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Global BV Solutions to a p-System with Relaxation
Journal of Differential Equations 162, 174198 (2000) doi:10.1006jdeq.1999.3697, available online at http:www.idealibrary.com on
Global BV Solutions to a p-System with Relaxation Tao Luo Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong and Institute of Mathematics, Academia Sinica, Beijing, 100080 People's Republic of China
Roberto Natalini Istituto per le Applicazioni del Calcolo ``M. Picone'', Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I-00161 Roma, Italia
and Tong Yang Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received September 23, 1998; revised May 3, 1999
In this paper, we study the global existence of BV solutions to a p-system with a relaxation source term using the Glimm's scheme. The pressure p is given by a #-law with #=1. By a suitable choice of the measure for the strength of the shock waves, we show that the total strength of the waves and the total variation of the solutions are uniformly bounded with respect to the relaxation parameter. Furthermore, when the relaxation parameter tends to zero, we show that the sequence of BV solutions eventually converges to a weak solution to the equilibrium equation. 2000 Academic Press
1. INTRODUCTION Let us consider the following p-system with relaxation v t &u x =0, 1 u t + p(v) x = ( f (v)&u), = 174 0022-039600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
(1.1)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
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for (x, t) # R_(0, + ), p$(v)<0 and where =>0 is a relaxation parameter. We are interested in the Cauchy problem for (1.1) with initial data v(x, 0)=v 0 (x),
u(x, 0)=u 0 (x)
x # R.
(1.2)
According to the discussion in [26, 45] and in order to have stable wave patterns, we assume the sub-characteristic condition, which in the present case reads | f $(v)| - & p$(v),
(1.3)
for all v under consideration. Under this condition we expect that, as = tends to zero, the solutions to problem (1.1)(1.2) are well described by the equilibrium equation given by the scalar conservation law v t & f (v) x =0.
(1.4)
To put this problem in the right perspective, let us recall that relaxation phenomena arise in a number of different contexts. A general survey of the recent issues in the theory of relaxation hyperbolic problems can be found in [33]. Here we just mention the models for gases not in thermodynamic equilibrium and extended thermodynamics [8, 43, 30], the kinetic theories [7], and more general wave models, including chromatography, viscoelasticity with vanishing memory, phase transitions with small transition time, river flows and traffic flows [36, 45, 39, 42]. More recently the study of various classes of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws [20, 12, 3, 4]. The basic linear theory and results for local smooth solutions can be found in [45, 47, 17]. The nonlinear theory essentially starts with the fundamental paper by T.-P. Liu [26] for the general 2_2 system t v+ x h(v, u)=0, 1 t u+ x g(v, u)= ( f (v)&u). =
(1.5)
Here v, u are both scalars and we assume that the system is strictly hyperbolic with two real eigenvalues * 1 <* 2 . The (formal) reduced equation is given by the scalar law t v+ x f *(v)=0,
(1.6)
where f *(v)=h(v, f (v)). The equilibrium speed is **(v)=h v (v, f (v))+ f $(v) h u (v, f (v)). In that paper nonlinear stability for (smooth) expansion
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and traveling waves was established. The main stability criterion was the sub-characteristic conditiona sort of causality principle: the speed ** of the equilibrium equation has to be included between the two speeds * 1 and * 2 of the full relaxing system. In the present case this yields condition (1.3). However, it is well known that in the generic situation solutions become discontinuous in finite time. The rigorous investigation of the relaxation limit for weak solutions was first studied by T.-P. Liu and his collaborators in [10] and [9], using the methods of compensated compactness [40]. Other results by the same methods have been given in [23, 22, 11, 42]. For 2_2 strictly hyperbolic systems and under the sub-characteristic condition, uniformly bounded sequences of solutions to relaxation systems converge almost everywhere, as = Ä 0 +, to some weak equilibrium solutions of the related scalar conservation law. More recently, D. Serre [38] proved, still by using the methods of compensated compactness, the convergence of the (semi-linear) JinXin's relaxation approximation [20] and some other discrete kinetic approximations to (one dimensional) genuinely nonlinear hyperbolic systems of conservation laws having a positively invariant domain. Let us recall that a different approach was considered in [5], where the averaging lemmas of the kinetic theory [16] have been used to investigate a special class of hyperbolic systems with relaxation having an equivalent kinetic formulation. Unfortunately the techniques involved with the above results seem to be not effective to deal with more general