatialdimensioninthesteadyNavier-Stokesequations.

1 The Compactness theorem

Definition 1 Let Ω be a open set in R5. We say that a pair (u,p)isasuitableweaksolutiontotheNavier-StokesequationsonthesetΩ×(-T1,0)ifitsatisfiestheconditions:

(i)

u∈L2,∞(Ω×(-T1,0))∩L2(-T1,0;H1(Ω)),

(3)

(ii)uandpsatisfytheNavier-Stokesequationsinthedistributionsense;

(iii)uandpsatisfythelocalenergyinequality

(4)

Theorem 1[5]LetX0,XandX1bethreeBanachspacesandXi(i=0,1)isreflectivesuchthat

X0?X?X1

theinjectionofXintoX1beingcontinuous;andtheinjectionofX0intoXiscompact.LetTbeafixednumber,andletα0,α1betwofinitenumberssuchthatαi≥1,i=0,1.

Weconsiderthespace

AndthespaceΥisprovidedwiththenorm

ThentheinjectionofΥintoLα(0,T;X)iscompact.

Lemma 1 Let (u,p)isaweaksolutionoftheCauchyproblemsoftheNavier-StokesequationsinΩwithu∈L2,∞(Ω×(-T1,0))∩L2(-T1,0;H1(Ω)).Inaddition,

u∈L4,∞(Ω×(-T1,0))

(5)

Proof First by using Holder inequality and Young inequality,

(6)

In fact, by interpolation inequality,

Andthenweknow

(7)

inanyopensetΩ?R5fora.e.t∈(-T,0).

By the elliptic regularity theory,

Theorem 2 Let (un,pn)isasequenceofweaksolutions(1)-(2)inΩ×(-T,0)satisfying:

(c) (un,pn)satisfy(4),whereE,E1somepositiveconstants.

Supposethat(u,p)isaweaklimitof(un,pn),then(u,p)isalsoasuitableweaksolutionof(1)-(2).

Proof In fact, we can choose a subsequence

(8)

(?tun,φ)=-(un·▽un,φ)-(▽un,▽φ)≤

Hence

In the following we prove in two steps.

asδ→0,o(1)→0

And

ο(1)asn→0,ο(1)→0

Accordingtotheweakcontinuousint,

asδ→0,ο(1)→0isindependentofn.

Hence,

FinallybyTheorem1,

un→u

(9)

convergesstronglyinL2(Ω×(-T,0)). Also,u∈L4,∞(Ω×(-T,0)),byinterpolationinequality,

Hencefrom(9),

un→u

(10)

convergesstronglyinL3(Ω×(-T,0)). Since (u,p)istheweaklimitof(un,pn), for any smoothφ>0compactlysupportedinΩ×(-T,0), we have that

From Lemma 1 and (10), the theorem is proved.

2 The Regularity theorem

Using the compactness theorem in the last section, we show the partial regularity of the weak solutions of (1)-(2). Here we give a result which characterizes H?lder continuous functions by the growth of their local integrals.

Theorem 3 Supposeu∈L2(Ω)satisfies

(11)

foranyBr(x)?Ωandα∈(0,1),where

thenu∈Cα(Ω).

Proof DenoteR0=dist(Ω′,?Ω),Ω′?Ω. For anyx0∈Ω′and0

andintegratingwithrespecttoxinBr1(x0)

from(11),

(12)

andthereforeforh

with

forany0

for anyx∈Ω′ andR≤R0. Henceuis bounded in Ω′withtheestimate

Then we have

The first two terms on the right sides are estimated in (11). For the last term we write

and integrating with respect toζoverB2R(x)∩B2R(y),whichcontainsBR(x),yields

Therefore,wehave

Inthefollowingweassume(u,p)isasuitableweaksolutionofNavier-StokesequationsinΩ×(-T1,0).

Lemma 2 Suppose (u,p)isasuitableweaksolutionof(1)-(2),ifthereexiststwopositiveconstantε0suchthat

(13)

and

u∈L4,∞(Ω×(-T,0))

(14)

then

(15)

for-θ2≤t≤0.DenoteQθ=Bθ×(-θ2,0).

Proof Suppose that Lemma 2 is false, then there is a subsequence of weak solutions (ui,pi)with

(16)

whereQ1=B1×(-1,0),andsuchthat(15)isnotvalidfor(ui,pi).Let

then

(17)

▽vi)

(18)

inQ1. By Fatou Lemma,

Sinceun→uisstrongconvergeinL3(Q), we have

(19)

for all sufficiently enoughi.

(20)

Here

(21)

and

Denote

thenbyCalderon-Zygmundestimateand(20),

(22)

Hencefrom(20),(22),(23),weget

(24)

Itisobviousfrom(24)that

(25)

Combining (19) and (25), we obtain a contraction and the lemma is proved.

Theorem 4 Under the assumptions of Lemma 2, then for any numberk,▽k-1uisH?ldercontinuousinsubsetK??Ω×(-T,0)andthefollowingboundisvalid:

wherec0isaconstantonlydependingonk.

Proof Let (u,p)beasuitableweaksolutionsuchthat

Let

Asimplecomputationyieldsthatis(u1,p1)asuitableweaksolutionof

▽u1+▽p1=0

Moreover,Lemma2impliesthat

WerepeatthesameargumentsasLemma2 ,itisconcludedthat

isboundedbyanabsoluteconstant.

Thecasek>1istreatedwiththehelpoftheregularitytheoryfortheStokesequationsandbootstraparguments.

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五維空間Navier-Stokes方程的正則性*

2016-09-19 基金項目：國家自然科學(xué)基金 (11671045)

馬西霞(1990年生)，女；研究方向：流體方程 ；E-mail:kfmaxixia@163.com

馬西霞

(中國工程物理研究院研究生院，北京 100088)

五維空間；Navier-stokes方程；緊性定理；H?lder連續(xù)

O175.26；O175.29

A

0529-6579(2017)01-0096-06

10.13471/j.cnki.acta.snus.2017.01.016