THE BLOWUP OF SOLUTIONS FOR 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS

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1 H. Yin Q. Qiu Nagoya Math. J. Vol. 154 (1999, THE BLOWUP OF SOLUTIONS FOR 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS HUICHENG YIN QINGJIU QIU Abstact. In this pape, fo thee dimensional compessible Eule equations with small petubed initial data which ae axisymmetic, we pove that the classical solutions have to blow up in finite time give a complete asymptotic expansion of lifespan. 1. Intoduction Thee ae many esults on the lifespan of classical solutions fo nonlinea wave equations with small initial data ([1, [2 etc.. Howeve, few papes have teated the poblem of the lifespan of solutions fo highe dimensional compessible Eule equations. In fact, it is vey difficult to detemine whethe the smooth solutions of compessible Eule equations blow up o not. In [3, T. Sideis gave an uppe bound fo the lifespan of thee dimensional Eule equations unde appopiate conditions. In the pape, we will discuss 3-D poblem with spheically symmetic initial data. Geneally speaking, the spheically symmetic initial data don t satisfy the conditions in [3, so we can t give the uppe bound of lifespan in light of the esult in [3. Fo the 2-D isentopic Eule equations with otationally invaiant data which ae a petubation of size ε of a est state, S. Alinhac [4 has established the lifespan of solution. Fo the geneal iotational initial data (not spheically symmetic which is a small petubation, we can also detemine the lifespan of classical solutions. This will be given in anothe pape. Conside the following initial data poblem fo 3-D compessible Eule Received Mach 26,

2 158 H. YIN AND Q. QIU equations: (1 t div(v = 0 t v (v v = p( = c2 ( t=0 = ε 0 (x, v t=0 = εv 0 (x whee > 0 is a constant, c 2 ( = dp/d > 0, p( C fo > 0, ε > 0 sufficiently small, v = (v 1, v 2, v 3, x = (x 1, x 2, x 3, 0 (x, v 0 (x C (R 3 have compact suppots in x R 0. Moeove we assume that v 0 (x = v 0 1 (xx, whee v0 1 is a smooth function in R3, v 0 1 (x, 0(x depend only on, = x 2 1 x2 2 x2 3. Main Theoem. fo 0 t < T ε, whee Unde the above assumptions, (1 has a C solution lim ε ln T ε = τ 0 ε 0 2 c = [ ( c ( c min q 2 q v1(q 0 3qv1(q 0 c (q q 0 (q 0 (q q R 0 c = c(, T ε denotes the lifespan of smooth solution. Remak. It is easy to know min q R0 [q 2 q v1 0(q3qv0 1 (q c(q q 0 (q 0 (q/ < 0 unless v1 0 0, 0 0. Moeove c ( c > 0 is known (see [5. Fo poving Theoem, we note that the otations of v ae zeo, then (1 will be educed into a nonlinea wave equation. As in [2, by constucting an appoximate solution consideing the diffeence of exact solution appoximate solution, we easily get the lowe bound of lifespan. On the othe h, the spheical symmeticity of solution makes us to change (1 into a 2 2 system equation in two vaiables (, t. Hence by using the popeties of above appoximate solution imitating the poof in [4, [6, [7, we may obtain the estimate of uppe bound fo the lifespan. Theoem assets that the solution of (1 blows up in finite time unless v 0 0, 0 0 in spite of any small ε. 2. The lowe bound of lifespan T ε Unde the assumptions of Theoem, we know the solution of (1 has such a fom in t < T ε : (x, t = (, t, v(x, t = ṽ(, tx, whee ṽ(, t

