Vietnam J. Math. (2015) 43'283-295 DOI IO.IO07/S1OO13-OI4-0II7-8
On the Hausdorff Dimension of the Singular Set in Time for Weak Solutions to the Non-stationary Navier-^toites Equation on Torus
Dao Quang Khai • Nguyen Minh TVi
Received. 16 October 2013 / Accepted: 6 July 2014 / Published online: 18 January 2015
© Vietnam Academy of Science and Technology (VAST) and Springer Sc ien ce-i-Business Media Singapore 2015
Abstract In this note, we investigate the Hausdorff dimension of the possible time singular set of weak solutions to the Navier-Stokes equation on the three dimensional torus under some regularity conditions of Serrin's type (Arch. Rational Mech. Anal., 9, 187-195, 1962).
The results in the paper relate the regularity conditions of Serrin's type to the Hausdorff dimension of the time singular set More precisely, we prove that if a weak solution u belongs to L''(0, 7"; V^) then the 11 — '^ " " 1-dimensional Hausdorff measure of the time singular set of u is zero. Here, r is just assumed to be positive. We also establish that if a weak solution u belongs to £.''(0, T; VV'-^) then the (1 - '^'"^7" M-dimensional Hausdorff measure of the time singular set of u is zero. When r = 2, a — 1, or r = 2,q = 2.
we recover a result of Leray (Acta Math 63, 193-248, 1934), Scheffer (Commun. Math Phys. 55, 97-112, 1977), Foias and Temam (J. Math. Pures Appl. 58, 339-368, 1979), and Temam (Navier-Slokes equations and nonlinear functional analysis. SIAM, Philadelphia, 1995). Our results in some way aLso relate to the regularity results obtained by Giga (J Differ Equ. 62, 186-212, 1986).
Keywords Navier-Stokes equations • Semn's type conditions Hausdorff measure • Singular set in time
Mathematics Subject ClassiHcation (2010) 35Q30 76D05 76N10
D, Q. Khai N M. Tn {[i?)
Insuiuie of Mathematics, Vietnam Academy of Science and Technology, IS Hoang Quoc Viet. 10307 Hanoi, Vietnam
e-maiP [email protected]
•a Spm
D-Q Khai, N.M. Tri
1 Introducrion
In this paper we consider the initial value problem for the non-stationary Navier-Stoka equation on torus T ' = I ' / Z ' . Of •" °*«" '">"'<' >" * ' ' * ' * P " ' " * ' : boundary conditions i ^ . - f „ . ! i ! i _ A „ , + i ^ - / ; = u o n T ' r ~ T ' x ( 0 , T ) , i = l , 2 , 3 , (1)
St ^^ '3Xj Sx,
''"M = E ^ = » "" T|, (2)
uU.O) = « ° W i n T ^ x l O ) , (3) where fix. t) = (fi(x, I), fi(x. I), / 3 U , 0 ) . »°(*) are given functions with u°U) sans-
fying die condition div(H'*) = 0. Denote by flj?) the space of all infinitely differentiable solenoidal vector fields with zero averaging on T ' ; by i'(j\) the space of all compactly supported in T\ infinitely differentiable solenoidal vector fields with zero averaging o n ' t for each f E [0, T"]; H, V are the closures of die set r (T^) in the spaces L^(T'), H' (T=), respectively. Assume that / e L*=(0, T\ V ) , wo e H. where V is the dual space of V. A weak solution of the problem (1 -3) in T^ is a vector field such that
» E £.^(0. T; V) n L°°(0, T; H) n C([0, T\, L J )
/ ^-^ 9u, ^ 9M, 9U, . dui
d i d / = {/.«) V U E C ^ C T J ) , f / ^ 3o, i ^ Sn, 80, Su, \
i/^,|:i»,u,.)i^d.+/^^^__^_^_E|S:r^*^iX.D-<-'°"'-'-
Vfo € [0, r ] \ E , /| € [ro. T], where £ has Lebesgue measure zero and 0 ^ S,
\\uix, t) - uoix)\\ii7fY^f -^0 as I ^ 0.
where (-, •) is the pairing between V and V It was proved by Leray that there exists at least one weak solution of the problem (1-3).
