On the global stability of contact discontinuity for compressible Navier-Stokes equations

On the Global Stability of Contact Discontinuity for Compressible

Navier-Stokes Equations.

FEIMIN HUANG(*) - HUIJIANG ZHAO(**)

ABSTRACT- The asymptotic behavior of the solutions toward the contact disconti- nuity for the one-dimensional compressible Navier-Stokes equations with a free boundary is investigated. It is shown that the viscous contact discontinu- ity introduced in [3] is asymptotic stable with arbitrarilylargeinitial pertur- bation if the adiabatic exponentgis near 1. The case the asymptotic state is given by a combination of viscous contact discontinuity and the rarefaction wave is further investigated. Both the strength of rarefaction wave and the in- itial perturbation can be arbitrarily large.

1. Introduction.

W e s t u d y i n p r e s e n t p a p e r t h e l a r g e t i m e b e h a v i o r o f t h e s o l u t i o n s f o r t h e o n e - d i m e n s i o n a l c o m p r e s s i b l e N a v i e r - S t o k e s ( N S ) e q u a t i o n s . I t i s k n o w n t h a t t h e r e h a v e b e e n a lo t o f w o r k s o n t h i s s u b j e c t . M o s t o f t h e s e r e s u l t s a r e c o n c e r n e d w i t h t h e r a r e f a c t i o n w a v e a n d v i s c o u s s h o c k w a v e . W e r e f e r t o [ 4 , 6 - 9 , 1 1 - 1 5 , 1 7 - 1 9 ] a n d r e f e r e n c e s t h e r e i n . I t i s n o t e d t h a t N i s h i h a r a , Y a n g a n d Z h a o r e c e n t l y e s t a b l i s h e d i n [ 1 7 , (*) Indirizzo dell’A.: Institute of Applied Mathematics, AMSS, Academia Si- nica, Beijing 100080, China. E-mail: fhuangHmail.amt.ac.cn

(**) Indirizzi dell’A.: Laboratory of Mathematical Physics, Wuhan Institute of Physics and Mathematics, Academia Sinica, Wuhan 430071, China. E-mail:

hhjjzhaoHhotmail.com

School of Political Science and Economics and Graduate School of Science and Engineering, Waseda University, Tokyo 169-8050, Japan.

1 8 ] t h e g l o b a l s t a b i l i t y o f s t r o n g r a r e f a c t i o n w a v e f o r t h e 333 eq u a - t i o n s . H o w e v e r f e w r e s u l t i s k n o w n f o r t h e c o n t a c t d i s c o n t i n u i t y e x - c e p t [ 3 ] d u e t o v a r i o u s d i f f i c u l t i e s . A s a n e l e m e n t a r y h y p e r b o l i c w a v e , t h e a s y m p t o t i c s t a b i l i t y o f c o n t a c t d i s c o n t i n u i t y s h o u l d b e i n v e s t i g a t - e d . L i u a n d X i n [ 1 0 , 2 0 ] f i r s t s t u d i e d i n 1 9 9 5 t h e n o n l i n e a r s t a b i l i t y o f c o n t a c t d i s c o n t i n u i t y f o r a n a r t i f i c i a l v i s c o s i t y s y s t e m . I t w a s s h o w n i n [ 1 0 , 2 0 ] t h a t t h e c o n t a c t d i s c o n t i n u i t y c a n n o t b e t h e a s y m p t o t i c s t a t e a n d a li n e a r d i f f u s i v e w a v e c a l l e d v i s c o u s c o n t a c t w a v e ( o r v i s c o u s c o n t a c t d i s c o n t i n u i t y ) , w h i c h a p p r o x i m a t e s t h e c o n t a c t d i s c o n t i n u i t y o n a n y f i n i t e t i m e i n t e r v a l , i n s t e a d d o m i n a t e s t h e l a r g e t i m e b e h a v i o r o f t h e s o l u t i o n s . M o t i v a t e d b y [ 1 0 , 2 0 ] , H u a n g , M a t s u m u r a a n d S h i [ 3 ] i n v e s t i g a t e d t h e c o n t a c t d i s c o n t i n u i t y c a s e f o r a ph y s i c a l s y s t e m – c o m - p r e s s i b l e N S e q u a t i o n s , w i t h a fr e e b o u n d a r y . U n l i k e t h e a r t i f i c i a l v i s - c o u s s y s t e m , t h e c o n t a c t d i s c o n t i n u i t y f o r c o m p r e s s i b l e N S e q u a t i o n s i s a p p r o x i m a t e d b y a no n l i n e a r d i f f u s i v e w a v e . T h e n o n l i n e a r s t a b i l i t y o f t h e v i s c o u s c o n t a c t w a v e w a s e s t a b l i s h e d i f t h e i n i t i a l p e r t u r b a t i o n i s s m a l l ( s e e [ 3 ] ) . T h i s m e a n s t h e n o n l i n e a r s t a b i l i t y o f [ 3 ] i sl o c a l. It is w o r t h y t o p o i n t o u t t h a t t h e s t a b i l i t y o f [ 1 0 , 2 0 ] i s a l s ol o c a l. Th u s , a n a t u r a l q u e s t i o n i s w h e t h e r t h e v i s c o u s c o n t a c t w a v e c o n s t r u c t e d i n [ 3 ] i s s t i l l s t a b l e o r n o t u n d e r a r b i t r a r i l yl a r g e p e r t u r b a t i o n . O u r m a i n p u r p o s e o f t h i s p a p e r i s t o g i v e a po s i t i v e a n s w e r t o t h i s q u e s t i o n w h e n t h e a d i a b a t i c e x p o n e n t g i s n e a r 1 . F u r t h e r m o r e , t h e c a s e t h e a s y m p t o t i c s t a t e i s g i v e n b y a su p e r p o s i t i o n o f t h e v i s c o u s c o n t a c t d i s - c o n t i n u i t y a n d t h e r a r e f a c t i o n w a v e i s a l s o t r e a t e d . I n t h i s s i t u a t i o n , b o t h t h e s t r e n g t h o f r a r e f a c t i o n w a v e a n d t h e i n i t i a l p e r t u r b a t i o n c a n b e a r b i t r a r i l y l a r g e.