3 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 159 is a smooth function of (, t. Because of ot v(x, t = 0, thee exists a function ω(x, t C, such that v(x, t = ω(x, t, moeove ω(x, t has compact suppot in x (by the finite popagation speed we know v has compact suppot, depends only on. If we denote ω(, t = ω(x, t, then ω(, t = ṽ(, t. Substituing v(x, t = ω into the second equation in (1, then we have: t ω ( 1 2 ω 2 = h( whee h ( = c 2 (/, h( = 0. Hence t ω ω 2 /2 = h(. Noting h ( > 0, by implicit function theoem, we know that = h 1( ( t ω 1 2 ω 2, = h 1 (0 Theefoe, fom the fist eqution in (1, it tuns out Set 2 t ω 2 3 k=1 k ω t k ω h 1 ( ( t ω 1 2 ω 2 (h 1 ( ( t ω 1 2 ω 2 ω 3 i ω k ω i k ω = 0. i,k=1 Now we detemine the initial data ω t=0 t ω t=0. Obviously, ω t=0 = ε R 0 sv1 0 (s ds. Since t ω t=0 ( ω 2 t=0 /2 = c 2 ( t=0 /, then we have g(x, ε = t ω t=0 1 2 ε2 [ 2 (v 0 1( 2 = ε c2 0 ε 2 1 R 0 [ (c 2 ( = θε0 dθ 0 0. (c 2 ( = θε0(s dθ 0 (s s 0 (s ds (v1 0 (2, then g(x, ε is smooth in x, ε has compact suppot in R 0. So t ω t=0 = ε c 2 0 / ε 2 g(x, ε.

4 160 H. YIN AND Q. QIU Then fo consideing the lowe bound of lifespan fo (1, we only discuss the lowe bound of lifespan fo the following poblem: (2 2 t ω 2 ω t=0 = ε 3 k=1 k ω t k ω h 1 ( ( t ω 1 2 ω 2 (h 1 ( ( t ω 1 2 ω 2 ω 3 i ω k ω i k ω = 0 i,k=1 sv1 0 (s ds R 0 t ω t=0 = ε c2 0 ε 2 g(x, ε. It is easy to know h 1 ( ( tω ω 2 /2 (h 1 ( ( tω ω 2 /2 = c2 2 c ( t ω/ c O( x,t ω 2. we set t 2ω 0 c 2 ω 0 = 0 (3 ω 0 t=0 = sv1 0 (s ds R 0 t ω 0 t=0 = c2 0 ( U(s, q = c ( c U(s, q s 2 c 2 q U(0, q = F 0 (q. Whee F 0 (q is the Fiedle adiation field of (3. By [2, (6.2.12, we have F 0 (q = 1 [ c d ( q q sv q 2 dq 1(s 0 ds q c2 0(q R 0 = c ( q sv 2 1(s 0 ds q 2 v1(q 0 c q 0(q. R 0 Set ω a = ε[χ(εtω 0 (1 χ(εt 1 U(ε ln(εt, ct, whee χ C (R deceases, equals to 1 in (, 1, equals to 0 in (2,. Then we have the following conclusion: 2 Lemma 1. Fo sufficiently small ε > 0, ε ln T < b < τ 0, then

5 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 161 (i (2 has a C solution fo 0 < t T. (ii β x,t (ω ω a C β,bε 2 ln(1/ε, β 2. 1 t (iii β x,t x,tω a C β,bε, β 0. 1 t Poof. (i Noting 2 F 0 (q/ q 2 = c[q 2 q v 0 1 (q 3qv0 1 (q cq q 0 (q/ c 0 (q/ /2, as in [2, Theoem 6.5.7, we know (i holds. (ii By [2, Lemma S.Klaineman s inequality in [8, we easily know (ii holds. (iii Its poof is simila to that in [2, Lemma Fom Lemma 1 (i, it is easy to know that lim ε 0 ε ln T ε τ 0. Remak. We state that min q M [q 2 q v 0 1 (q3qv0 1 (q cq q 0 (q/ < 0 unless v In fact, if min q M [q 2 q v 0 1 (q 3qv0 1 (q cq q 0 (q/ 0, then 2 F 0 (q/ q 2 0. Because F 0 (q has compact suppot in q M, then F 0 (q/ q 0, hence F 0 (q 0. By [2, Theoem 6.2.2, we have v 0 1 0, The uppe bound of lifespan T ε Assume that ω a, b ae defined as above, when ε ln T b, we set a = h 1 ( ( t ω a ω a 2 /2. Since a = 1 0 (h 1 ( θ ( t u 1 2 u 2 (1 θ ( t u a 1 2 u a 2 dθ [ t (u u a 1 2 ( u u a (u u a then by lemma 1, then we obtain (4 β x,t ( a C β,bε 2 ln(1/ε, β 2 1 t (5 β x,t ( a C β,bε 1 t, β 0 Set α(, t = ω(, t, α a (, t = ω a (, t. Fo the smooth solution of (1, we have

