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Viscous solutions of quadratic conservation laws with umbilic points
Nonlinear AfloWs, Theory, Printed in Great Britain.
Methods
Vol. 21, No. 7, pp. 485-499,
& Applications,
1993. 0
0362-546X/93 S6.a) + .OQ 1993 Pergamon Press Ltd
VISCOUS
SOLUTIONS OF QUADRATIC CONSERVATION WITH UMBILIC POINTS
Wuhan Institute
of Mathematical
LTJ YUNGUANG,
ZHU CHANGJIANG
Sciences,
Academia
Sinica,
and
LAWS
ZHAO HUIJIANG
Wuhan,
430071,
People’s
Republic
(Received 10 January 1991; received in revised form 4 August 1992; received for publication Key words and phrases: Riemann invariants, uniformly bounded estimates, viscous regions, quasi-convexity, energy method, existence and asymptotic behavior.
of China
1 March 1993)
solutions,
invariant
1. INTRODUCTION THE GENERAL 2 * 2 SyStCm Of COmXVatiOn II, + &a,
kiWS
flUX fUIICtiOnS
with quadratic
u2 + 2bl uu + Cl uqx = 0
t u, + +(a,~’
+
(1.1)
2b,uu + c2 02)X = 0
is of interest because solutions of (1.1) may approximate solutions of a general 2 * 2 system of conservation laws in a neighborhood of an isolated point at which strict hyperbolicity fails. In [l] Schaeffer and Shearer show that when system (1 .l) is hyperbolic, there is a nonsingular linear change of dependent variables which transforms system (1.1) into the normal form u, + +(bu2 + 2uv), = 0 (1.2)
i u, + +(au2 + 2buv + u~)~ = 0. Let F be the mapping from R2 into R2 defined by F: (u, u) + (+(bu2 + 2uu), &u’
+ 2buu + u2))
then two eigenvalues of dF are ~_ = ((a - 1)u + bu) f d((a + ? with corresponding
1)u +
bu)2 + 4(bu + tQ2 (1.3)
right and left eigenvectors
rT =
bu + u,
I, =
bu + u,
-((a -
1)u +
bu) T ~/((a - 1)u + bu)2 + 4(bu + u)’ = 2 >
-((a -
1)u +
bu) T t&a 2
1)u +
bu)2 + 4(bu + u)~ >-
By simple calculation, we have VA, * rT = (bu + u) ; (a + 3) f
(3 - a)(@ d((a 485
1)u +
1)u +
bu) T 3b(bu + u)
bu)2 + 4(bu + u)~ 1 ’
(1.4)
486
Lu YUNGUANG
etal.
Therefore, it follows from (1.3) that A+ = A_ at points satisfying (a - 1)~ + bv = 0 and bu + u = 0 at which strictly hyperbolicity fails to hold. That the first and the second characteristic fields are linearly degenerate on bu + v = 0 follows from (1.4). As a model of a nonstrictly hyperbolic system, (1.2) has been studied by many scholars, especially for the symmetric case (i.e. b = 0). The Riemann problem for the system (1.2) is first considered in [2, 31 when b = 0, a E (1,2) or a E (2, +m), and the interactions of elementary waves are given in [4] when b = 0, a > 2. Recently, using the vanishing viscosity method and theory of compensated compactness, Yunguang Lu [5] and Kan [6] obtained the global weak solutions to system (1.2) when b = 0, a = 3. (Rubino [7] also studied the global weak solutions to system (1.2) when b = 0, a > 2, but his work seems incomplete.) In their work, in order to use the theory of compensated compactness, a fundamental step is to obtain the L--priori estimates independent of the viscosity coefficients on the viscous solutions. So it is of interest to consider the P-priori estimates on the viscous solutions. In this paper, first, we study the uniformly bounded estimates on the solution to the following parabolic system u, + &bU2 + 2UV), = &U** (1.5)
U, + *(au2 + 2buu + vqx = EU, with initial value
(1.6)
(v, @It=0 = (Ql(x), %(X)).
When a, b satisfy a - 1 1 b2, (16b3 + 9( 1 - 2a)b)2 # 4(4b2 - 3(a - 2))3 and b # 0 or a 1 1, b = 0, we get the uniformly bounded estimates (which are independent of a) on the solutions of the Cauchy problem (1.5), (1.6). On the other hand, there have been many papers [8, 91 about the study of solutions to strictly hyperbolic systems of two conservation laws, with viscosity, especially to the gas dynamics system [lo-131 and elasticity system [14], but it is more difficult and important for systems of nonstrictly hyperbolic systems. Since both nonstrictly hyperbolic and linearly degenerate points occur in our model, we also consider the existence and asymptotic behaviors of solutions to the Cauchy problem (1.5), (1.6) for arbitrary constants a, 6. The program of this paper is as follows: in Section 2, for a - 1 L b2, (16b3 + 9(1 - 2~)b)~ # 4(4b2 - 3(a - 2))3 and b # 0 or a 1 1, b = 0, we study the existence of the invariant domain and obtain the L--priori estimates on the solutions of the Cauchy problem (1.5) and (1.6); in Section 3, we consider the existence and asymptotic behaviors of the solutions to the Cauchy problem (1.5) and (1.6) for arbitrary constants a, b. It is worth pointing out that when b = 0, our results coincide with that of [15], hence, our work is an extension of [15]. 2. INVARIANT
REGIONS
AND
THE
UNIFORMLY
BOUNDED
ESTIMATES
The main purpose of this section is to obtain the L”-priori estimates, independent of E, to system (1.5) with initial value (1.6). In order to establish such a priori estimates, we shall use the theory of invariant regions due to Chueh et al. [16]. The result of [16] can be summarized in the following theorem.
Quadratic conservation laws
487
THEOREM.Let gT be two smooth functions, g,: R ’ + R and C = ((v, u): g,(u, u) 5 0). Assume that for any t > 0 and (13,fi) E 8X, the following conditions hold: (a) Vgr is a left eigenvector of VF(D, ii); (b) gr is quasi-convex in (0, ii), i.e. for all r E R2, < * VgT = 0 * V2g7(<, <) 2 0 then X is an invariant region for (1.5) for all E > 0. Namely, if (v,, uO) E E for all X, then (V&(X,t), u&(x, t)) E E:, for all (x, t). We say that w_ E C’(w+, respectively) is a first (second) Riemann invariant for (1.2) if for all (u, u) (Vw_(v, UK r-) = 0 Therefore, Further, tively, the rarefaction
((Vw+(v, UK r,) = 0).
Vw,//l,. let us recall that the integral curves R, of rT in the state space are called, respecfirst and second rarefaction wave curves. Thus, in our case, a first (second) wave curve satisfies the ordinary first order differential equation du dv=
((a - 1)~ + bu) f d((a - 1)~ + bv)* + 4(bu + v)* 2(bu + v)
(2.1)
From (2.1), we can conclude that R_, R, are symmetric on the origin. We can, hence, restrict our attention to the R, family only. Now we turn to analyze the properties of the R, curves. If a, b belong to region IV (i.e. (16b3 + 9(1 - 2a)b)* - 4(4b* - 3(a - 2))3 > 0, see [2]), then we have the following lemma. LEMMA2.1. Under the above hypothesis, if u 2 uO, then R+(v,, u,)-the R, curve through (u,,, u,J (where (u,,, uO) satisfy: bu, + u,, = 0, u. 2 0) has the following properties (see Fig. 1): (a) the R, curve R+(q,, u,,) contains only three parts: except the half line ((u, u): bu + u = 0, u 2 01, one is in the half space bu + u 2 0, the other is in bu + u I 0; (b) the R+(u,, u,,) curves are in one to one correspondence with the points of the half line ((u, u): bu + u = 0, u L 01;
bu+v=O
Fig. 1.