We now formulate our main results. The 1-dcompressible NS equa- tions reads in Lagrangian coordinates:

. ` / `

´

vt2ux40 , u_t1p_x4m

g

^uv^x

h

_x^,

g

^e1u2²

h

_t^1(pu)x4

g

^kuv^x 1muu_x v

h

x

, (1.1)

whereu(x,t) is the velocity, v(x,t)D0 the specific volume, u(x,t) the absolute temperature, mD0 the viscosity constant and kD0 the coeffi-

cient of heat conduction. Here we consider the perfect gas so that the pressure

p4 Ru

v 4Av^2ge

g21 R s

, (1.2)

and the internal energye4_g^R

21u1const., where s is the entropy, gD1 is the adiabatic exponent andA, Rare positive constants. By (1.2)₂, the entropys can also be regarded as a function ofv andu. Our initial and boundary conditions are

. /

´

uNx404u₂,

g

^p(v,u)2muv^x

h

^{( 0 , t)4p0, tD0 ,}

(v,u,u)(x, 0 )4(v₀,u0,u0)(x)K(v₁,u₁,u₁) as xK 1Q, (1.3)

where (1.3)2means the gas is attached at the boundaryx40 to the atmo- sphere with pressurep₀(see [19]),v₁D0 ,u₁,u₁D0 are constants and u0( 0 )4u₂ holds as a compatibility condition.

It is well known that the contact discontinuity is a unique Riemann solution

(V,U,U)(x,t)4^./

´

(v₂,u₂,u₂), xE0 , (v₁,u₁,u₁), xD0 , (1.4)

of the following Riemann problem

. ` / `

´

v_t2u_x40 , ut1px40 ,

g

^e1u2²

h

_t^{1(pu)x40 ,}

(v,u,u)(x, 0 )4(v₂,u₂,u₂), xE0 , (v,u,u)(x, 0 )4(v₁,u₁,u₁), xD0 , (1.5)

ifu₂4u₁andp₂4^Ru_v ²

2

4p₁4^Ru_v ¹

1

. In view of [3], the contact discon- tinuity (V,U,U) in the half spacexD0 can be approximated by a viscous contact wave (V,U,U) satisfying

V(V2V,U2U,U2U)V_Lp( 0 ,1Q)4O(k)( 11t)^{1 /2p}, all pF1 .

Here

V(x,t)4 R

p₁U(x,t), U(x,t)4u₁1 k(g21 ) gR

U(x,t)_x U(x,t) , (1.6)

and U(x,t)4U(j), j4 ^x

k11t is the unique self similarity solution of (1.7) U_t4a

g

^UU^x

h

_x^{,U(0,t)4u2,U(1Q,t)4u1, a4kp1}gR^{(g22 1)}D0.

