Huang, X. and Matsumua, A. Osaka J. Math. 52 (215), 271 283 A CHARACTRIZATION ON BRAKDOWN OF SMOOTH SPHRICALLY SYMMTRIC SOLUTIONS OF TH ISNTROPIC SYSTM OF COMPRSSIBL NAVIR STOKS QUATIONS XIANGDI HUANG and AKITAKA MATSUMURA (Received May 13, 213, evised Septembe 27, 213) Abstact We study an initial bounday value poblem on a ball fo the isentopic system of compessible Navie Stokes equations, in paticula, a citeion of beakdown of the classical solution. Fo smooth initial data away fom vacuum, it is poved that the classical solution which is spheically symmetic loses its egulaity in a finite time if and only if the concentation of mass foms aound the cente in Lagangian coodinate system. In othe wods, in ule coodinate system, eithe the density concentates o vanishes aound the cente. Fo the latte case, one possible situation is that a vacuum ball appeas aound the cente and the density may concentate on the bounday of the vacuum ball simultaneously. 1. Intoduction and main esults We ae concened with the isentopic system of compessible Navie Stokes equations which eads as (1.1) t div(u), (U) t div(u Å U)Ö P U ()Ö(div U), whee t, x ¾ Ê N (N 2, 3), (t, x) and U U(t, x) ae the density and fluid velocity espectively, and P P() is the pessue given by a state equation (1.2) P() a with the adiabatic constant 1 and a positive constant a. The shea viscosity and the bulk one ae constants satisfying the physical hypothesis (1.3), N 2. 21 Mathematics Subject Classification. 35Q3, 76N1.
272 X. HUANG AND A. MATSUMURA The domain is a bounded ball with a adius R, namely, (1.4) B R {x ¾ Ê N Á x R ½}. We study an initial bounday value poblem fo (1.1) with the initial condition (1.5) (, U)(, x) (, U )(x), x ¾, and the bounday condition (1.6) U(t, x), t, x ¾, and we ae looking fo the smooth spheically symmetic solution (,U) of the poblem (1.1), (1.5), (1.6) which enjoys the fom (1.7) (t, x) (t, x), U(t, x) u(t, x) x x. Then, fo the initial data to be consistent with the fom (1.7), we assume the initial data (, U ) also takes the fom (1.8) (x), U u (x) x x. In this pape, we futhe assume the initial density is unifomly positive, that is, (1.9) (x), x ¾ fo a positive constant. Then it is noted that as long as the classical solution of (1.1), (1.5), (1.6) exists the density is positive, that is, the vacuum neve occus. It is also noted that since the assumption (1.7) implies (1.1) U(t, x)u(t, x), x ¾, we necessaily have U(t, ) (also U () ). Thee ae many esults about the existence of local and global stong solutions in time of the isentopic system of compessible Navie Stokes equations when the initial density is unifomly positive (efe to [1, 5, 6, 7, 13, 14, 15, 18, 19] and thei genealization [1, 11, 12, 17] to the full system including the consevation law of enegy). On the othe hand, fo the initial density allowing vacuum, the local well-posedness of stong solutions of the isentopic system was established by Kim [8]. Fo stong solutions with spatial symmeties, the authos in [9] poved the global existence of adially symmetic stong solutions of the isentopic system in an annula domain, even allowing vacuum initially. Howeve, it still emains open whethe thee exist global stong solutions which
BRAKDOWN CRITRION OF CLASSICAL SOLUTION 273 ae spheically symmetic in a ball. The main difficulties lie on the lack of estimates of the density and velocity nea the cente. In the case vacuum appeas, it is woth noting that Xin [2] established a blow-up esult which shows that if the initial density has a compact suppot, then any smooth solution to the Cauchy poblem of the full system of compessible Navie Stokes equations without heat conduction blows up in a finite time. The same blowup phenomenon occus also fo the isentopic system. Indeed, Zhang Fang ([21], Theoem 1.8) showed that if (, U) ¾ C 1 ([, T ]Á H k ) (k 3) is a spheically symmetic solution to the Cauchy poblem with the compact suppoted initial density, then the uppe limit of T must be finite. On the othe hand, it s unclea whethe the stong (classical) solutions lose thei egulaity in a finite time when the initial density is unifomly away fom vacuum. Theefoe, it is impotant to study the mechanism of possible blowup of smooth solutions, which is a main issue in this pape. In the spheical coodinates, the oiginal system (1.1) unde the assumption (1.7) takes the fom (1.11) t (u) (N 1) u, (u) t u 2 P() (N 1) u2 u (N 1) u whee 2. Now, we conside the following Lagangian tansfomation: (1.12) t t, y Then, it follows fom (1.1) that (t, s)s N 1 ds. (1.13) y t u N 1, t u, y ( N 1 ) 1, and the system (1.11) can be futhe educed to (1.14) t 2 ( N 1 u) y, 1 N u t p y (( N 1 u) y ) y whee t, y ¾ [, M ] and M is defined by (1.15) M R () N 1 d R accoding to the consevation of mass. Note that (1.16) (t, ), (t, M ) R. We denote by the initial enegy (1.17) R u 2 (t, ) N 1 d, a N 1 d, 2 1