(*p*z LIE~~O~O3aas
‘ b ‘ ELI‘?I 30 suo!~~u~3apay$ _IOJ) *Llaapz)adsal
‘(0 7 n :(n ‘0)) u (0 7 nkJ - R :(n ‘0)) u (0 5 rzEJ - n :(n ‘0)) = ,I\1 ‘(0 7 n :(n ‘n)) u {O7 ?zEcJ- n :(n ‘n)) u (0 5 R + nq :(n ‘0)) = ,111 ‘(0 7 n :(n 72)) u (0 5 Yzz3- n put? 0 7 R + nq :(n ‘n)] = ,I1 ‘(0 5 nb - R :(n ‘(2))n {O7 nz3 - R :(n ‘0)) = ,I 01 ssuolaq yrea ‘s$wd .uIo3 syeu~o~ sa&InD+v aqy 30 auo ya?za (I?) :(Z .f$a) sayado.Id OU~MOIIOJ aql aneq saMn3 +y aql ‘sysayloddq aAoqt? ayl lapun ‘Z’Z vvuwa~ w_w.ua~%y~ol103 ayi aAEy ah4 uaql ‘(0 > ,((Z - @E - ,4P)P - ,(S@Z - I)6 + Eq91) put2 zq < 1 - D *a*!)111uoi%al 01 FiuoIaq q 9 31 ‘0 + R SB 0 + n ~3spw (0 5 R + nq) 0 z R + nq uf pauyuo3 saAln3 +g ayl (a) f(0 > R ‘0 = R + nq :(n ‘0)) awl 3p2y aq4 ~I!M lDaslalu! IOU saop (On‘OR)+~ saam3 +v ayl 30 au0 yea (“Ip) too- + R SE w+ e n (“‘p) fxaauo3 s! (On‘%)+y saA.nv +x aql30 au0 y2ea (“p) !sast?alDu! OCI5 R s12sasEal3ap (On‘On)+8 saA.uv +g ayl 30 au0 y3sa (*p) :sa!yadold S!UIMOIIOJ ayl s~q 0 5 n + nq u! (On ‘0n)+a aAln3 ayL (p) (*p*Zb.11)?110~03 aas ‘3 30 uoyw3ap ayl 10~) ‘(0 < n ‘rz13 = n :(n ‘rz)] lDaslalu! IOU saop (On‘%)+g sar\.uw +x ayl 30 au0 qxa (A13)
awl 31y
aql
qq~
fco+
t
n SE 03+
c
n (III,)
fxaAuo3 sy (On‘On)+y sa.4ln3 +u aql30 au0 y3va (“3) fsasaalaur O0 7 ci se sastza.cw! (On‘%)+y sakuv +x ayl30 au0 ycwa (5) :sapladold %I~MO~~O~ aql sey 0 z R + nq u! (On ‘On) +y a&In3 ayl (3)
JO U8!S 3q;L
‘lUE?JSUO3 E Sy (I ‘0)
3 I3 al3yM
‘(0
7
YZ) II I3 =
0 ‘(0
7
0) 0 =
f-4 +
?lq
:SjOO.l I&?al
Pue SvY 0 = (n ‘fl)V ‘JW ‘0 < ,((Z- o)E- ,4P)P- ,(S(nZ - I)6+ $q91)31 (E) ‘&l + nq)p + @q + n(r - a>>pn + mz(E - 0) + pqz - pq = (n ‘n)y ‘p’z nxv-IToxo~
OMI SW LlUO
WOOJ pad w~gs!p aa.yl sey 0 = b + Xd + EL uopEnba uayl ‘0 > &Id) + &Z/b)31 (q) ‘100.1pzal auo lip.10sey 0 = B + rCd+ & uoyenba uayl ‘0 < &f/d) + ,(bf) 31 (e) ‘fez VYYPEI~ *sllnsaJ %U!MO~~OJ ayl paau afi ‘z-z pur? 1*z swutual allold 01 lap10 UI TI$QIO aql u! IdaDxa n ? = R 10 () = 0 + nq saug I#!M pas.Ialuy IOUsaop Sahln3 +g ayl30 au0 yDE?a(“‘p) too-- t R se CO+ + n ‘0 c n se () c n (“‘p) fxaAuo3 s! saAln3 +g ayl30 au0 yzea (up) fsasealzy 0 5 n se sasea.I3ap saam3 +g aql30 au0 yDea (Ip) :sa!yadold Ouy~ol~o3 ayl aAt?y ,111 UO!%aJ u! Sahln3 +v ayl (p) !uyS~~oaqi uy IdaDxa (nap = n) 0 = n ~0 (n 9 = n) n Z3 = n sag aql ~JIM y3aslalu! 10u saop saAln3 +g ayl30 au0 ycea (h) f(cO- + n) oo+ t R St? oo+ + n ‘0 + 0 SE 0 + n (Yl) laAe3uo3 s! saAln3 +g ayl30 au0 qcea (“3) fsasr?a.wy (0 5 0) 0 z R se (sasEarDap) sastza.w_ysalz.rn3 +8 ayl30 au0 yst?a (‘3) :sagladoId %U~MO~IO~ aq$ ahzy (,AI) ,II uoy8al III sahln3 +x aql (3) to + n se 0 + n L3spes (0 5: R + nq) 0 z n + nq uy pauyuo3 saam3 +x ayl (Iuq) 1(() < n ‘n? = R :(?I ‘0)) 10 (0 < n ‘n? = n :(n ‘0)) saug 31~29 ~J!M 13aslaly IOU saop (On‘“n)+~ sakrn3 +z aql 30 au0 yrea (“9) f 1-z w.n.ua~ uy (lllp)-(Ip) ‘(W)-(3) ‘(q) ‘(e) 30 wq$ 01 ~e~y.uyssa!yadold ay$ w?y (On‘On)+8 sah.uw +y ayl 30 auo yDt?a (‘9) :sapladold Ouy~o~~o3 ay$ aAeey(On z n :(?I ‘0)) u ,I uor?ial u! (0 z (4-1‘0 = Ocz+ Onq A3sges (On‘On) alayM) (On‘On) y?inoryl saAJn3 +y ayl-(On ‘ On)+8 (q) 68P
‘p?~~g ale Isal ayL
‘5
‘%I
0=&Z-m-Z9P)tz(9(“Z-I)6+F991)
zzm + xl
=
$
‘(Iup) put? (up) ‘(1%) ‘(I$) aAold 01 paau Quo aM *I-Z wuruq Jo Joodd
(‘0 < 4 uoy~puo~ ayl lapun are p-Z d.re~lo~o~ put2 p-1 Sam%& *3oold OU~MO~[O~ aql u! 0 < 4 asE3 aql JapysuoD Quo aM ‘[gr] uy paJapyuo2 uaaq st?y 0 = 4 ase3 ayl axus ‘0 z 9 ‘dyyxaua8 30 SSOIInoyyM ‘azunsse 6txu ah ‘(s %d) spz-w ay3 uo 3p~aunuLs an2 01 Ouolaq cj ‘w y~yl suor8al ayl axiS *ICBM atuBs aqy us paAold aq u’t?3Z’Z sunua[ ‘1 ‘Z sun.uaI aAold Quo aM ‘Z’Z put! 1‘Z srxnua~ aAOld 01 ulna aM MON ‘p *8y u! paD!dap sy (n ‘n)y 30 uSis ayL ‘slue~suo~ are (fA(w- Z)f + zwp + qz)- ‘c-1 3 ?J ‘(q- ‘f/((Z + w-)f + zqvp + qz)-) 3 E3 ‘(1 ‘0) 3 z3 ‘(0 7 n) nb = 0 ‘(0 7 n) nE3 = n ‘(0 7 n) nZ3 = R ‘(0 7 n) 0 = n + nq :qooJ pal Ino3 AIuo Pue seq 0 = (n ‘ah uaql ‘0 > ,((Z - w)f - ,4P)P - ,(S(wZ - 06 + ,991) 31 (9)
aJaw
W
‘P
‘$!d
06P
Quadratic
conservation
491
laws
where Y = (a - 1)~ + bu, Z = bu + u, X = n, in order to study the convexity of the R, curves we compute the second derivative. Through tedious but simple computation, we get g
= (a - 1 - P)(X + Y)h(u,
u)/4Z3X.