From (1.7), U(j) satisfies (1.8) 21

2jU8 4a

g

^UU⁸

h

^{8, U( 0 )4u2, U(1Q)4u1, 84} dj^d . By the same lines as in [2, 3], we have

(1.9)

. / ´

C₁(g21)^21/2Nu₁2u₂NGNU8(0)NGC₂(g21)^21/2Nu₁2u₂N,

N g

^NUxxN,kg¹21NU_xN, 1

g21NU2u₁N

h N

^GC3e2C4j

2

g21, as jKQ, whereC_i,i41 , 2 , 3 , 4 are positive constants depending onu₆. By (1.9), the lemma 1.1 of [3] reads in the following style.

LEMMA 1.1. If Nu₁2u₂N GM(g21 ), then

(1.10)

0 Q

U⁴_xdxGC(g21)^5/2(11t)^23/2,

0 Q

U_xx² dxGC(g21)^1/2(11t)^23/2,

(1.11)

0 Q

U²_xxxdxGC(g21)^21/2(11t)^25/2,

0 Q

x(U²_x1NU_xxN)dxGC(g21), where the constant C only depends on M.

The main aim of present paper is to show the global stability of the viscous contact wave and the superposition of a viscous contact wave and a rarefaction wave. The precise statement of our first result is

THEOREM 1.2. Assume that p₀4p₁ and Nu₁2u₂N GM(g21 ) holds for some constant M. Assume that(V,U,U) is the viscous con-

tact wave constructed in (1.6) and (1.7) and the initial data satis- fies

./

´

0EM₀²¹Gv₀(x),u0(x)GM₀, u₀(x)2U(x, 0 )H¹( 0 , Q), (v₀(x)2V(x, 0 ),s₀(x)2S(x, 0 ) )H₀¹( 0 , Q)

(1.12)

for some positive constant M₀, where S4_g^R

21

g

^lnRUA 1(g21 ) lnV

h

and s0(x)4s(x, 0 ). Then there exists a positive constant d0D0 such that if gE11d0, the problem (1.1)-(1.3) has a unique global solution (v,u,u)(x,t) satisfying

(v2V,u2U,u2U)(x,t)C( 0 ,1Q; H¹( 0 ,1Q) ), (1.13)

(v2V)_x(x,t)L²( 0 ,1Q; L²( 0 , 1Q) ), (1.14)

(u2U,u2U)x(x,t)L²( 0 ,1Q;H¹( 0 , 1Q) ), (1.15)

and

sup

xF0N(v2V,u2U,u2U)(x,t)NK0 , as tK 1Q. (1.16)

REMARK 1.3. The difference Nu₁2u₂N is naturally bounded by (g21 ) multiplying some constant M from the physical point of view.

Whenp₀cp₁, there are two subcases: the superposition of a viscous contact wave and a 3-rarefaction wave or that of a viscous contact wave and a 3-viscous shock wave. Here we only consider the previous one. In this situation, by the basic theory of hyperbolic system of conservation laws, there exists a unique point (v_m,u_m,um) such that p₀4p_m4^Ru_v ^m

m

and (v_m,u_m,u_m) belongs to the 3-rarefaction wave curve R(v₁,u₁,u₁) in the phase plane, where

(1.17) R(v₁,u₁,u₁)4

{

^{(v,u,u)Ns4s1,u4u12}_v

1 v

l(h,s₁)dh,vDv₁

}

^,

l(v,s)4

o

Agv^2g21e

g21 R s

. (1.18)

The precise statement of our second result is

THEOREM 1.4. Assume that there exists a unique point (v_m,u_m,um) such that p₀4p_m and (v_m,u_m,um)R(v₁,u₁,u₁) and Nu₁2umN 1 Num2u₂N GM(g21 ) holds for some constant M. As- sume that (V^cd,U^cd,U^cd) is the viscous contact wave constructed in (1.6) and (1.7), where (v₁,u₁,u₁) is replaced by (v_m,u_m,u_m) and (V^r,U^r,U^r) is the smooth rarefaction wave constructed in (3.4) satis- fying S^r4s(V^r,U^r)4s(v₁,u₁)4s₁. Let

V4V^cd1V^r2v_m,U4U^cd1U^r2u_m,S4S^cd. (1.19)

Assume that the initial data satisfies(1.12). Then there exists a positi- ve constantd0D1 such that if gE11d0,the problem (1.1)-(1.3) has a unique global solution (v,u,u)(x,t) satisfying (1.13)-(1.15). Further- more,

sup

xF0N(v2V^cd2v^r1vm,u2U^cd2u^r1um,s2S^cd)(x,t)N K0 , (1.20)

where (v^r,u^r,u^r), (s^r4s(v^r,u^r)4s₁) is the 3-rarefaction wave uniquely determined by (3.1).