274 X. HUANG AND A. MATSUMURA and define a cuboid Q t,y, fo t and y ¾ [, M ], as (1.18) Q t,y [, t] [y, M ]. Ou main esult is stated as follows. Theoem 1.1. Assume that the initial data (, U ) satisfy (1.8), (1.9) and (1.19) (, U ) ¾ H 3 (). Let (, U) be a classical spheically symmetic solution to the initial bounday value poblem (1.1), (1.5), (1.6) in [, T ], and T be the uppe limit of T, that is, the maximal time of existence of the classical solution. Then, if T ½, it holds the following. 1. In ule coodinate system (1.1), fo any ¾ (, R), (1.2) lim sup sup (t, x) 1 tt x (t, x) ½. 2. In Lagangian coodinate system (1.14), fo any y ¾ (, M ), (1.21) lim sup tt sup y¾[,y ] (t, y) ½. Moeove, fo system (1.14) in Lagangian coodinate, thee exists a constant C depending only on a,, N such that fo any given y ¾ (, M ) it holds (1.22) (t, y) C sup 1( 1) exp(c 1 H) inf M y exp CT sup ( 1) exp(c 1 H), (t, y) inf M y ¾ Q T,y, whee (1.23) H y (N 1) (N( 1)) M 12 (N N 2)(2N( 1)) y ( 1) ( 1) T. RMARK 1.1. The local existence of smooth solution with initial data as in Theoem 1.1 is classical and can be found, fo example, in [8] and efeences theein. So the maximal time T is well defined. RMARK 1.2. Thee ae seveal esults on the blowup citeion fo classical solutions to the system (1.1) (efe to [2, 3, 4, 16] and efeences theein). specially, the authos in [3] established the following Sein-type blowup citeion:
BRAKDOWN CRITRION OF CLASSICAL SOLUTION 275 When N 3, (1.24) lim T T ( L ½ (,T Á L ½ ) u L (,T Á L s )) ½, fo any ¾ [2, ½] and s ¾ (3, ½] satisfying (1.25) When N 2, 2 3 s 1. (1.26) lim T T L ½ (,T Á L ½ ) ½. RMARK 1.3. Theoem 1.1 assets that the fomation of singulaity is only due to the concentation of the mass aound the cente in Lagangian coodinate system. Moe pecisely, the mass anywhee away fom the cente is bounded up to the maximal time. On the hand, in ule coodinate system, eithe the density concentates o vanishes aound the cente. Fo the latte case, one possible situation is that a vacuum ball appeas aound the cente and the density may concentate on the bounday of the vacuum ball simultaneously. 2. Poof of Theoem 1.1 We only pove the case when N 3 since the case N 2 is even simple. Thoughout of this section, we assume that (, U) is a classical spheically symmetic solution with the fom (1.7) to the initial bounday value poblem (1.1), (1.5), (1.6) in [, T ], and the maximal time T, the uppe limit of T, is finite, and we denote by C geneic positive constants only depending on the initial data and the maximal time T. We fist have the following basic enegy estimate. Since the poof is standad, we omit it. (2.1) Lemma 2.1. It holds fo any t T, U2 o equivalently, (2.2) a 2 1 R u2 R a 2 1 u 2 dx a 2 1 t t N 1 d ÖU 2 dx d R N 1 d. U 2 a dx, 2 1 u 2 u2 N 1 d 2 d
276 X. HUANG AND A. MATSUMURA Next, using the basic enegy estimate above, we can efine the blowup citeion (1.24) in the pesent case of spheically symmetic solutions as follows. Lemma 2.2. (2.3) lim T T L ½ (,T Á L ½ ) ½. Poof. Due to the Sein type condition (1.24), it suffices to show that (2.4) sup t¾[,t ) implies (t, ) L ½ ½ (2.5) lim T T u L 4 (,T Á L 12 ) ½. To do that, making use of the identity (2.6) U Ö div U u (N 1) u we ewite the equation of momentum as (2.7) (U) t div(u Å U)Ö P U, 2. Multiplying the equation (2.7) by 4U 2 U and integating it ove, we have (2.8) d dt U 4 dx (ÖU 2 ) 2 dx C C x, PU ÖU 2 dx U 4 dx C ÖU 2 dx whee we used the assumption (2.4). Then, it easily follows fom the Gonwall s inequality and Lemma 2.1 that (2.9) t (ÖU 2 ) 2 dx d C, t ¾ [, T ), which implies the desied estimate (2.5) by Sobolev s embedding theoem H 1 () L 6 (). Thus, the poof of Lemma 2.2 is completed. Due to the efined citeion above, to pove Theoem 1.1, it emains to show that the density away fom the cente stays bounded up to the maximal time T. To do that, we pepae the next lemma which gives a elationship between and y.