6 162 H. YIN AND Q. QIU (6 Lemma 2. (a α, satisfy the following system: t α ( α 2 α = 0 t α α α c2 ( = 0 (b Set = A(, t/, α = B(, t/, Z 1 = ( A/ B/c/2, Z 2 = ( A/ B/c/2, then (Z 1, Z 2 satisfy the following system: { t Z 1 (α c Z 1 = Q (7 1 t Z 2 (α c Z 2 = Q 2 whee Q 1 = Z2 1 (c c Z 1Z 2 ( 3c B Z [ 1 2 3B ca c A 2 c B ( c 2 ( 2c Z 2 2 [ B ca ( c 2 ( A c c B 2c 1 ( 2 ca 2 B B 2 ABc 2 2 4, Q 2 = Z 1Z 2 ( 3c 2A B Z [ 1 ( c 2 ( 2 Z 2 2 [ 1 ( 2 ca 2 B B 2 ABc Z 2 ( 2 B c 1 [ B2 3 c ( c 2 ( A c B 2 AB 2 c 2 ( ca c A 2 B 2 cb 2 c 2 c 2 A 2 Z 2 ( 2 B 2A c 2c 3AB 2 cb 2 2 A c c 2 c B c A 2c 2 B 2 2 Bc 2 2B c 2 ( c 2 ( A c ca c B c A 2c 2 B 2 c 2 Bc 2 1 [ B2 3 c ( c 2 ( A 2 2c 3AB 2 Bc 2 2 Remak. It is vey impotant to appea the facto c( /2 in the coefficients of Z 1 / 2 in Q 1, o we can t give the uppe bound of lifespan T ε of solution to (1.

7 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 163 Poof. (a It can be veified diectly, we omit it. (b Since Z 1 Z 2 = B/c = (α α/c, Z 1 Z 2 = ( /, thus α = c(z 1 Z 2 2 α = c (Z 1 Z 2 2 = (Z 1 Z 2 then (8 (9 B 2, = (Z 1 Z 2 A 2, c (Z1 2 Z2 2 2 Ac (Z 1 Z 2 3 2c(Z 1 Z 2 2 2B 3, (Z 1 Z A(Z 1 Z 2 3 2(Z 1 Z 2 2 2A 3. t (Z 1 Z 2 = B (Z 1 Z 2 c (Z 1 Z 2 c(z 1 Z 2 c (Z 1 Z 2 2 B 2 (Z 1 Z 2 2B2 c 3 2B(Z 1 Z 2 2 ( c 2 ( [ 2 (Z 1 Z 2 2 2A(Z 1 Z 2 c c 2 A2 c 3 c(z 1 Z 2 2 c (Z 1 Z 2 2 ca(z 1 Z 2 2 2c(Z 1 Z 2 ca 2 c B(Z 1 Z 2 c 2 t (Z 1 Z 2 = B (Z 1 Z 2 c (Z 1 Z 2 2c (Z2 1 Z2 2 2cA(Z 1 Z 2 2 3AB 3 B(Z 1 Z B(Z 1 Z 2 2 c (Z2 1 Z2 2 c A(Z 1 Z 2 2 2B 2 B (Z 1 Z 2 2 Bc 2 3 c(z 1 Z 2 B 2 B(z2 1 Z2 2 2 Bc(Z 1 Z 2 3 c(z 1 Z 2 2 B(Z 1 Z 2 2 ABc 2 4 Ac(Z 1 Z 2 2. Fom (8 (9, it is easy to obtain (7. Hence Lemma 2 is poved. Denote Γ ± λ as the integal cuves of t ± (α c passed though (λ, 0 in the plan (, t, D as the stip domain bounded by Γ R 0 Γ σ 0 1 whee