The classical results on local existence of strong solutions and global existence of weak solutions to the initial boundary value problems for the Navier-Stokes equation were obtained in [5, 7, 9] (see also the monographs [8, 17]). The study of Navier-Slokes equa- tion on the torus is presented in [18]. The Hausdorff measures of singular sets of weak (or suitably weak) solutions lo the Navier-Stokes equation were investigated in [2, 3, 11, 12, 16] (see also the references therein). The uniqueness and regularity of weak solutions with some additional assumptions, say the Serrin conditions L'^ L'' or L''W^-'', were proved in numerous papers (see [I, 14] and the references dierein). In this paper, we study the Hausdorff dimension of the possible singular set in time of weak solutions under additional assumptions that the solutions belong lo L'" / / " or L*" IV' •<?. In order to obtain the results, we follow the method proposed in [3, 18] with some generahzation. In general, we will use the notations introduced in [18]. In the paper, p and po are used to denote the Lebesgue and D-dimensional Hausdorff measures, respectively, of a set in iR'. The paper is organized as follows. In Section 2, we consider weak solutions which belong to L"" H". In Section 3, we consider weak solutions which belong to L'' VV' •'. Throughout the paper, we denote by C a
^ Springer
On ihe Hausdorff DimenMon of the Singular Set in Time for Weak Solutions
general constant which may vary from place to place and it can take different value e one line.
2 Weak Solutions i n / , ' • / / ' '
Let ^ = - A H with DiA) = [u e H, Au e H) and G be the orthogonal complement of H in L-(T^). The operator A can be seen as an unbounded positive hnear self-adjoint operator on H. and we can define the powers A", a e R, with domain DiA"). Denote Va = DiA"^-). Then, A is an isomorphism from Va+i onto Vg. The norm of an element u e Va will be denoted by \u\a.
For u.v.w e / ( T ^ ) , we set
biu.v.w) = y ^ / u,DiVjUjjdx.
Lemma I Suppose that a e [ ^, 2]. Then there exists a constant C such thai
\hiu.V,w)\<C\u\a\vU+i\w\i^a iA)
forail u.v.w eriT^).
Proof First, we consider the case ^ < a < 1. By applying Lemma 2.1 of [17] with mi =a,m2 = ce,mi = 1 - a, we get (4).
Now, we consider the case 1 < a <2.By integration by parts, using the Stokes formula, we get
biu,v.w) = biu,v.AA^'w)^ V / u,D,kVjDk{A~'u;)jdx
+ I ] \DkUiD,VjDkiA-'w)jdx. (5)
Again, by applying Lemma 2.1 of [18] with m\ —a.mi—a-- l.m^ = z —a for the first temi on the right hand side of (5); with mj — a — 1. IUT — a, m^ — 2 — a for the second term on the right hand side of (5), we get
IMw. V. w)\ < C | H | „ | u U i | A - ' « > | 3 _ „ < C|M|„tl-|„+,|l«|,_„.
This proves the lemma. D Using the property of trilinearity of the form b, from Lemma 1 for a e [\. 2]. we can
extend b from T^iT^) to Va x K^+j x ¥,_£, satisfying
|fc(H.u,w.)| < C ! M | „ | U | „ + , | I ^ | | - „ forail (M, V. w)eVaX V„+i x Vi_„.
F o r u e riJ^)p\Hbi.a(u) = biu,u, A^u).
^ Springer
D.Q Khai, N.M.Tri Lemma 2 Suppose that o £ [ 5, co). Then
\bi.aiu)\ < ClHli+^WlJ;" ifae [ ^ - l ) -
\h.3iu)\ <Ci£)\u\]-'\u\l+' forO<:s<-, (fi) '•2 3 2 2
\biaiu)\ < C\u\l\uUi ifa>- forallu El^(¥-•').