BRAKDOWN CRITRION OF CLASSICAL SOLUTION 277 Lemma 2.3. such that Thee exists a positive constant C depending only on a,, N (2.1) (t, y) Cy (N( 1)) and 1(N( 1)) 1( 1) (2.11) R N (t, y) N C(M y) ( 1), (t, y) ¾ [, T ] [, M ]. (2.12) Poof. By the enegy inequality (2.2), we have y s N 1 ds s N 1 ds C N(1 1 ) 1, s (N 1) s N 1 (N 1) ds 1 s N 1 ds 1 1 which implies (2.13) (t, y) Cy (N( 1)) Similaly, we have 1(N( 1)). (2.14) M y R s N 1 ds R R s N 1 ds 1 R C(R N (t, y) N ) 1 1 1, s (N 1) s N 1 (N 1) ds s N 1 ds 1 1 which implies (2.15) R N (t, y) N C(M y) ( 1) Thus, the poof of Lemma 2.3 is completed. 1( 1). We ae now in a position to establish the pointwise estimates of the density away fom the cente. Lemma 2.4. Fo any given y ¾ (, M ), thee exists a constant C exactly as in Theoem 1.1 such that (2.16) (t, y) C, (t, y) ¾ [, T ] [y, M ].
278 X. HUANG AND A. MATSUMURA (2.17) Poof. In view of (1.14), it holds (log ) ty t y (( N 1 u) y ) y 1 N u t p y ( 1 N u) t p y (N 1) u2 N. Thus, fo y y, integating (2.17) ove (, t) (y, y), we deduce that (2.18) log (t, y) (t, y ) log (y) y (y ) t y (( 1 N u)(, z) ( 1 N u)(t, z)) dz (p(s, y ) p(s, y)) ds t y (N 1) u2 (s, z) y N dz ds, which is equivalent to (2.19) (t, y) y (y) (t, y ) (y ) exp 1 (( 1 N u)(, z) ( 1 N u)(t, z)) dz y t exp 1 (p(s, y ) p(s, y)) ds t y exp 1 (N 1) u2 (s, z) dz ds. y N We can ewite (2.19) as (2.2) (t, y) P(t)U(t, y) exp 1 t p(s, y) ds whee (2.21) P(t) (t, y ) (y ) exp 1 t p(s, y ) ds and (2.22) y U(t, y) (y) exp 1 exp (( 1 N u)(, z) ( 1 N u)(t, z)) dz y t y 1 (N 1) u2 (s, z) dz ds. y N
BRAKDOWN CRITRION OF CLASSICAL SOLUTION 279 Hence, it follows fom the equation of mass, the enegy inequality (2.2) and Lemma 2.3 that (2.23) M y 1 N u dy R (y ) R u d C 1 N (y ) u N 1 d (y ) R C 1 N (y ) N 1 d C 1 N (y )M 12 12 (N 1) (N( 1)) Cy 12 R u 2 N 1 d M 12 (N N 2)(2N( 1)) 12 and (2.24) T M Consequently, we have u 1 N 2 y dy dt T R (y ) C N (y ) u 2 T R d dt u 2 N 1 d dt C N ( 1) ( 1) (y ) T Cy T. (2.25) U(t, y) C sup x¾ (x) exp { C 1 y (N 1) (N( 1)) M 12 (N N 2)(2N( 1)) } and (2.26) U(t, y) 1 C inf (x) x¾ 1 exp { C 1 y (N 1) (N( 1)) M 12 (N N 2)(2N( 1)) C 1 y ( 1) ( 1) T }, (t, y) ¾ Q T,y. If we set (2.27) H y (N 1) (N( 1)) M 12 (N N 2)(2N( 1)) y the above estimates fo U(t, y) can be simply witten as (2.28) U(t, y) C U 1 (t, y) C sup x¾ (x) inf (x) x¾ exp(c 1 H), 1 exp(c 1 H). ( 1) ( 1) T,