8 164 H. YIN AND Q. QIU σ 0 is taken such that (q 2 q v1 0(q 3qv0 1 (q cq q 0 (q/ c 0 (q q=σ0 / is minimum. D ε = D {t 1/}. As in [4, [6, [7, fo any T satisfying 1/ T e b/ε, b < τ 0, we define J(t = V (t = sup 1/ s t (,s D Z 1 (, s d, M(t = sup sup s Z 2 (, s. 1/ s t (,s D 1/ s t (,s D ( A(, s B(, s, Lemma 3. Thee exist some constants J 1, M 1, V 1 > 0, fo sufficiently small ε > 0 any t satisfying 1/ t T e b/ε, such that J(t J 1 ε, M(t M 1 ε, V (t V 1 ε 1/2. Moeove ct/2 in D ε. Since Poof. Fistly we veify the lemma fo 1/ t 1/ε. Set = a. It tuns out Noting that 2Z 1 = ( a 2Z 2 = ( a t 2Z 2 ( α a c t ( cte ( ε ε ln 1 ε. a α α a c α ( a α c c α c ( a α. c ( a α c α a c a = ( α a c (c c a c α a c a c α a c a = c ε( c ( t ωa F 0( ct [ 2F0 ( ct ε 3 qf 0 ( ct 2 O( x,t ω a 2 O( ω a ω a.

9 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 165 Thus, by β ( x,t ωa F 0( ct C β (1 t 2, β 0, ( c 2 t we know i.e., t Z 2 t α a c a cte ( ε ε ln 1 cteε 1/2 ε V (t cteε 1/2. On the othe h, owing to the width of D is finite Z 1 cteε, then J(t cteε, M(t = sup ( α ( cteε. 1/ s t (,s D We choose M 1 = (2 c 2 2 c J 1, V 1 = J 1, s.t. J(1/ J 1 ε/4, M(1/ M 1 ε/4 V (1/ J 1 ε 1/2 /4. Now we veify the lemma as 1/ t T T. Fo this aim, as in [4, we fist claim that: (i On Γ λ D, then ct λ cte. In paticula, fo t 1/ ε small enough, then 3 ct/4. In fact, along Γ λ D, d cdt = (α c c dt, then (10 ct λ cte cte t 0 t 0 α c c dt ε dt cteε ln(1 t cte. 1 t (ii If (, t Γ µ D ε, (, t Γ µ D ε, t t, then t t cte. In paticula, t t/2 fo sufficiently small ε. In fact, d dt ( ct µ cteε 1 t (t ct µ ((t ct µ cteε ln 1 t 1 t cte. Then imitating the poof in [4, we know the statement (ii holds.

10 166 H. YIN AND Q. QIU Choosing µ = α c, along the integtal cuve in D, we wite d(z 1 (d µdt = [(α c µ Z 1 Z 1 µ Q 1 d dt = Qd dt whee Q = Z 1Z 2 ( c 2c B Z 2 2 Z [ 1 2 2B ca ca 2 c B 2c B ca ( c 2 ( A Z 2 2 [ 1 ( ca 2 2 B B 2 cab ( B c ( c 2 ( c c B 2c 1 [ 3 B2 c ( c 2 ( As in [4, we have (i Z 1 (, t d J ( 1 (,t D (ii cte Z 1 (, t d J ( 1 Γ (x,t D ε A c B 2 AB 2 c 2 ( ca ca 2 B 2 Bc 2 c 2 c 2 1/ s t (,s D A 2 2c 3AB 2 Bc 2 2 Q dds 1/ s t (,s D Q dds whee Γ (x, t is the integal cuve of t (α c passed though (x, t D ε. In D ε, fo sufficiently small ε, we have Q cte [ Z 1 ( V 1 ε 1/2 Hence 1/ s t (,s D 1 ε t 3 V 2 t 2 ( 1 M 1 ε t M 1ε t 2 M 1 2 t 3 M 1V 1 ε 3/2 t 3 ε t 2 V 1 ε 1/2 ( M1 t 3 ε t M 1ε t 2 M 2 1 ε2 t 3 M 2 1 ε2 t 4. Q dds ctej 1 ε 3/2, J(t J ( 1 Q dds 1 1/ s t 2 J 1ε. (,s D In ode to estimate M(t, we note that t A (α c A = 2cZ 1 AB 2 B t B (α c B = 2c 2 Z 1 B2 2 c2 ( A.