Our plan of this paper is as follows. In sect. 2, the single contact dis- continuity case is investigated. In sect. 3, the case including the rarefac- tion wave is treated.

NOTATIONS. Throughout this paper, several positive generic con- stants are denoted by c,C without confusions. For function spaces, H^l(V) denotes the l-th order Sobolev space with its norm

VfV_l4

g

j

^!

40 l

V¯^j_x fV²

h

^{1 /2, when VQV»4VQVL2(V).}

(1.21)

The domain V will be often abbreviated without confusions.

2. Global stability of viscous contact wave.

PROOF OF THEOREM 1.2. We put the perturbation (f,c,z,W)(x,t) by

. /´

v(x,t)4V(x,t)1f(x,t), u(x,t)4U(x,t)1c(x,t), u(x,t)4U(x,t)1z(x,t), s(x,t)4S(x,t)1W(x,t), (2.1)

where (V,U,U)(x,t) is the viscous contact wave constructed in (1.6)

and (1.7). By (1.6) and (1.7), we have

. ` / `

´

V_t2U_x40 , U_t1

g

^RUV

h

x

4m

g

^UV^x

h

_x^1F,

R

g21U_t1p₁U_x4k

g

^UV^x

h

_x^1mUV^x2 1G, (2.2)

where

F4U_t2m

g

^UV^x

h

_x^4k(gRg^{21 )}

k

^{( lnU)xt2m}

g

RU^p1 ( lnU)_xx

h

_x

l

^,

(2.3)

G4 2mUx2

V 4 2mp₁

RU

g

^k(gRg^{21 )}( lnU)_xx

h

^2.

(2.4)

From lemma 1.1, we have

VFV_L1GC(g21 )( 11t)²¹, VGV_L1GC(g21 )^{5 /2}( 11t)^{23 /2}. (2.5)

Substituting (2.2) into (1.1) and (1.3) yields

. `

` ` / `

` `

´

ft2cx40 ,

c_t1

g

^R(UV1¹f^z)2 RU

V

h

_x^4m

g

^UV^x11^cf^x 2 U_x V

h

_x^2F,

R

g21zt1 R(U1z)

V1f (U_x1cx)2p₁U_x,

4k

g

^UV^x1¹f^zx

h

_x^2k

g

^UV^x

h

_x^1m(UV^x11^cf^x)22mU_x² V 2G,

g

V^Ru1²f 2mU_x1cx

V1f

h _N

^x404p1,

z( 0 , t)40 ,

(f,c,z)(x, 0 )4(v₀2V,u₀2U,u2U)(x, 0 ).

(2.6)

We shall prove theorem 1.2 by the local existence and the a priori esti- mate. We look for the solution (f,c,z) in the solution spaceX(0,1Q),

where

(2.7) X(0,T)4

m

^{(f,c,z) : (f,z)C(0,T;H01),cC(0,T;H1), 1}2M₁²¹G GvG2M1, 1

4M021

GuG4M0, f_xL²(0,T;L²), (c,z)_xL²(0,T;H¹)

n

for some 0ETG 1Q, where the constant M₁ will be determined later.

Since the local existence has already been established in [3], we only con- sider the a priori estimate here. We have

PROPOSITION 2.1. (A Priori Estimate). Assume that the conditions of theorem1.2 hold, then there exists a positive constantd₀such that if gE11d0and(f,c,z)X( 0 ,T)is a solution of(2.6)for some positive T, then the followings hold

M₁²¹Gv(x,t)GM₁, 1

2M₀²¹Gu(x,t)G2M₀, (2.8)

(2.9) V(f,c,z)V₁²1

0 t

]Vf_xV²1V(c_x,z_x)V₁²(dtGC(M₀)(11V(f₀,c0,W0)V₁²), where the constant C(M0) depends on M,M0 and the initial data, but does not depend on M1.

Proposition 2.1 is proved by a series of lemmas.