28 X. HUANG AND A. MATSUMURA On the othe hand, it follows fom (2.2) that d t (2.29) dt exp p(s, y) ds a (t, y) exp a (P(t)U(t, y)), t p(s, y) ds which implies 1 (2.3) exp t p(s, y) ds 1 a t (P(s)U(s, y)) ds 1. Next, we ae in a position to estimate P(t). Fist, obseve that (2.31) M y dy R (t, y) (y ) N 1 d R N (y ) N C(M y ) ( 1) N 1( 1). In view of (2.2) and (2.3), we have (2.32) (t, y) P(t)U(t, y) 1(a ) Ê t (P(s)U(s, y)) ds 1. Then, P(t) can be estimated as (2.33) R N N (y ) P(t) M N y M C y M 1 y a C P(t) (t, y) dy 1(a ) Ê t (P(s)U(s, y)) ds 1 U(t, y) dy C(M y ) C(M y ) 1 sup U(t, y) Q T,y U(t, y) sup U 1 (t, y) Q T,y sup Q T,y U(t, y) dy sup Q T,y U 1 (t, y) sup U 1 (t, y) Q T,y M t y t P(s) ds 1dy P(s) ds 1.
BRAKDOWN CRITRION OF CLASSICAL SOLUTION 281 Using (2.31) and taking -th powe on both sides of (2.33), we have (2.34) M y ( 1) P(t) C sup U 1 (t, y) Q T,y C sup U 1 (t, y) sup U(t, y) Q T,y Q T,y Theefoe, by Gonwall s inequality, we deduce fom (2.34) that P(t) C 1( 1) sup U 1 (t, y) M y Q T,y (2.35) exp CT sup U 1 (t, y) sup U(t, y) Q T,y Q T,y Finally, ecalling (2.32), we have (2.36) (t, y) P(t) sup U(t, y) Q T,y, t. P(s) ds. and plugging the estimates (2.25), (2.26) and (2.35) into (2.36), we can deduce the desied pointwise estimate (2.16). Thus the poof of Lemma 2.4 is completed. 3. Poof of Theoem 1.1 Note that (1.21) and (1.22) ae diect consequences of Lemma 2.2 and Lemma 2.4. Now we ae in a position to pove (1.2). We ague by contadiction. Suppose (1.2) fails to hold. In ule coodinate system (1.11), thee exist positive constants C, ( T ) and 1 ( R) such that (3.1) C 1 (t, ) C, fo all (t, ) ¾ [T, T ) [, 1 ]. Then we claim that in Lagangian coodinate system (1.14), thee exist a positive constant y 1 ( M ) such that (3.2) (t, y) C fo all (t, y) ¾ [T, T ) [, y 1 ]. In fact, by vitue of (3.1), it holds (3.3) y(t, 1 ) 1 (t, ) 2 d 1 C 1 2 d 1 3 C 1 1 3, t ¾ [T, T ) which immediately implies (3.2). Now, it follows fom (3.2) and Lemma 2.4 that the density is bounded on [, T ), which contadicts to the blowup citeion Lemma 2.2. Thus the poof of Theoem 1.1 is completed.
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BRAKDOWN CRITRION OF CLASSICAL SOLUTION 283 Xiangdi Huang NCMIS, AMSS Chinese Academy of Sciences Beijing 119 P.R. China and Depatment of Pue and Applied Mathematics Gaduate School of Infomation Science and Technology Osaka Univesity Toyonaka, Osaka 56-43 Japan e-mail: xdhuang@amss.ac.cn Akitaka Matsumua Depatment of Pue and Applied Mathematics Gaduate School of Infomation Science and Technology Osaka Univesity Toyonaka, Osaka 56-43 Japan e-mail: akitaka@math.sci.osaka-u.ac.jp