11 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 167 Then, fo ε small enough, we have ( A B (, t 3 4 M 1ε. Similaly, V (t V 1 ε 1/2 /2. Theefoe, by continuous induction, we know that Lemma 3 holds. Along Γ σ 0 D, we define w(t = Z 1 ((t, t. It is easy to know that w satisfies the equation: (11 w (t = a 0 (tw 2 a 1 (tw a 2 (t whee a 0 (t = ( c c ((t, t, a 1 (t = Z 2 ( 3c B a 2 (t = Z 2 B 2 AB 2 c ( c B 1 [ 2 3B ca c A 2 c B 2c 2 (, Z 2 [ 2 B ca B 2 Bc 2 c 2 c 2 1 [ 3 B2 c ( c 2 ( A 2 ( c 2 ( A c (c 2 ( A c cb 2c ca c A 2 1 ( ca 2 2 B B 2 2c 3AB 2 Bc 2 2 cab Lemma 3 the above calculation give out the following estimates a 0 (t = c ( c ct O(t 2, a 1 (t cte ε1/2 t 2, a 2(t cte ε1/2 t 2 (12 w ( 1 1 [ (a (( 1 =, 1 ( 1 a (( 1, 1 2 a (( 1, 1 α a(( 1, 1 ( 1 α a (( 1, 1 c( a (( 1, 1 o(ε.

12 168 H. YIN AND Q. QIU As ε is small enough, fom the poof in Lemma 3, we know (1/ ( c/2(1/. So it tuns out: β x,t ( ω0 F 0( ct =(1/,t=1/ cte (1 1 cteε2, β 0 2 ( α a ( 1 1 [ q F 0 (( 1, = ε c ( 1 F 0(( 1 c 2 ( 1 a = ε c o(ε, q F 0 (( 1 c ( 1 ε c ( 1 [ 2 q F 0 (( 1 c ( 1 qf 0 (( 1 c 2 ( 1 o(ε. Substituting them into (12, we obtain w ( 1 = ε 2 w ( 1 ε q 2F 0(( 1 = c o(ε. c By (10, (1/ c/ σ 0 cteε ln(1 1/, then [ σ0 2 q v1(σ 0 0 3σ 0 v1(σ 0 0 c (σ 0 q 0 (σ 0 0 (σ 0 o(ε Denote K = ( T 1/ a 2(t dt exp( T 1/ a 1(t dt, note the equation (11 satisfies the conditions of [2, Lemma in 1/ t T, so we obtain: that is ( T 1/ ( a 0 (t dt exp T 1/ a 1 (t dt < ( w ( 1 1 K c ( c (ln T ln O(εe cteε3/2 c [ < ε ( σ q v1(σ 0 0 3σ 0 v1(σ 0 0 c σ 0 q 0 (σ 0 c 0(σ 0 o(ε cteε 3/2 1. Let ε 0, we get (13 lim ε ln T ε τ 0. ε 0 Combine (13 with the conclusion in 2, we know Theoem holds.

13 3-D AXISYMMETRIC COMPRESSIBLE EULER EQUATIONS 169 Refeences [1 S. Klaineman, Unifom decay estimate the Loentz invaiance of the classical wave equation, Comm. Pue Appl. math., 38 (1985, [2 L. Höme, Nonlinea hypebolic diffeential equations, Lectues, [3 T. Sideis, Fomation of solution singulaities in thee dimensional compessible fluids, Comm. Math. Phys., 101 (1985, [4 S. Alinhac, Temps de vie des solutions eguliees de equations d Eule compessibles axisymetiques en dimendion deux, Invent. Math., 111 (1993, [5 R. Couant, K. O. Fiedichs, Supesonic flow shock waves, Wiley Intescience, New Yok, [6 L. Höme, The lifespan of classical solutions of nonlinea hypebolic equations, Mittag-Leffle epot No. 5 (1985. [7 F. John, Blow-up of adial solutions of u tt = c 2 (u t u in thee space dimensions, Mat. Apl. Comput., 4 (1985, No. 1, [8 S. Klaineman, Remaks on the global Sobelev inequalities in the Minkowski space R n1, Comm. Pue Appl. Math., 40 (1987, Huicheng Yin Depatment of Mathematics Nanjing Univesity Nanjing P. R. China Qingjiu Qiu Depatment of Mathematics Nanjing Univesity Nanjing P. R. China