LEMMA 2.2. It follows that

(2.10)

V u

⁽

o

^F

g

V^v

h

^,c, _k¹_d^z

^v

^(t)

^V

²¹

0

t

m ^V g

v^{c1 /2x} , z_x

v^{1 /2}

h

^(t)

^V

²

n

^dtG

GC(M₀)

m

^{V(f0,c0,kdW0)V121C(M1)}

k

^d1d

0 t

V g

v^{f3 /2x} , zx

v^{1 /2}

h ^V

^2dt1

1

0 t

d²( 11t)^{28 /5}

V g

^c, k¹dz

h ^V

^2(t)dt

^ln

^,

where d4g21E1.

PROOF. Due to [3], we have

(2.11)

g

¹2c²1RUF

g

V^v

h

^1RdUF

g

U^u

hh

_t^{1 kU}vu²zx2

1 mUcx2

vu 1 1H_x1Q4 2Fc2 zG

u , where

(2.12) F(s)4s212lns, C(s)4s²¹211lns, (2.13) H4R

k g

^U1v ^z2U

V

h

^c

l

^2m

k g

^Ux1v^cx2U_x V

h

^c

l

²

2kz

u

g

^uv^x 2 Ux

V

h

^,

(2.14) Q4p₁C

g

V^v

h

^Ux1pd¹C

g

U^u

h

^Ux1mcxUx

g

¹v 2 1 V

h

²

2z

u(p₁2p)Ux2k U_x

u²vzzx2kz_xfUU_x

u²vV 1k zf u²vVU²x2 2mzU_x²

u

g

¹v 2 1

V

h

²²vu^mzc_xU_x. By the formula of F(s), we know that F( 1 )4F8( 1 )40 , C( 1 )4 4C8( 1 )40 and F(s) is a strictly convex function. This yields that F(s)D0 and

. /

´

C₁(M₀)C₁(M₁)f²GF

g

V^v

h

^{GC2(M0)C2(M1)f2,}

N

^C

g

U^u

h _N

^{GC(M0)C(M1)z2,}

(2.15)

for some constants Ci(M0),Ci(M1),i41 , 2 . Since (2.16) NQN G kU

4vu²zx2

1 mU 4vucx2

1

1C(M₀)C(M₁)

k

^NUxN

g

^{f21 1}dz²

h

^1(f21z2)U2x

l

^,

and

(2.17)

0 Q

NFcN 1

N

^Gzu

N

^{dxG 43mM}02

0 Qcx2

v dx1 k 4⁴M₀³

0 Qzx2

v dx1

1C(M₀)C(M₁)

y

^{d( 11t)26 /51d2( 11t)28 /5}

^V g

^c, k¹dz

h ^V

²

^z

due to (2.5), integrating (2.11) over R₁3( 0 , t) gives (2.18)

V u o

^F

g

V^v

h

^,c, _k¹_d^z

^{v V}

²¹

0 t

0 Qcx2

v 1z2x

v dx dt2

0 t

H( 0 , t)dtG

GC(M₀)C(M₁)

m

0 t

0 Q

(z²1f²)(NUxxN 1U2x)dx dt1d1

1

0 t

d²( 11t)^{28 /5}

V g

^c, k¹dz

h ^V

^2dt

ⁿ

^1C(M0)

^V g

^f0,c0, k¹dz0

h ^V

^2.

On the other hand, the boundary condition (1.2) exactly gives the value of f( 0 , t). From [3] and the fact that f₀(x)H₀¹( 0 ,Q), we have

f( 0 , t)4f0( 0 )e²

p1 m t

40 , (2.19)

which yields that H( 0 , t)40 . Note that

Nz(x,t)N Gx^{1 /2}VzxV, Nf(x,t)N Gx^{1 /2}VfxV, (2.20)

due to [16], applying Lemma 1.1 and the boundary condition (2.19), we have

(2.21)

0 t

0 Q

(z²1f²)(NUxxN 1U2x)dx dtG

GC(M₀)C(M₁)d

0 t

V g

v^{f3 /2x} , zx

v^{1 /2}

h ^V

^2dt.

By (1.2) and (2.1), we have

Vz0V₁GCdV(f₀,W0)V₁. (2.22)

Thus, combining (2.18)-(2.22) yields lemma 2.2.