Under the condition (16b3 + 9(1 - 2a)6)2 - 4(4b2 - 3(a - 2))3 > 0, from corollary 2.4, we can conclude that each one of the R, curves R+(u,, u,,) has at most two turning points, which are on the half line {(u, u): u = ciu, u 2 0) or ((u, u): bu + u = 0, u r 0), respectively. Note that u = ciu (u 2 0) itself is an integral curve of du/du = (X + Y)/2Z. Hence, the theory of ordinary differential equations (the uniqueness of the solutions) indicates that each one of the R, curves R+(u,, u,,) on bu + u > 0 does not intersect with the half line ((u, u): u = c1 u, u 2 0). The above facts and (d,,) deduce that (cn) and (d,,) are true. As to (cm), from the monotonicity, Lim u = M exists, we assume M < +oo, otherwise, v-2t43
is proved. Furthermore, Then we have
(cIII)
we have A4 > uO.
du (a - l)M + 2bu, + d((a - l)M + 2bu,)2 + 4(bA4 + 2u,)2 = d > o 1 9 z W0.M) = 2(bM + 2u,) since for all u 2 2u, du
du
=d,>O
-du > G (2V~.lw we have u - A4 2 d,(u - 2v,) which indicates u -+ +oo as u + +03, (cm) is proved. (din) can be proved in the same way. Finally, for (1.2), we can construct Riemann invariants w_, W, so that W_(U,u) 5 0 I W+(U,2.4). Since wy is constant along every RT curve, if we prescribe w,(u,, -u,/b) follows: Case 1. (16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 > 0 w-(uo 9 -u,/b)
=
w+(uo 9 -u,/b)
=
QJ, uo < 0 1 0, u#JL 0 uo, uo > 0 i 0, ug I 0.
Case 2. (16b3 + 9(1 - 2~)b)~ - 4(4b2 - 3(a - 2))3 < 0;
in region I’ w-(uo 9 -u,/b)
=
w+(uo 9 -Q/b)
=
uo, UIJ< 0 i 0, ug 2 0 &I I uo > 0 I 0, ug I 0;
(2.2) or w,(O, 0) as
492
Lu
YuNGUANG et al.
in region II’, IV’ w-(0,0)
= w_(uo, -U,/@
w+(O, 0) = w+(vo, -v,/b) and let w_(u, u), w+(v, U) satisfy &,/au
> 0;
in region III’ w-(0,0) = w_(u,, -u,/b)
w+(O,0) = w+(vo,-v,/b) and let w_(u, u), w+(u, U) satisfy c?w,/& < 0. Furthermore,
we set
w-l((“,.):“=.,.,.so,= w-l,(“,.):.=o,.50) = -co, then w_(v, U) 5 0 I w+(u, U). From the construction
of the Riemann invariants, we also have the following corollary.
2.5. The Riemann invariants constructed as before satisfy: (a) if (16b3 + 9(1 - 2a)b)’ - 4(4b2 - 3(a - 2))3 > 0, then C~W+/C~U < 0; (b) if (16b3 + 9(1 - 2a)b)’ - 4(4b2 - 3(a - 2))3 < 0, then aw,/au < 0 in region I’ or III’, a~,/& > 0 in region II’ or IV’. COROLLARY
From now on, we shall use the Riemann invariants w_, w, provided by the previous construction. Since by definition Vw7 are left eigenvectors for VF, the Riemann invariants are the functions used to define the invariant regions following [16]. Therefore, in this way we will find a family of invariant regions (Figs 6 and 7 corresponding to lemmas 2.1 and 2.2, respectively) given by XC = (w_ + c 2 01 n (w, - c 5 01,
c > 0.
These regions Xi, C,“, Ci (see Figs 6 and 7) are bounded by the wave curves of the two families. On the other hand, CJ strictly increases in c and spans the whole state space as c -+ 00. Cz (Ef) strictly increases in c and spans the whole region [(u, 24):u 2 CJU, u 2 O)(((u, 2.4):u I c3u, u I 0)) We have the following lemma by the above construction.
as c --* +co.
493
Quadratic conservation laws
/
/-
I
b;+v=O
Fig. 6.
Fig. I
LEMMA
2.6. The Riemann invariants w, satisfy *V2w+(r*, r+) 2 0
namely, w, (w_) are quasi-convex (respectively quasi-concave) in the sense of [16]. Proof.
We only prove V2w, (r+, r+) 2 0 when (16b3 + 9(1 - 2a)Q2 - 4(4b2 - 3(a - 2))3 > 0.
The rest can be done similarly.
(2.3)
494
YUNGUANG et al.
Lu
In our case, w, satisfies f(v,
aw aw u)L + g(v, 2.42 = 0 au at4
(2.4)
wheref(v, U) = bu + v, g(v, U) = -i((a - 1)~ + bv - d((a - 1)~ + bv)’ + 4(bu + v)‘). Differentiating (2.4) with respect to u
and multiplying by (aw+/av)2/f,
then
au
On the other hand, if we differentiate
(2.5)
f au
avauav
(2.4) with respect to v, it follows that
Then we multiply by (-l/f)(aw+/au)(aw+/av)
and, therefore,
a2w+aw+ aw++(f_!_?&=&$$%+~~?$%.
auav
au
(2.6)
av
Since one has
a2w+
v2w +(r 4-3r +) =
av2-z (
aw,
a2w+ aw, 2 aw a2w+ + -a2w+ -au au + au2 ( au > au a0 auav
2 -2_2+aw >
and aw+
a0 - -g
f aw+ au
adding (2.5) and (2.6), we get pw
+ (r
3 -- -i-+g-B)=-(g$ln,. +,+ (aw, au>( g g r
)
=
_
ffu
f2&
f"
fg"
From (err), (d,,) (lemma 2.1) and corollary 2.5, we have V2w+(r+, r,) 2 0. Because of the quasi-convexity
property, we can conclude with the following theorem.
THEOREM 2.7. If a > 1 + b2, (16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 + 0 and b f 0 or a L 1, b = 0. Assume that (v,,(x), U,,(X))E L”(R) * L”(R) (when
(16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 < 0, we assume further that v0 2 c3u0, v,, L 0 or v,, I c3u0, v0 5 0), then solutions (v&(x, r), W,
r))
of the Cauchy problem (1.5), (1.6) has an a priori L” estimate independent of E.
Quadratic
conservation
495
laws
Remarks.