Proof First, we consider the case 4 < o < 1. By applying Lemma 2.1 of [18] with Hji :^ ff + i , m2 — \ — a. HI3 :^ 0 and then using an interpolation inequality for the Sobolev norms, we get
\bi..iu)\ < C\u\^^i\u\2-.\U\2. < ClHli+^lMlJ^^
Now, we consider the case I < or < 5. By integration by parts, using the Stokes formula, we get
bi,a(u) = biu.u.AA''-ht) = Yi / ''iD,kUjDkiA'"^u)jdx
+ Y [ DkU,D,UjDkiA"-^u)jdx. {7}
, TkLi •^•'^'
Again, by applying Lemma 2 I of [18] with;w| =a,m2 = 0 , / n 3 = | - a for the fim term on the right hand side of (7), with mi - a - | , m 2 ^ a - | , m 3 = 3 - 2 « for the second term on the right hand side of (7) and then using an interpolation inequality for the Sobolev norms, we get
\bhaiu)\ <C (\u\a\u\2\u\^^, +\ul^l_\ul^iju\2) < C|H|i"^"|«|J;".
Finally, we consider the case o > | . Denote by [a] the integer part of a and by [a] the fraction part o f a . By integration by parts, using the Stokes formula, we get
bi.aiu) - / > ( H . H , A I " U ' " 1 W ) (8) J2 ciai.a2,ci3) Y lD"'uiD"'-D,UjD''^iA^"^u)jdx,
where D is a subset of {(ai, 02, aj) : |ail + |a2| - [a], lasl - [a]}. Applying Lemma 1 for each term of the nght hand side of (8) withm, ^ a - [a]], mj = [a] - [0:2], m3 = 0 if ff >j,m-i = e.O<E<i;ifa — \, w e get
\bi.aiu)\ < C | H | „ | H | [ „ | + I 1 M U , „ I if ff > ^ , h.aiu)\ < Cie)\u\.\u\2\u\2+, if a - ^ .
and then using an interpolation inequality for Sobolev norms, we get the desired inequalities.
This proves the lemma Q 0 Springer
On the Hausdorff Dimension of ihe Singular Sei in Time for Weak Solutions 287 From Lemma 2, f o r a e h^. cc ), we can extend fti from / ' ( T ' ) to VQ+I satisfying the
inequality (6).
Lemma 3 Assume that a e (^. ^\,u e L^(0, T. V„+i) n L ^ ( 0 . T: V^), the funciion {l t-f \uil)[^] is absolutely continuous and almost everywhere sati'^fies the equality
where f e L'^iO, T; V^-i). Then there exist conslanls T*, n.,z, depending only on
|w(0)|a.af, supo<,<j- \fil)\a-\ such thai
JO
Proof By applying the Hblder and Young inequalities, we get
|(/(f),A"H(f))| < ]/(Olt.-i|A"M((}|,_„ < C | / ( f ) | ^ _ i | A ^ » ( r ) |
< ^ l » ( r ) l ^ + i + C l / ( r ) | ^ _ | < ^ | H ( t ) | ^ ^ i + C ( / , a ) . (10) where C ( / . a) is a constant depending only on supo<,<7- |/(f)lo_i.
From Lemma 2, by applying Young's inequality, we have
\bi.Auii))\ < C|H(r)|i"^''|M(f)|^;" 5 -\uil)\l+i+Cia)\uil)\^ (11) Since ^ ^ ^ > 2 from (9), (10). and (11), we get
— IfiDll + lnml+i < Cia) \uil)\^ -^ Cifa)
< C ( / , f f ) ( | M ( r ) | ^ - n ) ' ^ ^ (12) If we set
y ( / ) - | H ( O I ^ + l from (12), It follows that
y'U) < 2CiN.a)y'-^ = Cy'-^. (13) where C ^ 2 0 ( ^ , 0 ! ) is a constant. By integrating (13). we obtain
1 \ " " « I yit) < iCa)-
\Cay"iO) where a — T^TJ- Now, we define
r ^ - L z i ; , ^<°-"' ^ . „5)
°<^-''°'"' (|«(0)|J + 1)
When I equals 7"*, then the right hand side of (14) is equal to 2y(0). From (14) and (15), we have
y(t) < 2y(0) Vr E [0, T'l (16)
D. Q. Khai. N. M. Tii Using(12)and(]6).weget
\iiit)\l<2\uiO)\l + l V ; e [ 0 , r ] . (17) Integrating bolh sides of the (12) and using (17), we get
r l M ( r ) | ^ + | d / < 2 r * C ( i V , a ) ( 2 | M ( 0 ) | ^ + 2 ) - " " + | H ( 0 ) | ^ . (18)
The lemma is proven by (18) and (17). D In the following theorem, we use the notion of strong solutions in [15] (see Definition
4.1.1 there). Note diat when a > ^. we have the embedding Va C L^^^, widi j ^ > 3.