LEMMA 2.3. There exists a small positive constant d0 such that if dGd0, then

(2.23)

V g

^f,c, k¹dz

h

^(t)

^V

^21VfxV21

0 t

]V(fx,cx,zx)(t)V²(dtG

GC(M0)]11V(f0,c0,W0)V₁²(. PROOF. Following [14], we introduce a new variable vA4_V^v . Then (2.6)₂ can be rewritten by the new variable as

g

^m

g

^vAvA^x

h

^2c

h

_t^2px4F.

(2.24)

Multiplying (2.24) by ^vAx

vA , we have

(2.25)

g

^m2

g

^vAvA^x

h

^22cvAvA^x

h

_t¹

g

^cvAvA^t

h

_x^1Ruv

g

^vAvA^x

h

^{22 R}v zx

vA_x vA 1 1Ru

v

g

U¹ 2 1

u

h

^{Ux vA}vA 4^x c²_x

v 1c_xU_x

g

¹v 2 1

V

h

^1FvAvA^x . The Cauchy inequality yields that

(2.26)

N

^Rv zx

vA_x

vA

N

¹

N

^cxUx

g

¹v 2 1

V

h _N

^{G Ru}4v

g

^vAvA^x

h

²¹

1C(M0)

g

^zv^2x 1c_x²

v 1C(M₁)f²U_x²

h

^,

(2.27)

N

^Ruv

g

U¹ 2 1

u

h

^{Ux vA}_vA^x

N

¹

N

^FvAvA^x

N

^{G Ru4v}

g

^vAvA^x

h

²¹

1C(M₀)C(M₁)(z²U2x

1 NFN²), and

(2.28) c₁ fx2

v² 2C(M₀)C(M₁)f²U2x

G

g

^vAvA^x

h

^{2Gc3 f}v^2x2 1

1C(M0)C(M1)f²U_x².

Similar to (2.19), we also have

g

^vAvA^t

h

^{( 0 ,t)4}

g

^fv^t

h

^{( 0 ,t)4}0 . Note that the right hand sides of (2.26)-(2.28) have already been investigated in Lem- ma 2.2. Integrating (2.25) overR₁3( 0 , t) and using Lemma 2.2, (2.15), (2.26)-(2.28) and the fact that

N

^cvAvA^x

N

^{G m4}

g

^vAvA^x

h

^21Cc2,

we get

(2.29)

V u o

^F

g

V^v

h

^,c, _k¹_d^z

^v

^(t)

^V

²¹

1

V

^vAx

vA

V

²¹

0

t

m ^V g

v^{f3 /2x} , cx

v^{1 /2} , zx

v^{1 /2}

h

^(t)

^V

²

n

^dtG

GC(M0)

m

^{V(f0,c0,kdW0)V121C(M1)}

k

^d1d

0 t

V g

v^{f3 /2x} , z_x

v^{1 /2}

h

^(t)

^V

^2dt1

1

0 t

d²( 11t)^{28 /5}

V g

^c, k¹dz

h ^V

^2(t)dt

^ln

^,

We now choose a small positive constant d0E1 so that C(M₀)Q QC(M₁)d0E¹₂. Then for any dEd0, the Gronwall’s inequality yields

(2.30)

V u o

^F

g

V^v

h

^,c, _k¹_d^z

^v

^(t)

^V

²¹

1

V

^vAx

vA

V

²¹

0

t

m ^V g

v^{f3 /2x} , cx

v^{1 /2} , zx

v^{1 /2}

h

^(t)

^V

²

n

^dtG

GC(M0)]11V(f0,c0,kdW0)V₁²(. We note here that the constant C(M₀) does not depend on M₁.

We now use the method of [14] and (2.30) to show (2.8)₁. To this end, let

h(vA)4

1 vA

k

F(s)

s ds, F(s)4s212lns. (2.31)

Since

h(vA)K ./

´

2Q, vAK01, 1Q, vAK1Q, (2.32)

and

(2.33) Nh(vA(x,t) )N 4

4

N

1Q x

¯

¯yh(vA(y,t) )dy

N

^G

R₁

g

^F

^g

V^v

h

¹

g

^vAvA^x

h

²

h

^(x,t)dx,

the inequality (2.30) yields that there exists a positive constantM₂which only depends on M,M₀ and the initial data such that

M₂²¹Gv(x,t)GM₂. (2.34)

LetM₁4M₂, then (2.8)₁is proved. After we obtain the a priori estimate (2.34), it is easy to imply (2.23) from (2.15) and (2.30).

LEMMA 2.4. It follows that (2.35) V(c_x,z_x)(t)V²1

1

0 t

]V(c_xx,z_xx)(t)V²(dtGC(M₀)( 11V(f₀,c₀,z₀)V₁²).