(1) When a - 1 = b’, (1.2) can be reduced to a single conservation law, hence, the existence of a global weak solution can be obtained. (2) The cases we consider are of interest, because regions Zf (i = 1,2,3) contain the origin at which system (1.2) is not strictly hyperbolic. 3. EXISTENCE
AND
ASYMPTOTIC
BEHAVIORS
OF THE
VISCOUS
SOLUTIONS
In this section, we study the existence and asymptotic behaviors of the solutions to the Cauchy problem (1.5) and (1.6) for arbitrary a, b. The priori estimate that guarantees the global existence of a solution is obtained by the energy method (hence, they are dependent on E). We assume (v,(x), u,,(x)) E I+‘2S”(R)
(3.1)
(Q,(X), u,(x)) E H2(R)
(3.2)
Lim (V,(X), u&)) = (0,O). x-+00
(3.3)
Using the iteration method as in [lo], we have the following lemma. LEMMA 3.1. (Local existence theorem.) If the initial value satisfies (3.1), then for any fixed E > 0, the Cauchy problem (1.5) and (1.6) admits a unique smooth local solution (v, u) and (v, 24)satisfies
(3.4) where M(t,) is a positive constant that depends only on t, and I, on /Iu~//~, JJv,JJ,, the nonnegative integer i = 0, 1,2. LEMMA 3.2. (v, u) is the local solution obtained in lemma 3.1, if we assume further that the initial value satisfies (3.3), then
Lim (u, 2.4)= (0,O) x-+m
(3.5)
Lim (v,, u,) = (0,O) x+*4,
(3.6) (3.7)
uniformly in t E [0, ti]. Proof. We only prove (3.9, the rest can be done similarly.
Let ((v”, u”)) be the iteration sequence of the integral equation corresponding problem (1.5), (1.6). We have, as in [lo], that (un, u”) -+ (u, u)
(as n --t +co) uniformly in [0, ti] *R.
to the Cauchy
(3.8)
496
Lu
YUNCUANG
et al.
For any fixed IZ, using (3.3), we can prove easily Lim (v”(x, t), un(x, t)) = (0, 0) x-+*=
uniformly in t E [0, ti].
(3.9)
Since I@, 01 5 IV%, 0
+ IW, t) - 0X, 01
and lu(x, 01 5 IU%, t>l + IU%, t) - U(X,Ol from (3.8), (3.9), we have (3.5). LEMMA3.3. If the initial value satisfies (3.1)-(3.3), then the solution (u, u) obtained in lemma 3.1 satisfies f+\
rt, r+-
I
J--m
+
+oO
dx+c\‘\
r+m
Jo J--m
+ u&.&lx + ;(a2+b2
5
-ca
(u,“+u,‘)dwdt=
(3.10)
\ +(u,2+u,$Lx J--m
+ l)(llull; + ~~u~~~)
+-(u:;
+ &) dwdt.
-co
(3.12) Proof. (3.10)-(3.12) are standard energy inequalities. Using the estimates (3.4)-(3.7), with the method given in [17], we can immediately get (3.10)-(3.12).
From the estimates (3.10), (3.1 l), we can obtain the P-norm estimate that guarantees the existence of the global smooth solution for the Cauchy problem (1.5), (1.6). LEMMA3.4. If the conditions of lemma 3.3 are satisfied, then the solution obtained in lemma 3.1 satisfies
I14cm + ll4lm5 ~(IIUOllHI9 II~OIIH’, E) < +a* Proof. Since u2
+
u2
=
2
sx
(vu, + uu,) dx 5 2
--m
l/2
+m
(1 --m
(u2 + f42)du >
+m
(S --m
l/2
(u,” + u,“) d.X >
Quadratic
conservation
497
laws
and +O”(u2 + u2) dx 5 ~,dl~Oll2,
-ca
i
+moJ;
--co
+ 4w
5 ~2(ll~O*ll2~
Il~Oll,)
Il~Oxll2)
+ w(ll~Oll2,
bOll2,
M4l:
+
II43
If we set Y = llullt + Ilull:, then, we have Y 5 ~1(ll~Oll2~
Il~,ll2m42(11~,ll2~
II~Oxl12) + ~dll~Oll2,
ll~Ol12,dY)1’2
so
Y 5 ~dIuolLl9
II~OIIN~ 9 E) < +a.
Now we give the main theorem of this section. THEOREM3.5. If the initial value satisfies (3.1)-(3.3), then, the Cauchy problem (1.5), (1.6) has a unique global smooth solution (u, u) and (u, u) satisfies tL_imm (x tFEuf*R (IW +
01 + I@,
aI
= 0.
(3.13)
The former part of the theorem follows from lemmas 3.1 and 3.4. In order to prove the asymptotic behavior, we need the following result. LEMMA 3.6. For the continuous and differentiable IO+”If’(s)1 d.s L M, then Lim f(t) = 0.
function f(t) L 0, if SO+” f(s) & I M,
t-t+=
Proof. From
If@)-fOl = Ij:f’(y)dyl
5 ~~~If’(Y)b’~
and the absolutely continuity of the Lebesgue integral, we conclude that f(t) continuous. Using the hypothesis SO+” f(s) d.s I M, we have Lim f(t) = 0. t++a, Now we turn to prove the asymptotic behavior. From estimates (3.1 l), since 3 j?z (vi + u,“) dx is independent of ti , we get
is uniformly
(3.14) Similarly, from system (1.5), using integration by parts, the Cauchy inequality and estimates (3.10)-(3.12), we have
498
Lu YUNGu.i~Getal.
+s
Thus, from lemma 3.6 and estimates (3.14) and (3.15)
s +m
Lim I-+m
Lim I-+-
u;dx = 0, _-m
f&.x = 0. _-m
(3.16)
In addition, from estimate (3.10), we get +m (V2 + 2) dx I s
+m (u,” + U,“)dx.
(3.17)
i
so v2 =
* (~2)~~~I(i~~~2~~‘2~~~~~~~‘2 .I--oo
+ 0
as t*
+w.
Similarly u2 + 0
ast++oo.
Therefore, Lim f++m
SUP (Iu@, (x,t) ER *R,
01
+
lu(.G
01)
= 0.
Theorem 3.5 is proved.
Acknowledgement-The
authors
are grateful
to Professor
Ding Xiaxi for his encouragement
and guidance.
REFERENCES 1. SCHAEFFER D. G. & SHEARER M., The classification of 2 * 2 systems of nonstrictly hyperbolic conservation laws with application to oil recovery; appendix with D. Marchesin and P. .I. Paes-Leme, Communs pure appl. Math. 40, 141-178 (1987). 2. ISAACSON E., MARCHESION D., PLOHR B. & TEMPLE B., The classification of solutions of quadratic Riemann problem (I). MRC Report, University of Wisconsin (1985). 3. ISAACSON E. & TEMPLE B., The classification of solutions of quadratic Riemann problem (II). MRC Report, University of Wisconsin (1985). 4. YUNGUANG Lu & JINGHUA WANG, The interactions of elementary waves of nonstrictly hyperbolic system, J. math. Analysis Applic. 166(l), 136-169 (1992). 5. YUNGUANG Lu, Convergence of the viscosity method for a nonstrictly hyperbolic system, Acta math. Sci. 12(2), 230-239 (1992). 6. KAN P. T., On the Cauchy problem of a 2 * 2 system of nonstrictly hyperbolic conservation laws, Ph.D. thesis. Courant Institute (1989). 7. RUBINO B., On the vanishing viscosity approximation to the Cauchy problem for a 2 + 2 system of conservation laws (preprint) (1989). 8. LIU T. P., Nonlinear stability of shock waves for viscous conservation laws, Mem. Am. math. Sot. 32B, I-108 (198%. stability of rarefaction waves for 2 * 2 viscous hyperbolic conservation laws, J. dty$ 9. .?HO&ING XIN, Asymptotic Eqns 73, 45-77 (1987). IO. DING X. X. & WANG J. H., Global solutions for a semilinear parabolic system, Acta math. Sci. 3, 397-414 (1983). 11. YUNGUANG Lu, Asymptotic behavior of solutions of initial problems to systems of gas dynamics with viscous term, Chin. Sci. Bull. 34, 7-11 (1989). 12. YUNGUANG Lu, The existence and asymptotic behavior of solutions to gas dynamics system with viscosity E # p, Chin. Sci. Bull. 35, 785-786 (1990). 13. YUNGUANG Lu & GUILIANG XIE, The asymptotic behavior of solutions of initial value problem to system of gas dynamics with viscosity, Acta math. Sci. 8, 185-198 (1988).