Theorem 1 Assume thai
I 3 / 6 L ^ ( 0 , T; Va-i), HO e V„, 2 "^ " "" 2 ' Then there exists a unique strong solution to the Navier-Stokes equation, satisfying
( ( € L - ( 0 , r " ; V „ + i ) n C ( [ 0 . r * * ] ; Va). (19) where T** — min(7", T*), T* is given by (15).
Proof To prove the existence of a strong solution, we use the standard Galerkin method (see for example [18]) combining with estimates proven m Lemmas 1—3
To prove the uniqueness, we note that
i«(oi„+i < i » ( o i l i « a ) i i + | . (20) From (19), (20), we get M e L^iO.T": V^_|_i). Sincea > ^ by Sobolev'sembeddmg theo-
rem, wehave L^(0, T**; V^^i) Ct^(0,T**;L'*(T^)). Therefore M€/.**(0,r**;/,''(T^)) satisfies Serrin's uniqueness condition. This completes theproof of the theorem. D
Let ff e ( 5 - f ) We say that a weak solution M is M''(T^)-regular on ((1, ri) if M ^ C ( ( / | , ( 2 ) . H " ( T 3 ) ) We say that an H " regularity interval Ui.h) is maximal for w if there does not exist any interval of H " regulanty stnctly containing (?|, I2). From Theorem I, we can easily prove that if (f 1,12) is a H^-maximal interval of a solution u, then
limsup|M(f)|(, — -j-co.
( 4 . l ) - UO e H. f € Z.°^(0, r , Va-i) Assume that u is a \x Theorem 2 Let a e
solution ofihe Navier-Stokes equation. Then, u is W iJ^)-regular on an open set o/(0, T) whose complement has Lebesgue measure 0,
Proof Since u is weakly continuous from [0. T] into H. uit) is well defined for every ( and we can define
£0 - {( e [ 0 . r i , uii)^ Va],
£2„ - {( e [0, T], uil) e Va],
a - { ' e ( 0 , r ) , a £ > 0 , uit)eCiit-s.i+e).v^)].
a Spnnger
On the Hausdorff Dimension of liie Singular Sei m Time for Weak Solutions 289 it is clear Oa isopen.First, we claim that w e L ' ( 0 , T: VQ). Indeed, if a e (\.l the claim
followsfrom the very definition of weak solutions. If a € ( 1 , 5 ) from the proof of Theorem 4.2 of [18], we note that H e £.'(0, 7"; Vs), hence the claim follows. From the just proved statement, we have p. ( £ „ ) = 0. Thus, fli'La U S ) = 0 (E was introduced in connection with the energy inequality in the deftnihon of weak solutions). If fo e ^a\iOa U S ) then, according to Theorem 1 and Che uniqueness theorem of Sather and Serrin (see [17]), IQ is the left end point of an interval of M^-reguIanty, i.e., one of connected components of 0 ^ . Thus, ^a\iOa U E) is a set no more than countable and therefore ixi^a\iOa U E)) ^ 0.
By noting that [0, T] \ 0 „ c i^a\iOa U E)) U E^ U E. we get M([0. 7"] \ 0 „ ) = 0. The
theorem is proved. D Lemma 4 LeiT > 0. Assume ihal a set O C [0, T] is open. Denote by S the complement of
0 in [0, T] and assume that fiiS) — 0 Suppose that there exists a funciion yil) e L^|0, 7"]
satisfying the following conditions
[ Uf) > 0 Vf e O.
j ( r , m i n | ; - F ^ , r | ) c O V f E O . f^"
(our convention is ^ — +00J, ivhere M and a are constants satisfying the condition M >0. a > s >0.
Then, we have Pi_s.iS) — 0.