PROOF. We first estimate the term cx. Multiplying (2.6)₂by 2cxx, we have

(2.36)

g

¹2cx2

h

_t^2(ctcx)x1mcv^xx2 2p_xcxx1 1m

g

^Uv^x 2 U_x

V

h

_x^cxx1mcx

g

¹v

h

_x^cxx4Fcxx.

Since p₁ is constant, we have Np_xcxxN G m

4vcxx2

1C(M₀)(f²1z²)Ux2

1C(M₀)(f²_x1zx2).

(2.37)

On the other hand, (2.38)

N

^m

g

^Uv^x 2U_x

V

h

_x^cxx

_N

¹

_N

^mcx

g

¹v

h

_x^cxx

_N

^G

G m

4vc_xx² 1C(M₀)d²(U⁴_x1U²_xx1U²_xxx)1C(M₀)(f²_x1c²_x)1 1C(M₀)Nf_xVc_xVc_xxN. We compute by lemma 2.3

0 Q

NfxVcxVcxxNdxGsup]NcxN(VfxV VcxxV (2.39)

Gk2Vc_xxV^{3 /2}Vc_xV^{1 /2}Vf_xV Gm

V

₂^c_v^xx1 /2

V

^{21C(M0)VcxV2.}

Since cx( 0 , t)4ft( 0 , t)40 , integrating (2.36) over R₁3( 0 ,t) and using (2.37)-(2.39) and Lemmas 1.1 and 2.3 imply

Vcx(t)V²1

0 t

Vc_xx(t)V²dtGC(M₀)( 11V(f₀,c₀,z₀)V₁²) . (2.40)

We now estimate z_x. Multiplying (2.5)₃ by 2z_xx, we have

g

2^Rdzx2

h

_t²

g

^Rdz_tz_x

h

_x^1kzv^xx2 1m

g

^uv^x2 2Ux2

V

h

^zxx1QA4Gzxx,

(2.41) where (2.42) QA

42(p2p₁)U_xzxx1kzx

g

¹v

h

_x^zxx2pcxzxx1k

g

^Uv^x2Ux

V

h

_x^zxx.

By the same method for the estimate on cx, we obtain

V

_d^z1 /2^x (t)

V

²¹

0 t

Vzxx(t)V²dtGC(M₀)( 11V(f₀,c0,z0)V₁²).

(2.43)

We omit the details here. Combining (2.40) and (2.43) yields lemma 2.4.

By lemmas 2.3 and 2.4, we have

Nz(x,t)N^LQGCVzV₁GC(M₀)d^{1 /2}( 11V(f₀,c₀,z₀)V₁) . (2.44)

Choosing d suitably small yields 1

2M₀²¹Gu(x,t)4U(x,t)1z(x,t)G2M₀. (2.45)

Thus proposition 2.1 is obtained from (2.34), (2.45) and lemmas 2.3 and 2.4. Theorem 1.2 is easy from the local existence and proposition 2.1.

3. Superposition of viscous contact wave and rarefaction wave.

PROOF OF THEOREM 1.4. In this case, there exists a unique point (vm,um,um)R(V₁,u₁,u₁) such that p04pm and the superposition of the viscous contact wave connecting (v₂,u_m,u₂) with (v_m,u_m,u_m) and the 3-rarefaction wave connecting (v_m,u_m,u_m) with (v₁,u₁,u₁), where v₂4^Ru_p²

0

. The 3-rarefaction wave (v^r,u^r,u^r)

g

^xt

h

^connecting

(v_m,u_m,um) and (v₁,u₁,u₁) is the weak solution of the Riemann problem

. ` / `

´

v_t2u_x40 , u_t1p(v,u)_x40 ,

u

^{e(v,u)1 u}₂²

v

t

1(p(v,u)u)_x40 , (v₀,u₀,u₀)(x)4^./

´

(v_m,u_m,um),xE0 , (v₁,u₁,u₁),xD0 . (3.1)

Since the rarefaction wave is a weak solution, it is necessary to construct a smooth approximate rarefaction wave (V^r,U^r,U^r)(x,t) of

(v^r,u^r,u^r)

g

^xt

h

^{inR13( 0 , 1Q}). To this end, we apply the idea of [4]

to construct the smooth rarefaction wave. The advantage of this kind smooth wave is that the boundary effect can be exactly eliminated. We first construct the solution w(x,t) of the following problem

. ` / `

´

w_t1ww_x40 , (x,t)R3( 0 , 1Q),

wN^t404

. /

´

w₂,

w₂1wAK_q

0 ex

z^qe^2zdz,

xE0 , xF0 , (3.2)

where w₆4l(v₆,s₁), s₁4s(v₁,u₁), wA 4w₁2w₂, K_q is a constant such thatKq

s

0 1Q

z^qe^2zdz41 for large constantqF8 andeis a small posi- tive constant. We have the following properties of w(x,t) due to [4].