Quadratic
conservation
laws
499
14. YUNGUANGLu, Existence and asymptotic behavior of solutions to inhomogeneous equations of elasticity with little viscosity, Nonlinear Analysis 16, 199-207 (1991). 15. YUNCUANG Lu & HUIJIANC ZHAO, Existence and asymptotic behavior of solutions to 2 * 2 hyperbolic conservation laws with small viscosity, Nonlinear Analysis 17(2), 169-180 (1991). 16. CHUEH K. N., CONLEY C. C. & SMOLLER J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. math. J. 26, 373-392 (1977). 17. SMOLLER J., Shock Waves and Diffusion Equation. Springer, New York (1983).
Methods
Vol. 21, No. 7, pp. 485-499,
& Applications,
1993. 0
0362-546X/93 S6.a) + .OQ 1993 Pergamon Press Ltd
VISCOUS
SOLUTIONS OF QUADRATIC CONSERVATION WITH UMBILIC POINTS
Wuhan Institute
of Mathematical
LTJ YUNGUANG,
ZHU CHANGJIANG
Sciences,
Academia
Sinica,
and
LAWS
ZHAO HUIJIANG
Wuhan,
430071,
People’s
Republic
(Received 10 January 1991; received in revised form 4 August 1992; received for publication Key words and phrases: Riemann invariants, uniformly bounded estimates, viscous regions, quasi-convexity, energy method, existence and asymptotic behavior.
of China
1 March 1993)
solutions,
invariant
1. INTRODUCTION THE GENERAL 2 * 2 SyStCm Of COmXVatiOn II, + &a,
kiWS
flUX fUIICtiOnS
with quadratic
u2 + 2bl uu + Cl uqx = 0
t u, + +(a,~’
+
(1.1)
2b,uu + c2 02)X = 0
is of interest because solutions of (1.1) may approximate solutions of a general 2 * 2 system of conservation laws in a neighborhood of an isolated point at which strict hyperbolicity fails. In [l] Schaeffer and Shearer show that when system (1 .l) is hyperbolic, there is a nonsingular linear change of dependent variables which transforms system (1.1) into the normal form u, + +(bu2 + 2uv), = 0 (1.2)
i u, + +(au2 + 2buv + u~)~ = 0. Let F be the mapping from R2 into R2 defined by F: (u, u) + (+(bu2 + 2uu), &u’
+ 2buu + u2))
then two eigenvalues of dF are ~_ = ((a - 1)u + bu) f d((a + ? with corresponding
1)u +
bu)2 + 4(bu + tQ2 (1.3)
right and left eigenvectors
rT =
bu + u,
I, =
bu + u,
-((a -
1)u +
bu) T ~/((a - 1)u + bu)2 + 4(bu + u)’ = 2 >
-((a -
1)u +
bu) T t&a 2
1)u +
bu)2 + 4(bu + u)~ >-
By simple calculation, we have VA, * rT = (bu + u) ; (a + 3) f
(3 - a)(@ d((a 485
1)u +
1)u +
bu) T 3b(bu + u)
bu)2 + 4(bu + u)~ 1 ’
(1.4)
486
Lu YUNGUANG
etal.
Therefore, it follows from (1.3) that A+ = A_ at points satisfying (a - 1)~ + bv = 0 and bu + u = 0 at which strictly hyperbolicity fails to hold. That the first and the second characteristic fields are linearly degenerate on bu + v = 0 follows from (1.4). As a model of a nonstrictly hyperbolic system, (1.2) has been studied by many scholars, especially for the symmetric case (i.e. b = 0). The Riemann problem for the system (1.2) is first considered in [2, 31 when b = 0, a E (1,2) or a E (2, +m), and the interactions of elementary waves are given in [4] when b = 0, a > 2. Recently, using the vanishing viscosity method and theory of compensated compactness, Yunguang Lu [5] and Kan [6] obtained the global weak solutions to system (1.2) when b = 0, a = 3. (Rubino [7] also studied the global weak solutions to system (1.2) when b = 0, a > 2, but his work seems incomplete.) In their work, in order to use the theory of compensated compactness, a fundamental step is to obtain the L--priori estimates independent of the viscosity coefficients on the viscous solutions. So it is of interest to consider the P-priori estimates on the viscous solutions. In this paper, first, we study the uniformly bounded estimates on the solution to the following parabolic system u, + &bU2 + 2UV), = &U** (1.5)
U, + *(au2 + 2buu + vqx = EU, with initial value
(1.6)
(v, @It=0 = (Ql(x), %(X)).
When a, b satisfy a - 1 1 b2, (16b3 + 9( 1 - 2a)b)2 # 4(4b2 - 3(a - 2))3 and b # 0 or a 1 1, b = 0, we get the uniformly bounded estimates (which are independent of a) on the solutions of the Cauchy problem (1.5), (1.6). On the other hand, there have been many papers [8, 91 about the study of solutions to strictly hyperbolic systems of two conservation laws, with viscosity, especially to the gas dynamics system [lo-131 and elasticity system [14], but it is more difficult and important for systems of nonstrictly hyperbolic systems. Since both nonstrictly hyperbolic and linearly degenerate points occur in our model, we also consider the existence and asymptotic behaviors of solutions to the Cauchy problem (1.5), (1.6) for arbitrary constants a, 6. The program of this paper is as follows: in Section 2, for a - 1 L b2, (16b3 + 9(1 - 2~)b)~ # 4(4b2 - 3(a - 2))3 and b # 0 or a 1 1, b = 0, we study the existence of the invariant domain and obtain the L--priori estimates on the solutions of the Cauchy problem (1.5) and (1.6); in Section 3, we consider the existence and asymptotic behaviors of the solutions to the Cauchy problem (1.5) and (1.6) for arbitrary constants a, b. It is worth pointing out that when b = 0, our results coincide with that of [15], hence, our work is an extension of [15]. 2. INVARIANT
REGIONS
AND
THE
UNIFORMLY
BOUNDED
ESTIMATES
The main purpose of this section is to obtain the L”-priori estimates, independent of E, to system (1.5) with initial value (1.6). In order to establish such a priori estimates, we shall use the theory of invariant regions due to Chueh et al. [16]. The result of [16] can be summarized in the following theorem.
Quadratic conservation laws
487
THEOREM.Let gT be two smooth functions, g,: R ’ + R and C = ((v, u): g,(u, u) 5 0). Assume that for any t > 0 and (13,fi) E 8X, the following conditions hold: (a) Vgr is a left eigenvector of VF(D, ii); (b) gr is quasi-convex in (0, ii), i.e. for all r E R2, < * VgT = 0 * V2g7(<, <) 2 0 then X is an invariant region for (1.5) for all E > 0. Namely, if (v,, uO) E E for all X, then (V&(X,t), u&(x, t)) E E:, for all (x, t). We say that w_ E C’(w+, respectively) is a first (second) Riemann invariant for (1.2) if for all (u, u) (Vw_(v, UK r-) = 0 Therefore, Further, tively, the rarefaction
((Vw+(v, UK r,) = 0).