and more realistic situations. Actually, even without the singular perturbation term, the theory for global weak solutions to n_n hyperbolic systems of conservation laws (n 2) is mainly developed for BV solutions [15, 6], but for the moment very few results have been obtained in this framework for relaxation problems. Actually, all the existing results which use strong convergence techniques (i.e.: the FrechetKolmogorov's compactness theorem), have been obtained only when the limit equation is scalar and the approximating system is semi-linear or weakly coupled. In particular let us recall the results on the convergence of the JinXin's relaxation approximation to conservation laws and related models, see [41, 21, 31, 32, 44]. As far as we know, before the present paper, uniform BV estimates for strongly coupled systems with relaxation were proved only for the simple model in viscoelasticity considered in [27, 48]. Notice that, for systems with relaxation source term, including system (1.1), even the global existence of BV solutions for a fixed value of = needs to be established. In order to explain this point let us consider a quasilinear n_n system with a source term t U+ x F (U)=G(x, U),
(1.7)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
177
with U # R n, under the condition that matrix DF (u) has real, distinct eigenvalues. In general, the presence of a source term distorts the pattern of elementary wave interactions and induces amplification of waves. As a result of this process the total variation of the solution will generally increases with time and thus eventually leave the range of applicability of the Glimm's scheme. However, for source terms coming from geometrical or physical effects, we can expect the global existence of BV solutions, as for example for the quite different case considered in [25], where global existence of BV solutions to the Cauchy problem for (1.7) was established when the L 1 norms of G(x, U) and G U (x, U) are small and the source term have compact support in the x variable. Up to now the natural tool for more general source terms, including the relaxation case, has been the fractional step version of the Glimm's scheme, as used by C. Dafermos and L. Hsiao in [14]. Notice that, to prevent instability, one has to assume some dissipativity conditions on the source term G. In Theorem 2 of [14] the condition of total dissipativity of the source term was taken in account to prove the existence of global solutions, provided that the initial data are confined in a small neighborhood of an equilibrium state U and have small total variation. In that case the total variation of U decays to U exponentially fast as t Ä , see also [2]. Unfortunately it is easy to verify that for system (1.1), this dissipativity condition is not verified (even under the sub-characteristic condition, see [33]) and we cannot apply these results. Actually, as far as one is concerned with global existence of solutions for a fixed value of =, it is possible in principle to argue as in [13], by adopting a clever change of variable, at least for the case of the p-system (1.1) with f #0. Unfortunately, as noticed in [33], the estimates which are obtained by these methods are in general not uniform in the relaxation parameter =. The aim of the present paper is to present a direct approach by studying the wave interaction for system (1.1) when the pressure p is given by the relation p(v)=
a2 , v
(1.8)
for some a>0 and every v>0. What we show is that, although the relaxation term is only partially dissipative, the special structure of the system and the source term enable us to obtain the global existence of the BV solutions by using the fractional step version of Glimm's scheme and under the sub-characteristics condition (1.3), which now reads a | f $(v)| , v
(1.9)
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LUO, NATALINI, AND YANG
for all v>0 under consideration. Since the total variation of the solutions thus obtained is uniformly bounded with respect to the parameter =, the convergence to a BV solution of the equilibrium equation (1.4) follows immediately. Precisely, we shall prove the following results. Theorem 1.1.
Suppose that the initial data u 0 (x), v 0 (x) satisfy |u 0 (x)| M 1 ,
0<$ v 0 (x) M 1 ,
and TVv 0 (x)+TVu 0 (x) M 2 for some positive constants M 1 , M 2 , $ independent on =. There are some positive constants M 1 , $, which do not depend on =, such that, if sup
v | f $(v)| a,
(1.10)
1 $ v M
then there exists a global weak solution (v =, u = ) # L ([0, + ), BV(R)) & Lip([0, + ), L 1loc(R)) to the Cauchy problem (1.1), (1.2), with p given by (1.8). Moreover, the following estimates hold for any (x, t) # R_[0, + ): 0<$ v = (x, t) M 1 ,
|u = (x, t)| M 1 ;
(1.11)
there exists a positive constant C 1 , which does not depend on =, such that TV x v = (x, t)+TV x u = (x, t) C 1 (TVv 0 (x)+TVu 0 (x));
(1.12)
for any bounded real interval I R, there are two positive constants C 2 and C =3 , the first one not depending on =, such that, for every s, t 0,
|
|v = (x, t)&v = (x, s)| dx C 2 |t&s|
(1.13)
|
|u = (x, t)&u = (x, s)| dx C =3 |t&s|.