LEMMA 3.1 [4]. Let 0Ew₂Ew₁, then the problem (3.2) has a unique smooth solution w(x,t) satisfying

i) w₂Gw(x,t)Ew₁, w_xF0 , for xF0 ,tF0 .

ii) For any p( 1GpG 1Q),there exists a constant C_p,qsuch that for tF0 ,

Vw_x(Q,t)V_LpGC_p,qmin

(

wAe^{121 /p},wA^{1 /p}t^{2111 /p}

)

,

Vwxx(Q,t)V_LpGCp,qmin

(

wAe^{221 /p},wA^{1 /q}e^{121 /p11 /q}t^{2111 /q}

)

^.

iii) When xGw₂t, w(x,t)2w₂4wx(x,t)4wxx(x,t)40 . iv) lim sup

tK1Q,xDNw(x,t)2w^R(x,t)N 40 .

Here w^R(x,t) is the Riemann solution of the scalar equation (3.2) with the initial data w₀(x)4w₂, if xE0 and w₀(x)4w₁, if xD0 .

The smooth approximation (VA^r,UA^r,UA^r) to (v^r,u^r,u^r)(x,t) is given by

. ` / `

´

SAr(x,t)4s₁,

l(VA^r(x,t),s₁)4w(x,t), UA^r(x,t)4u₁2

v₁ VAr(x,t)

l(h,s₁)dh. (3.3)

Setting

(V^r,U^r,U^r)(x,t)»4(VA^r,UA^r,UA^r)(x,t1t₀)N^xF0, (3.4)

where the constant t0D1 will be given later. Then we have

. ` / `

´

V_t^r2U_x^r40 ,

U_t^r1p(V^r,U^r)_x40 ,

g

^{e(Vr,Ur)1 1}2(U^r)²

h

_t^{1(p(Vr,Ur)Ur)x40 ,}

(V^r,U^r,U^r)N^x404(v_m,u_m,um),

(V^r,U^r,U^r)N^t404(V₀^r,U₀^r,U^r0)(x)4(V^r,U^r,U^r)(x, 0 ).

(3.5)

Due to Lemma 3.1, (V^r,U^r,U^r) has the following properties.

LEMMA 3.2. The smooth rarefaction wave (V^r,U^r,U^r)(x,t)satis- fies, if qFp,

i) U_x^r(x,t)F0 , NU_x^rN GCe, for tF0 ,xF0 .

ii) For any p( 1GpG 1Q), there exists a constant C_p,q such that

V(Vx^r,Ux^r,U^rx)V_Lp(xF0 )GCp,qmin]e^{121 /p}, (t01t)^{2111 /p}(,

V(V_xx^r ,U_xx^r ,U^rxx)V_Lp(xF0 )GC_p,qmin]e^221/p, (t₀1t)^{2111 /q}(, tF0 . iii) (V^r,U^r,U^r)N^xGl(vm,s₁)t4(v_m,u_m,u_m),

(V_x^r,Ux^r,U^rx,Vxx^r ,Uxx^r ,U^rxx)N^xGl(vm,s₁)t40 . iv) lim sup

tK1Q,x]xF0(

N

^{(Vr,Ur,Ur)(x,t)2(vr,ur,ur)}

g

^xt

h _N

^{40 .}

Let (V^cd,U^cd,U^cd) be the viscous contact wave constructed in (1.6) and (1.7), where (v₁,u₁,u₁) is replaced by (vm,um,um). Then (V^cd,U^cd,U^cd) satisfies

. ` / `

´

V_t^cd2U_x^cd40 ,

U_t^cd1

g

^RUV^cdcd

h

_x^4m

g

^UV^xcdcd

h

_x^1Fcd,

R

g21U^cd_t 1p₀U_x^cd4k

g

^UV^cdcdx

h

_x^1m(UV^xcdcd)21G^cd, (3.6)