Vw,//l,. let us recall that the integral curves R, of rT in the state space are called, respecfirst and second rarefaction wave curves. Thus, in our case, a first (second) wave curve satisfies the ordinary first order differential equation du dv=
((a - 1)~ + bu) f d((a - 1)~ + bv)* + 4(bu + v)* 2(bu + v)
(2.1)
From (2.1), we can conclude that R_, R, are symmetric on the origin. We can, hence, restrict our attention to the R, family only. Now we turn to analyze the properties of the R, curves. If a, b belong to region IV (i.e. (16b3 + 9(1 - 2a)b)* - 4(4b* - 3(a - 2))3 > 0, see [2]), then we have the following lemma. LEMMA2.1. Under the above hypothesis, if u 2 uO, then R+(v,, u,)-the R, curve through (u,,, u,J (where (u,,, uO) satisfy: bu, + u,, = 0, u. 2 0) has the following properties (see Fig. 1): (a) the R, curve R+(q,, u,,) contains only three parts: except the half line ((u, u): bu + u = 0, u 2 01, one is in the half space bu + u 2 0, the other is in bu + u I 0; (b) the R+(u,, u,,) curves are in one to one correspondence with the points of the half line ((u, u): bu + u = 0, u L 01;
bu+v=O
Fig. 1.
(*p*z LIE~~O~O3aas
‘ b ‘ ELI‘?I 30 suo!~~u~3apay$ _IOJ) *Llaapz)adsal
‘(0 7 n :(n ‘0)) u (0 7 nkJ - R :(n ‘0)) u (0 5 rzEJ - n :(n ‘0)) = ,I\1 ‘(0 7 n :(n ‘n)) u {O7 ?zEcJ- n :(n ‘n)) u (0 5 R + nq :(n ‘0)) = ,111 ‘(0 7 n :(n 72)) u (0 5 Yzz3- n put? 0 7 R + nq :(n ‘n)] = ,I1 ‘(0 5 nb - R :(n ‘(2))n {O7 nz3 - R :(n ‘0)) = ,I 01 ssuolaq yrea ‘s$wd .uIo3 syeu~o~ sa&InD+v aqy 30 auo ya?za (I?) :(Z .f$a) sayado.Id OU~MOIIOJ aql aneq saMn3 +y aql ‘sysayloddq aAoqt? ayl lapun ‘Z’Z vvuwa~ w_w.ua~%y~ol103 ayi aAEy ah4 uaql ‘(0 > ,((Z - @E - ,4P)P - ,(S@Z - I)6 + Eq91) put2 zq < 1 - D *a*!)111uoi%al 01 FiuoIaq q 9 31 ‘0 + R SB 0 + n ~3spw (0 5 R + nq) 0 z R + nq uf pauyuo3 saAln3 +g ayl (a) f(0 > R ‘0 = R + nq :(n ‘0)) awl 3p2y aq4 ~I!M lDaslalu! IOU saop (On‘OR)+~ saam3 +v ayl 30 au0 yea (“Ip) too- + R SE w+ e n (“‘p) fxaauo3 s! (On‘%)+y saA.nv +x aql30 au0 y2ea (“p) !sast?alDu! OCI5 R s12sasEal3ap (On‘On)+8 saA.uv +g ayl 30 au0 y3sa (*p) :sa!yadold S!UIMOIIOJ ayl s~q 0 5 n + nq u! (On ‘0n)+a aAln3 ayL (p) (*p*Zb.11)?110~03 aas ‘3 30 uoyw3ap ayl 10~) ‘(0 < n ‘rz13 = n :(n ‘rz)] lDaslalu! IOU saop (On‘%)+g sar\.uw +x ayl 30 au0 qxa (A13)
awl 31y
aql
qq~
fco+
t
n SE 03+
c
n (III,)
fxaAuo3 sy (On‘On)+y sa.4ln3 +u aql30 au0 y3va (“3) fsasaalaur O0 7 ci se sastza.cw! (On‘%)+y sakuv +x ayl30 au0 ycwa (5) :sapladold %I~MO~~O~ aql sey 0 z R + nq u! (On ‘On) +y a&In3 ayl (3)
JO U8!S 3q;L
‘lUE?JSUO3 E Sy (I ‘0)
3 I3 al3yM
‘(0
7
YZ) II I3 =
0 ‘(0
7
0) 0 =
f-4 +
?lq
:SjOO.l I&?al
Pue SvY 0 = (n ‘fl)V ‘JW ‘0 < ,((Z- o)E- ,4P)P- ,(S(nZ - I)6+ $q91)31 (E) ‘&l + nq)p + @q + n(r - a>>pn + mz(E - 0) + pqz - pq = (n ‘n)y ‘p’z nxv-IToxo~
OMI SW LlUO
WOOJ pad w~gs!p aa.yl sey 0 = b + Xd + EL uopEnba uayl ‘0 > &Id) + &Z/b)31 (q) ‘100.1pzal auo lip.10sey 0 = B + rCd+ & uoyenba uayl ‘0 < &f/d) + ,(bf) 31 (e) ‘fez VYYPEI~ *sllnsaJ %U!MO~~OJ ayl paau afi ‘z-z pur? 1*z swutual allold 01 lap10 UI TI$QIO aql u! IdaDxa n ? = R 10 () = 0 + nq saug I#!M pas.Ialuy IOUsaop Sahln3 +g ayl30 au0 yDE?a(“‘p) too-- t R se CO+ + n ‘0 c n se () c n (“‘p) fxaAuo3 s! saAln3 +g ayl30 au0 yzea (up) fsasealzy 0 5 n se sasea.I3ap saam3 +g aql30 au0 yDea (Ip) :sa!yadold Ouy~ol~o3 ayl aAt?y ,111 UO!%aJ u! Sahln3 +v ayl (p) !uyS~~oaqi uy IdaDxa (nap = n) 0 = n ~0 (n 9 = n) n Z3 = n sag aql ~JIM y3aslalu! 10u saop saAln3 +g ayl30 au0 ycea (h) f(cO- + n) oo+ t R St? oo+ + n ‘0 + 0 SE 0 + n (Yl) laAe3uo3 s! saAln3 +g ayl30 au0 qcea (“3) fsasr?a.wy (0 5 0) 0 z R se (sasEarDap) sastza.w_ysalz.rn3 +8 ayl30 au0 yst?a (‘3) :sagladoId %U~MO~IO~ aq$ ahzy (,AI) ,II uoy8al III sahln3 +x aql (3) to + n se 0 + n L3spes (0 5: R + nq) 0 z n + nq uy pauyuo3 saam3 +x ayl (Iuq) 1(() < n ‘n? = R :(?I ‘0)) 10 (0 < n ‘n? = n :(n ‘0)) saug 31~29 ~J!M 13aslaly IOU saop (On‘“n)+~ sakrn3 +z aql 30 au0 yrea (“9) f 1-z w.n.ua~ uy (lllp)-(Ip) ‘(W)-(3) ‘(q) ‘(e) 30 wq$ 01 ~e~y.uyssa!yadold ay$ w?y (On‘On)+8 sah.uw +y ayl 30 auo yDt?a (‘9) :sapladold Ouy~o~~o3 ay$ aAeey(On z n :(?I ‘0)) u ,I uor?ial u! (0 z (4-1‘0 = Ocz+ Onq A3sges (On‘On) alayM) (On‘On) y?inoryl saAJn3 +y ayl-(On ‘ On)+8 (q) 68P
‘p?~~g ale Isal ayL
‘5
‘%I
0=&Z-m-Z9P)tz(9(“Z-I)6+F991)
zzm + xl
=
$
‘(Iup) put? (up) ‘(1%) ‘(I$) aAold 01 paau Quo aM *I-Z wuruq Jo Joodd
(‘0 < 4 uoy~puo~ ayl lapun are p-Z d.re~lo~o~ put2 p-1 Sam%& *3oold OU~MO~[O~ aql u! 0 < 4 asE3 aql JapysuoD Quo aM ‘[gr] uy paJapyuo2 uaaq st?y 0 = 4 ase3 ayl axus ‘0 z 9 ‘dyyxaua8 30 SSOIInoyyM ‘azunsse 6txu ah ‘(s %d) spz-w ay3 uo 3p~aunuLs an2 01 Ouolaq cj ‘w y~yl suor8al ayl axiS *ICBM atuBs aqy us paAold aq u’t?3Z’Z sunua[ ‘1 ‘Z sun.uaI aAold Quo aM ‘Z’Z put! 1‘Z srxnua~ aAOld 01 ulna aM MON ‘p *8y u! paD!dap sy (n ‘n)y 30 uSis ayL ‘slue~suo~ are (fA(w- Z)f + zwp + qz)- ‘c-1 3 ?J ‘(q- ‘f/((Z + w-)f + zqvp + qz)-) 3 E3 ‘(1 ‘0) 3 z3 ‘(0 7 n) nb = 0 ‘(0 7 n) nE3 = n ‘(0 7 n) nZ3 = R ‘(0 7 n) 0 = n + nq :qooJ pal Ino3 AIuo Pue seq 0 = (n ‘ah uaql ‘0 > ,((Z - w)f - ,4P)P - ,(S(wZ - 06 + ,991) 31 (9)
aJaw
W
‘P
‘$!d
06P
Quadratic
conservation
491
laws
where Y = (a - 1)~ + bu, Z = bu + u, X = n, in order to study the convexity of the R, curves we compute the second derivative. Through tedious but simple computation, we get g
= (a - 1 - P)(X + Y)h(u,
u)/4Z3X.