(1.14)
I
and
I
Finally, for any C 2 entropy flux pairs (', q) of system (1.1), with ' convex, this solution verifies the entropy inequality 1 t '(v =, u = )+ x q(v =, u = )& ' u ( f(v = )&u = ) 0, = in the sense of distribution.
(1.15)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
179
Theorem 1.2. Let (v =, v = ) be the sequence of solutions to problem (1.1), (1.2) given by Theorem 1.1. Then there exists a subsequence [(v =, u = )] which converges almost everywhere to a pair (v, u) as = Ä 0. Moreover, the limit functions satisfy u(x, t)= f (v(x, t)) and v is a weak solution to the Cauchy problem (1.4) with initial datum v 0 (x). A more detailed discussion would be necessary to check entropy inequalities for the limit solution v given by Theorem 1.2, as pointed out in [22]. The main difficulty here is to find appropriate extensions to the family ' k (v)= |v&k| of Kruz kov-type entropies for (1.4). In particular, one needs to show that, in a suitable fixed domain containing the equilibrium curve, for any k there exists an entropy ' k (v, u) for system (1.1) such that ' k (v, f (v))= |v&k| and which is strictly convex and strictly dissipative, namely ' kuu(v, u)>0. Actually this fact is not covered by the results of [9]. In [22] entropy inequalities are proven for an infinite family of smooth entropy functions which uniformly approach the Kruz kov's entropy functions. Nevertheless it is still not clear whether these inequalities imply in the limit the ``true'' Kruz kov's entropy inequalities t |v&k| & x (sign(v&k)( f (v)& f (k)) 0 , for any k # [&M, M] and in the sense of distributions. Let us explain better the main tools used in the present paper. Our analysis depends on the following observations. First of all we recall that, as first noticed by T. Nishida [34], the essential property of the case #=1 is that the wave curves for shock waves of the homogeneous part of system (1.1) are independent on the starting point and they are obtained one from the another by translations and symmetries. Therefore, we can define a sort of ``distance'' in the set of the states, which is not symmetric but satisfies the triangle inequality, and it is possible to avoid the analysis of the quadratic term in the Glimm's functional. This remark was used in many papers to deal with global weak solutions in presence of a source term, see for instance [46, 28, 19, 35]. On the other hand the sub-characteristics condition (1.9) guarantees that the total wave strength of the approximate solutions will not grow when the source term is added into the approximate solutions, if we measure the strength of waves in a convenient way. Actually the choice of the measure of the wave strength is somewhat subtle here. In [34] and [29], the wave strength of 1-shock (respectively 2-shock) was measured by the absolute value of the variation of rthe right Riemann invariant (respectively sthe left Riemann invariant). However, in our case, following the ideas of [24, 28], it is better to measure the wave strength as the absolute value of the variation of 2a log v, just to reveal the dissipative effect of the relaxation term. As we show in Lemma 3.5 below,
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with this choice of the wave strength, the relaxation term will not increase the total strength of the waves in the approximate solutions. The paper will be organized as follows. In Section 2 we briefly review the construction of the approximate solutions by using Glimm's scheme. The main wave interaction estimates are given in Section 3. Finally, in Section 4 we complete the proof of Theorems 1.1 and 1.2. It is easy to see that our results could be generalized to other systems sharing the Nishida's properties, as discussed in [37], Ch. 5. Finally, let us mention that after the completion of our work we received the preprint [1], where similar results have been obtained by using a modified version of the Bressan's front tracking algorithm.
2. THE DIFFERENCE SCHEME Let us recall that system (1.1) is strictly hyperbolic, provided that 0
a += . v
(2.1)
The Riemann invariants are given by r=u+a log v,
s=u&a log v.