Proof We have
0 =U,eiic,,d,).
where / is an index set no more than countable. The sets (Cj.d,) are connected components of O. Now take an arbitrary index / e I and a point r e ic,.di). From (21). it follows that
d, > m i n k - I - .T\. (22) I .v"(r) J
From (22), we deduce that
1 l / ( T ) 1 I ^ y ( r ) , 1 id,-r)o I A / O ( r - r ) " l MS ( r - r ) s foreveryr e (c,, rf,). Integrating the above inequality from Q to rf,. we obtain
( , , _ „ . - , (1 _ 1) („-s £'',.,r,d.+ £'•-ii_j).
Summing in i e / . we have
X ^ ( d , - c , ) ' - ^ < ( | - ^ ) ( M - ^ /" / ( r ) d r - F ^ — ^ j < + c x ) (23) For any s > 0 from (23) and the assumpUon that piS) — 0, it follows that there exists a finite set 1^ C I such that
Q Springer
290 D.Q. Khai.N.M.TVi It is easily seen that there exists a natural number m such that
[0, T]\U,g/, icd,) ^ Uj^i[aj,bj] - ^"j'^iBj,
where Bj U By = & if j ^ j ' . By denoting Ij theset of i e / \ / E such that (c,,rf,) c fl,, we have for every j
Bj = |J(c,,rf,)lJ(fi^n5). (24) From (24), we deduce that
diam(fi^) ^bj-oj^ ^ ( r f , - c ) < ^ (rf, - c ) < £.
By using the inequality
^i-i I (=1
: 0, (• — 1, ...tl), u < / < 1,
; j;diara(/ij)'-i = j i ^ ( r f , - c , )
By letting e -> 0, we conclude that M i - i ( 5 ) ^ 0 . • Theorem 3 / I s j ™ , fc, „ E ( i , J ) , « „ £ « , / £ i » ( 0 , J , l',_i) and i, ,s a weak
solution of tile NanerSlokes equation and satisfies Ihe following condition
iiiEL'I.O.T:V„), , - > 0 , r ( 2 o - l ) < 4 . (25) Tliai there exists a closed set S„ c (0, T'l such that u e C([0, T] \ S„: V„) and
Proof Denote S, = [0, T]\0,. where O . was introduced in the proof of Theorem 2, By Theoren, 2, we get „ ( S . ) = 0. From (25). we have , „ ( , ) E i i ( 0 , T). where , . ( , ) = l«(OI„ -I- 1. By applying Lemma 4 with 0 = 0„ y„(t) = lu(,)|2 + ] „ _ J _ , ^ f w e s e e t h a t M | _ 2 ! ^ ( 5 . ) = 0 Thisprovesthelheorem. ° ' ^ ~ ' ' Q fe».orf ; The condition r(2» - 1) < 4 in Theorem 3 is not essential since if K 2 . - 1) > 4 then „ sausfies the Sernn eondttton, hence . is smooth i f / ,s smooth. In this case, „ ( W = 0. This condition is assumed only in order diat ,i, „»,_„ make sense make sense.
RewiflrA 2 Theset E, where the energy inequality in the Hptln.ti,.., „f i , • fail i« 1 siih^... ..f c Tl, r I . ^ J " ' ^""^ oetinition of weak solutions may taiJ, IS a subset of 5 Therefore, under the hypotheses of Theonjm 3 die (1 ^ i S ^ l dimensional Hausdorff measure of E IS equal to zero em j , me (1 , ) fenwrtJ If,- = 2.<, = l . t h e n t h e c o n d i t i o n » s t 2 ( 0 r - V . l . n - n . , - j , since it IS assumed in the definition of weak s o l u t L s I^'th s " I ^ w ? ' " " " . T result of Scheffer and Leray (see [9, , 2 , 18,). When 2 > . ^ " T , , 1 Ti^T^Tu
« Springer
On Ihe Hausdorff Dimension of the Singular Set in Time for Weak Solutions 291 force is smooth, the condition u e /.'"(O, T; Va) is redundant, and in this case our result is
not new. However, even if 2 > r(2ff — 1), ff e ij, I], our result is new with respect to non-regular force / e L°°(0, 7"; V„-i).