Under the condition (16b3 + 9(1 - 2a)6)2 - 4(4b2 - 3(a - 2))3 > 0, from corollary 2.4, we can conclude that each one of the R, curves R+(u,, u,,) has at most two turning points, which are on the half line {(u, u): u = ciu, u 2 0) or ((u, u): bu + u = 0, u r 0), respectively. Note that u = ciu (u 2 0) itself is an integral curve of du/du = (X + Y)/2Z. Hence, the theory of ordinary differential equations (the uniqueness of the solutions) indicates that each one of the R, curves R+(u,, u,,) on bu + u > 0 does not intersect with the half line ((u, u): u = c1 u, u 2 0). The above facts and (d,,) deduce that (cn) and (d,,) are true. As to (cm), from the monotonicity, Lim u = M exists, we assume M < +oo, otherwise, v-2t43
is proved. Furthermore, Then we have
(cIII)
we have A4 > uO.
du (a - l)M + 2bu, + d((a - l)M + 2bu,)2 + 4(bA4 + 2u,)2 = d > o 1 9 z W0.M) = 2(bM + 2u,) since for all u 2 2u, du
du
=d,>O
-du > G (2V~.lw we have u - A4 2 d,(u - 2v,) which indicates u -+ +oo as u + +03, (cm) is proved. (din) can be proved in the same way. Finally, for (1.2), we can construct Riemann invariants w_, W, so that W_(U,u) 5 0 I W+(U,2.4). Since wy is constant along every RT curve, if we prescribe w,(u,, -u,/b) follows: Case 1. (16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 > 0 w-(uo 9 -u,/b)
=
w+(uo 9 -u,/b)
=
QJ, uo < 0 1 0, u#JL 0 uo, uo > 0 i 0, ug I 0.
Case 2. (16b3 + 9(1 - 2~)b)~ - 4(4b2 - 3(a - 2))3 < 0;
in region I’ w-(uo 9 -u,/b)
=
w+(uo 9 -Q/b)
=
uo, UIJ< 0 i 0, ug 2 0 &I I uo > 0 I 0, ug I 0;
(2.2) or w,(O, 0) as
492
Lu
YuNGUANG et al.
in region II’, IV’ w-(0,0)
= w_(uo, -U,/@
w+(O, 0) = w+(vo, -v,/b) and let w_(u, u), w+(v, U) satisfy &,/au
> 0;
in region III’ w-(0,0) = w_(u,, -u,/b)
w+(O,0) = w+(vo,-v,/b) and let w_(u, u), w+(u, U) satisfy c?w,/& < 0. Furthermore,
we set
w-l((“,.):“=.,.,.so,= w-l,(“,.):.=o,.50) = -co, then w_(v, U) 5 0 I w+(u, U). From the construction
of the Riemann invariants, we also have the following corollary.
2.5. The Riemann invariants constructed as before satisfy: (a) if (16b3 + 9(1 - 2a)b)’ - 4(4b2 - 3(a - 2))3 > 0, then C~W+/C~U < 0; (b) if (16b3 + 9(1 - 2a)b)’ - 4(4b2 - 3(a - 2))3 < 0, then aw,/au < 0 in region I’ or III’, a~,/& > 0 in region II’ or IV’. COROLLARY
From now on, we shall use the Riemann invariants w_, w, provided by the previous construction. Since by definition Vw7 are left eigenvectors for VF, the Riemann invariants are the functions used to define the invariant regions following [16]. Therefore, in this way we will find a family of invariant regions (Figs 6 and 7 corresponding to lemmas 2.1 and 2.2, respectively) given by XC = (w_ + c 2 01 n (w, - c 5 01,
c > 0.
These regions Xi, C,“, Ci (see Figs 6 and 7) are bounded by the wave curves of the two families. On the other hand, CJ strictly increases in c and spans the whole state space as c -+ 00. Cz (Ef) strictly increases in c and spans the whole region [(u, 24):u 2 CJU, u 2 O)(((u, 2.4):u I c3u, u I 0)) We have the following lemma by the above construction.
as c --* +co.
493
Quadratic conservation laws
/
/-
I
b;+v=O
Fig. 6.
Fig. I
LEMMA
2.6. The Riemann invariants w, satisfy *V2w+(r*, r+) 2 0
namely, w, (w_) are quasi-convex (respectively quasi-concave) in the sense of [16]. Proof.
We only prove V2w, (r+, r+) 2 0 when (16b3 + 9(1 - 2a)Q2 - 4(4b2 - 3(a - 2))3 > 0.
The rest can be done similarly.
(2.3)
494
YUNGUANG et al.
Lu
In our case, w, satisfies f(v,
aw aw u)L + g(v, 2.42 = 0 au at4
(2.4)
wheref(v, U) = bu + v, g(v, U) = -i((a - 1)~ + bv - d((a - 1)~ + bv)’ + 4(bu + v)‘). Differentiating (2.4) with respect to u
and multiplying by (aw+/av)2/f,
then
au
On the other hand, if we differentiate
(2.5)
f au
avauav
(2.4) with respect to v, it follows that
Then we multiply by (-l/f)(aw+/au)(aw+/av)
and, therefore,
a2w+aw+ aw++(f_!_?&=&$$%+~~?$%.
auav
au
(2.6)
av
Since one has
a2w+
v2w +(r 4-3r +) =
av2-z (
aw,
a2w+ aw, 2 aw a2w+ + -a2w+ -au au + au2 ( au > au a0 auav
2 -2_2+aw >
and aw+
a0 - -g
f aw+ au
adding (2.5) and (2.6), we get pw
+ (r
3 -- -i-+g-B)=-(g$ln,. +,+ (aw, au>( g g r
)
=
_
ffu
f2&
f"
fg"
From (err), (d,,) (lemma 2.1) and corollary 2.5, we have V2w+(r+, r,) 2 0. Because of the quasi-convexity
property, we can conclude with the following theorem.