(2.2)
Let us introduce, following [14], a modified version of the Glimm's scheme. For an equidistributed random sequence : 0 , : 1 , ..., : n , ... in (&1, 1), we selected a space mesh-length l and a time mesh-length h satisfying the following CFL condition lh>a
<\exp \
r 0 &s 0 2a
++ ,
(2.3)
where r 0 =inf x # R r(v 0 (x), u 0 (x)), s 0 =sup x # R s(v 0 (x), u 0 (x)). For any fixed =, we choose h such that 0
(2.4)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
181
Assuming that (v l, u l ) has already been determined on n&1 k=0 7 k , we extend (v l, u l ) to 7 n as the admissible solution to the Cauchy problem v t &u x =0,
(2.5)
u t + p(v) x =0, with the initial data at t=nh as v l (x, h)=v l ((m+: n ) l, nh&), h u l (x, nh)= u l + ( f (v l )&u l ) ((m+: n ) l, nh&), =
\
+
(2.6)
for (m&1) l
3. BOUNDS FOR THE APPROXIMATE SOLUTIONS It is well known (see [34, 28]) that all shock curves for the homogeneous system 2.5 in the (r, s)-plane are independent on their starting points. The 1-shock wave curve S 1 starting from (r R , s R ) can be expressed in the form s R &s= g(r R &r)
for
r r R ,
(3.1)
and the 2-shock wave curve S 2 starting from (r R , s R ) can be expressed in the form r R &r= g(s R &s)
for
s s R ,
(3.2)
where 0
g"(x)>0,
for x>0,
lim g$(x)=1,
xÄ
g$(0)= g"(0)=0. The 1-rarefaction wave curves R 1 can be expressed in the form s&s R =0,
for
r r R ,
(3.3)
and the corresponding expression of the 2-rarefaction wave curves R 2 is r&r R =0,
for
s s R .
(3.4)
Following [34], from (3.1), (3.2) we can obtain the following results, which play a key ro^le in our paper, concerning the geometry of the shock curves, see also the thorough discussion in [37].
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Theorem 3.1. (A) All the shock curves of equal strength are translates of one another. The 2-shock curve S 2 is the reflection of the 1-shock curve S 1 with respect with the line v=v R . (B) the quantity 2a log v increases along R 1 , S 2 curves and decreases along R 2 , S 1 curves. The Riemann solution of 2.5 with initial data (v 0 (x), u 0 (x))=
{
(v & , u & ), (v + , u + )
x<0 x>0
(3.5)
with v + , v & >0 is the building block of Glimm's scheme. We first give some lemmas for estimating the Riemann invariants in the Riemann solutions. Lemma 3.2. holds
Let (v, u) be the solution of (2.5), (3.5). Then the following
v(x, t)>0, r(x, t)#r(v(x, t), u(x, t)) min(r(v + , u + ), r(v & , u & )),
(3.7)
s(x, t)#s(v(x, t), u(x, t)) max(s(v + , u + ), s(v & , u & )). The above lemma was proved in [34]. Using this result, we can obtain a lower bound estimate for v, which ensures the CFL condition is verified and shows that our scheme is well defined. Lemma 3.3. Under condition (1.9), let [v l, u l ] be the approximate solutions constructed in Section 2, and set r 0 =inf x # R r(v 0 (x), u 0 (x)), s 0 =sup x # R s(v 0 (x), u 0 (x)). Then for t 0, 0
x#R
(3.7)
x#R
and v l (x, t) exp Proof.
\
r 0 &s 0 . 2a
+
(3.8)
Lemma 3.2 implies inf r l (x, t)&sup s l (x, t) x#R
x#R l
inf r (x, (n&1) h+ )&sup s l (x, (n&1) h+ ), x#R
x#R
(3.9)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
183
for (n&1) h t
_\
1&
r l &s l h l h l h r & s + f exp 2= 2= = 2a
+
\ \
++&
_((m+: n ) l, nh&),
(3.10)
and s l (x, nh+)=
_\
1&
r l &s l h l h l h s & r + f exp 2= 2= = 2a
+
\ \
++&
_((m+: n ) l, nh&),
(3.11)
for (m&1) l
h
h
h
r&s
_\1&2=+ r&2= s+ = f \exp \ 2a ++&
and q 2 (r, s)=
h
h
r&s
h
_\1&2=+ s& 2= r+ = f \\ 2a ++&& .
Then we have r&s r&s h h 1 q 1 = 1& exp + f $ exp r 2= 2= a 2a 2a
\
+
\ + \ \ ++
and q 1 h h 1 r&s r&s f $ exp = & exp s 2= 2= a 2a 2a
\ + \ \ ++ .