Remark 4 Under the conditions of Theorem 3 with <^ £ ( ^ • 5) replaced by a E | , OO 1 , the conclusion of Theorem 3 should be replaced by
for ff — - . p\_'.giSa) — 0 for any £ > 0,
To prove it, we use the estimate (6) in Lemma 2 for fe|_o,(w) with a e i . 3C 1 and then exactly follow the proof of Theorem 3.
3 Weak Solutions in U'W^''^
In this section, for ^ > 2, we use the following notations
4 ^f-LiJP \SXI3XJI \axj\
Lemma 5 If Ihe force and solution of the Navier-Stokes equation are smooth enough then we have
1 A 4 11 / . \ i i 2 V |Vi,|4
3'», I 3o, 1'-^
^ ^ f 3^|3«,[' j^Jv''3x]Ux,\
Proof First by integrating by parts, we note that
d „„ „. ^ c |a«,T"' d-ui . (du.
,?,x
'«! .„T., = « L L \-d-.\ i ^ s i s n h ; - K
^ . ' " . „ , T . , - » / - . J ^ , | 3 „ | 3^,3, -2 32„ 3^11, 3uj Sfl I Sxj dl '
^ Spnnger
D Q. Khai. N. M. Tri
. i r S-Ui\Su,\1-- a-U, . 1 .X f 3^u, 3 /\Su,\1-'\ . /Bu,\
1 ^ r 3'u, | 3 „ , I''-' / 3 a , \
V [ ('''"• V|'""r'dt-
^ - ^ l y T ' (3.ri3j:jy \dxj\
Now by muhiplying the (-equation by Y.i=i l | ^ l ' ' " ^ | ^ . summing them all together and integrating over T^ using the formulas (26). (27), we obtain the desired resuh. D Lemma 6 Assume that q e [2, 3). u e Z.-(0, T: W^-1) n Z.~(0, 7: W^t), the futictioti {t I--* ll^"(0[l^,,(^3|l is absolutely continuous and almost everywhere satisfies the equality
^ t 4^ f bu, d'-u, I 3i,, I ' - ' , 4^ t Up 3^Ui I 3«, 1'-^
where / s L'»(0. 7; £,«(¥•')) anrf
l | V p ( f ) | | „ , p , < C||(a. V ) « ( , ) | | „ , T , , . (28) Ttoi there exist constants T', K. L depending only on ||V«(0)|l'' , q SUO„„,T
ll/(l)IU.„i|.>«*(/,ar ' - " " ' ) ™-'~' sup ||Vo(/)||' < K,
n
V(|V»(t)|}) d» < L Proof By the Sobolev inequalityIIV«(Ollr:
I||V»(/)|?|I
l < c | | v ( | V » ( r ) | f ) For a function g E t ' (T') by the Holder inequality, we have, fiu, |3a, 1'-^
jix)—^\--\ dx\
•.' dx^ \<IXj\
3'a, [ 3«/ I V
» ^ « ( ' ) l l j ; p , l l « l l t ' ( T > ) -
S Springer
On the Hausdorff Dimension of the Singular Set in Time for Weak Solutions
f d'u, Su, « " ' / f:—rlT^l ^x\ <
\hi dx^ 3xj
3'a, | 3 B ^
dx] I 'ix.
¥
W^iUmJns^WfmWLnT')
; l i v « ( . ) i i , l ^ -1- ^ -l-C||/(OII,^.
lyr^l < l|li(l)lli.,P)IIVa(I)llt.,TJ| < W^umJ.j,, l|Va(I)|| J , , ^ , ,
"Xl llz.«(T3)
< NV"(')ii3;T,|||v(iva(t)i?)j|^:^^
3a, a'lij | 3 a i | '
, ^ i | A ' " ' 3 * ; 3.tJ IsJtJ
l a ^ l|V(|Vu(()|?)||J, ,
< iiv»(.)ii,;-„',, + j - ^ ^ . (30)
From (28), we also have
l|Vp(r)||„,T., < l|v«(()ll,^T,| ||v(iva(i)l=!)|;;|^,j
« j t ! l l|V(|Va(/)|§)||2 S M
< l|Va(r)|| J„',, + ^ '-"'-fCl.nOlljg; (31)
Summing inequalities (29), (30), (31), using (27) and reducing similar terms, we obtain
I; (iiv„(,)ii'„„.,) + j ^ ^ < tviim^i + cii/o)ii »y,
< c ( | | V a ( / ) | | ' ^ , l P l + l ) * " ' (32) As in the proof of Lemma 3, by putting
yit) = |lVa(r)||'[,„j, -1-1
from (32), we can get Ihe conclusion of Lemma 6. • Theorem 4 Assume Ihal
/ e i ° ° ( 0 , r ; Z . ' ( T ' ) ) , ao £ H ' ' » ( T ' ) . qeH.i).