THEOREM 2.7. If a > 1 + b2, (16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 + 0 and b f 0 or a L 1, b = 0. Assume that (v,,(x), U,,(X))E L”(R) * L”(R) (when
(16b3 + 9(1 - 2a)b)2 - 4(4b2 - 3(a - 2))3 < 0, we assume further that v0 2 c3u0, v,, L 0 or v,, I c3u0, v0 5 0), then solutions (v&(x, r), W,
r))
of the Cauchy problem (1.5), (1.6) has an a priori L” estimate independent of E.
Quadratic
conservation
495
laws
Remarks.
(1) When a - 1 = b’, (1.2) can be reduced to a single conservation law, hence, the existence of a global weak solution can be obtained. (2) The cases we consider are of interest, because regions Zf (i = 1,2,3) contain the origin at which system (1.2) is not strictly hyperbolic. 3. EXISTENCE
AND
ASYMPTOTIC
BEHAVIORS
OF THE
VISCOUS
SOLUTIONS
In this section, we study the existence and asymptotic behaviors of the solutions to the Cauchy problem (1.5) and (1.6) for arbitrary a, b. The priori estimate that guarantees the global existence of a solution is obtained by the energy method (hence, they are dependent on E). We assume (v,(x), u,,(x)) E I+‘2S”(R)
(3.1)
(Q,(X), u,(x)) E H2(R)
(3.2)
Lim (V,(X), u&)) = (0,O). x-+00
(3.3)
Using the iteration method as in [lo], we have the following lemma. LEMMA 3.1. (Local existence theorem.) If the initial value satisfies (3.1), then for any fixed E > 0, the Cauchy problem (1.5) and (1.6) admits a unique smooth local solution (v, u) and (v, 24)satisfies
(3.4) where M(t,) is a positive constant that depends only on t, and I, on /Iu~//~, JJv,JJ,, the nonnegative integer i = 0, 1,2. LEMMA 3.2. (v, u) is the local solution obtained in lemma 3.1, if we assume further that the initial value satisfies (3.3), then
Lim (u, 2.4)= (0,O) x-+m
(3.5)
Lim (v,, u,) = (0,O) x+*4,
(3.6) (3.7)
uniformly in t E [0, ti]. Proof. We only prove (3.9, the rest can be done similarly.
Let ((v”, u”)) be the iteration sequence of the integral equation corresponding problem (1.5), (1.6). We have, as in [lo], that (un, u”) -+ (u, u)
(as n --t +co) uniformly in [0, ti] *R.
to the Cauchy
(3.8)
496
Lu
YUNCUANG
et al.
For any fixed IZ, using (3.3), we can prove easily Lim (v”(x, t), un(x, t)) = (0, 0) x-+*=
uniformly in t E [0, ti].
(3.9)
Since I@, 01 5 IV%, 0
+ IW, t) - 0X, 01
and lu(x, 01 5 IU%, t>l + IU%, t) - U(X,Ol from (3.8), (3.9), we have (3.5). LEMMA3.3. If the initial value satisfies (3.1)-(3.3), then the solution (u, u) obtained in lemma 3.1 satisfies f+\
rt, r+-
I
J--m
+
+oO
dx+c\‘\
r+m
Jo J--m
+ u&.&lx + ;(a2+b2
5
-ca
(u,“+u,‘)dwdt=
(3.10)
\ +(u,2+u,$Lx J--m
+ l)(llull; + ~~u~~~)
+-(u:;
+ &) dwdt.
-co
(3.12) Proof. (3.10)-(3.12) are standard energy inequalities. Using the estimates (3.4)-(3.7), with the method given in [17], we can immediately get (3.10)-(3.12).
From the estimates (3.10), (3.1 l), we can obtain the P-norm estimate that guarantees the existence of the global smooth solution for the Cauchy problem (1.5), (1.6). LEMMA3.4. If the conditions of lemma 3.3 are satisfied, then the solution obtained in lemma 3.1 satisfies
I14cm + ll4lm5 ~(IIUOllHI9 II~OIIH’, E) < +a* Proof. Since u2
+
u2
=
2
sx
(vu, + uu,) dx 5 2
--m
l/2
+m
(1 --m
(u2 + f42)du >
+m
(S --m
l/2
(u,” + u,“) d.X >
Quadratic
conservation
497
laws
and +O”(u2 + u2) dx 5 ~,dl~Oll2,
-ca
i
+moJ;
--co
+ 4w
5 ~2(ll~O*ll2~
Il~Oll,)
Il~Oxll2)
+ w(ll~Oll2,
bOll2,
M4l:
+
II43
If we set Y = llullt + Ilull:, then, we have Y 5 ~1(ll~Oll2~
Il~,ll2m42(11~,ll2~
II~Oxl12) + ~dll~Oll2,
ll~Ol12,dY)1’2
so
Y 5 ~dIuolLl9
II~OIIN~ 9 E) < +a.
Now we give the main theorem of this section. THEOREM3.5. If the initial value satisfies (3.1)-(3.3), then, the Cauchy problem (1.5), (1.6) has a unique global smooth solution (u, u) and (u, u) satisfies tL_imm (x tFEuf*R (IW +
01 + I@,
aI
= 0.
(3.13)
The former part of the theorem follows from lemmas 3.1 and 3.4. In order to prove the asymptotic behavior, we need the following result. LEMMA 3.6. For the continuous and differentiable IO+”If’(s)1 d.s L M, then Lim f(t) = 0.
function f(t) L 0, if SO+” f(s) & I M,
t-t+=
Proof. From
If@)-fOl = Ij:f’(y)dyl
5 ~~~If’(Y)b’~
and the absolutely continuity of the Lebesgue integral, we conclude that f(t) continuous. Using the hypothesis SO+” f(s) d.s I M, we have Lim f(t) = 0. t++a, Now we turn to prove the asymptotic behavior. From estimates (3.1 l), since 3 j?z (vi + u,“) dx is independent of ti , we get
is uniformly
(3.14) Similarly, from system (1.5), using integration by parts, the Cauchy inequality and estimates (3.10)-(3.12), we have
498
Lu YUNGu.i~Getal.
+s
Thus, from lemma 3.6 and estimates (3.14) and (3.15)
s +m
Lim I-+m
Lim I-+-
u;dx = 0, _-m
f&.x = 0. _-m
(3.16)
In addition, from estimate (3.10), we get +m (V2 + 2) dx I s
+m (u,” + U,“)dx.
(3.17)
i
so v2 =
* (~2)~~~I(i~~~2~~‘2~~~~~~~‘2 .I--oo
+ 0
as t*
+w.
Similarly u2 + 0
ast++oo.
Therefore, Lim f++m
SUP (Iu@, (x,t) ER *R,
01
+
lu(.G
01)
= 0.
Theorem 3.5 is proved.
Acknowledgement-The
authors
are grateful
to Professor
Ding Xiaxi for his encouragement
and guidance.
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conservation
laws
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14. YUNGUANGLu, Existence and asymptotic behavior of solutions to inhomogeneous equations of elasticity with little viscosity, Nonlinear Analysis 16, 199-207 (1991). 15. YUNCUANG Lu & HUIJIANC ZHAO, Existence and asymptotic behavior of solutions to 2 * 2 hyperbolic conservation laws with small viscosity, Nonlinear Analysis 17(2), 169-180 (1991). 16. CHUEH K. N., CONLEY C. C. & SMOLLER J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. math. J. 26, 373-392 (1977). 17. SMOLLER J., Shock Waves and Diffusion Equation. Springer, New York (1983).