Also, we observe that &1 exp
r&s 1
r&s
\ 2a + a f $ \exp \ 2a ++ 1,
due to the sub-characteristic condition. Thus, q 1 0, r
q 1 0, s
(3.12)
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LUO, NATALINI, AND YANG
if 0 h<=. Similarly, we have q 2 0, r
q 2 0. s
(3.13)
Lemma 3.2, together with (3.10)(3.13), yields inf r l (x, nh+) x#R
\
h 2=
+ h inf + f exp \ = \
1&
inf r l (x, (n&1) h+)&
x#R
x#R
h sup s l (x, (n&1) h+) 2= x # R
r l (x, (n&1) h+)&sup x # R s l (x, (n&1) h+) , 2a (3.14)
++
and sup s l (x, nh+) x#R
\
h h inf r l (x, (n&1) h+) sup r l (x, (n&1) h+)& 2= x # R 2= x # R
+ h inf + f exp \ = \
1&
x#R
r l (x, (n&1) h+)&sup x # R s l (x, (n&1) h+) , 2a (3.15)
++
Hence inf r l (x, nh+)&sup s l (x, nh+) x#R
x#R
inf r l (x, (n&1) h+)&sup s l (x, (n&1) h+). x#R
(3.16)
x#R
Thus inf r l (x, nh+)&sup s l (x, nh+ ) r 0 &s 0 .
x#R
(3.17)
x#R
The conclusion follows from (3.9)(3.17).
K
Let us consider Riemann problem (2.5), (3.5). Suppose the two states (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 and S 2 and f (v 1 )&u 1 > f (v 2 )&u 2 . It is easy to check that two states (v 1 , u 1 +(h=)( f (v 1 &u 1 )) and (v 2 , u 2 +(h=)( f (v 2 &u 2 )) can still be connected by R 1 and S 2 if 0
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
185
(see Lemma 3.5 below). In this case, one can easily verify that, if one uses the absolute value of the variation of the Riemann invariants in the shock wave to measure the shock strength as in [34], the total wave strengths of the waves which connect the states (v 1 , u 1 +(h=)( f (v 1 &u 1 )) and (v 2 , u 2 +(h=)( f (v 2 &u 2 )) will be greater than those of the waves which connect the states (v 1 , u 1 ) and (v 2 , u 2 ). However, if we take, following [24, 28], for the wave strength the quantity |2(r&s)| = |2(2a log v)|, the absolute value of the variation of 2a log v across the wave, we have a regular parameterization of the wave curves. Then we can show (see Lemma 3.5 below) that the total wave strength will not increase after updating the value of u at t=nh on account of the effect of relaxation. In the following we shall use this new definition of the wave strength. For the Riemann problem (2.5), (3.5), we use P(v & , u & , v + , u + ) to denote the summation of the total strength of the solution. The following lemma, which can be found for example in [18, 35], will be often used in our analysis. Lemma 3.4.
Under the assumptions of Lemma 3.9, there holds
P(v 1 , u 1 , v 3 , u 3 ) P(v 1 , u 1 , v 2 , u 2 )+P(v 2 , u 2 , v 3 , u 3 ),
(3.18)
where v 1 , v 2 and v 3 are arbitrary positive constants and u 1 , u 2 and u 3 are arbitrary constants. Let us remark that it is possible to show, see [29], that inequality (3.18) still holds true if one uses the measure of the wave strength as in [34]. However, this fact is no longer true for the following result, actually the main ingredient of our analysis, which shows that the total wave strengths will not increase when we update the values of u in the scheme to take into account the effect of relaxation. Lemma 3.5.
If 0
h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) P(v 1 , u 1 ; v 2 , u 2 ), (3.19) = =
\
+
for any constants u 1 , u 2 and any positive constants v 1 , v 2 . Proof. The proof of this lemma is also based on properties (A) and (B) of Theorem 3.1. For illustration, we consider the following four cases. (I) (v 1 , u 1 ) and (v 2 , u 2 ) are connected by S 1 and S 2 , i.e., there exists a unique state (v, u ) such that the following Rankine-Hugoniot condition and entropy condition hold,
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LUO, NATALINI, AND YANG
p(v )&p(v 1 )
& v&v (v&v ), p(v )& p(v ) u &u =& & v &v (v &v), u &u 1 =
1
(3.20)
1
2
2
2
(3.21)
2
and v v 2 .
v v 1 , First, we show that
f (v 1 )&u 1 f (v 2 )&u 2 .