^ Springer
-.gj D Q. Khai, N M Tn Tiien there e-ust a constant T" depending only on ||VM(0)||^,^.J.JJ. q, supo<,^y ll/(Ollt?r-'l " " ' ^ " unique strong solution to the Navier-Stokes equation, satisfying
u e L-iO, r " : w~'0 nC([0, T**]: w^-").
Proof The proof of this theorem is similar to thai of Theorem 1 by using Lemma 6. We
omit the details. H Let q e [2,3) We say that a weak solution u is lV''^(T^)-regular on (f],(2) if
u e Ciiti.t2), W''^(T^)). We say that a W'''-regularity interval ili.tj) is maximal for (I if diere does not exist any interval of W''^-regulanty strictly containing (f|, 12). From Theorem 4, we can easily prove that if (/i, fi) is a VK''''-maximal interval of a solution ii, dien
limsup||H(f)||iyi,„C]r3, = + 0 0 . r 1-^/2-0
Theorem 5 Let q e [2, 3), UQ e H, f e Z.~(0, T; £,''(¥')}, Assume that u is a weak solution of the Navier-Slokes equation belonging lo L^ (0. T, W •''). Then u is W^-''-regular on an open set o/(0, T) whose complement has Lebesgue measure 0.
Theorem 6 Assume ihal q e [2, 3), MO & H, f e L'^(0, T; L''(T^}) and u is a weak solution of the Navier-Stokes equation and satisfies the following condition
ueL'i<d,T;W'-'<), '"^^^ ~ ^ ' < ] . 2q
Then there exists a closed set S,^ c [0, T] such that u e C([0, T] \ S^; W^-i) and U-, .12.-3) (5.) ^ 0 .
Proof By die hypothesis of the theorem )'(f)e Z,5(0, T), where >>(/) = ||Vw(f)||[,j.j,3j-|-l.
Now applying Lemma 4 with ^ — L a = j ^ , we get the desired result. D Remarks The condition '^'^^— < 1 in Theorem 6 is not essential since if ^ ^ ^ ^ ^ > 1 then H satisfies the Serrin condition, hence u is smooth if / is smooth. In this case poiSq) = 0.
This condition is assumed only in order that p r(2«-3} make sense.
Remark 6 Under the hypotheses of Theorem 6. the ^'"'-^~^'Vdimensional HausdoriT measure of E is equal to zero.
Remark 7 If r = 2. q ^ 2, then the conditions/ e Z.^(0. T: W^-'^) in Theorem 6 is redun- dant since it is assumed in the definition of weak solutions. In this case, we recover the early result of Scheffer and Leray (see [9. 12, 18J). When r < 2, g = 2, the condition u e L'"(0. T: W'--) is also redundant since it is weaker the condition u e /.^(O, T: Vj). In this case, the result in Theorem 6 is not the best possible. When q > ri2q - 3), our result is not new.
Remarks Fromtheproofsof Theorems 3 and 6, we see that the condition K e L''iO.T: Va) (with ff = 1) or M e L'"(0. T- IV''^) may be weakened by V« e L'"(0, T; H) or V i / e
^ S p m
On the Hausdorff Dimension of the Singular Set in Time for Weak Soluuons 295
L'iO, T; Ll), r e s p e c t i v e l y T h i s k i n d o f r e q u i r e m e n t s w a s s t u d i e d e a r l i e r in t h e n o t e [1] a n d n o w in n u m e r o u s p a p e r s ( s e e [6, 10] a n d t h e r e f e r e n c e s t h e r e i n ) .
Aeknowledgnients The authors express their thanks lo NAFOSTED for suppon.
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