(3.22)
In fact, due to the fact v v 1 , v v 2 and thanks to (3.20), (3.21) and the sub-characteristic condition (1.9), we have f (v 2 )& f (v 1 ) = f (v 2 )& f (v )+ f (v )& f (v 1 ) =
| |
1
f $(v +%(v 2 &v )) d%(v 2 &v )+ 0
|
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
1
&- & p$(v +%(v 2 &v )) d%(v 2 &v ) 0
+
|
1
- & p$(v 1 +%(v &v 1 )) d%(v &v 1 )
0 1
& p$(v+%(v &v)) d% (v &v) + | p$(v +%(v &v )) d%(v &v ) p(v )& p(v ) =& & v &v (v &v) p(v )& p(v ) + & v &v (v&v ) &
&
|
2
2
0
1
1
1
1
0
2
2
2
1
1
1
=u 2 &u +u &u 1 =u 2 &u 1 , which yields (3.22).
(3.23)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
187
FIGURE 1
In view of (3.22), one must consider the following subcases Subcase 1: f (v 2 )&u 2 f (v 1 )&u 1 0; Subcase 2: f (v 2 )&u 2 0 f (v 1 )&u 1 ; Subcase 3: 0 f (v 2 )&u 2 f (v 1 )&u 1 . In the following we just discuss the Subcase 1, since Subcases 2 and 3 can be handled similarly. Assume that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 and S 2 , as in Fig. 1. Since f (v 2 )&u 2 f (v 1 )&u 1 , we can take a point C in the straight line v=v 2 such that |CE| = |AB|. The point C is the point (v 2 , u 2 +(h=)[( f (v 2 )&u 2 )&( f(v 1 )&u 1 )]). It follows from property (A) that h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
h =P v 1 , u 1 ; v 2 , u 2 + [( f (v 2 )&u 2 )&( f (v 1 )&u 1 )] =
\
+
P(v 1 , u 1 ; v 2 , u 2 ). In the case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 and R 2 , since v decreases along S 1 and R 2 , due to property (B), we have
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LUO, NATALINI, AND YANG
h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
=a |log v 2 &log v 1 | a |log v 2 &log v | +a |log v 1 &log v | =P(v 1 , u 1 ; v 2 , u 2 ). The case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 and S 2 can be handled similarly. In the following, we show that that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) cannot be connected by two waves R 1 and R 2 . In fact, in this case h h s v 2 , u 2 + ( f (v 2 )&u 2 ) >s v 1 , u 1 + ( f (v 1 )&u 1 ) , = =
(3.24)
h h r v 2 , u 2 + ( f (v 2 )&u 2 ) >r v 1 , u 1 + ( f (v 1 )&u 1 ) . = =
(3.25)
\
and
+ \
\
+
+ \
+
Suppose (3.24) and (3.25) are true and let us set along this proof, 1 :=(1&(h=)). Assume v 2 v 1 . Then (1.9) yields 1 (s(v 2 , u 2 )&s(v 1 , u 1 )) =1 ((u 2 &u 1 )&(a log v 2 &a log v 1 ))
\
=1 (u 2 &u 1 )&
|
\
1 0
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
h 1 (u 2 &u 1 )& 1& =
=
|
1 0
|
+
1
f $(v 1 +%(v 2 &v 1 ) d%
0
|
1
0
a d% v 1 +%(v 2 &v 1 )
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
+
h =1 (u 2 &u 1 )& a log v 2 &a log v 1 & ( f (v 2 )& f (v 1 ) =
_
&
h h =u 2 + ( f (v 2 )&u 2 )&a log v 2 &u 1 & ( f (v 1 )&u 1 )+a log v 1 = = h h =s v 2 , u 2 + ( f(v 2 )&u 2 ))&s v 1 , u 1 + ( f (v 1 )&u 1 ) >0, = =
\
\
+
(3.26)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
189
which implies that s(v 2 , u 2 )>s(v 1 , u 1 ), since 0
\
=1 (u 2 &u 1 )+
|
\
1 0
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
h 1 (u 2 &u 1 )+ 1& =
>
|
1 0
|
+
1
f $(v 1 +%(v 2 &v 1 ) d%
0
|
1 0
a d% v 1 +%(v 2 &v 1 )
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
+
h =1 (u 2 &u 1 )+ a log v 2 &a log v 1 & ( f (v 2 )& f (v 1 ) =
_
&
h h =u 2 + ( f (v 2 )&u 2 )+a log v 2 &u 1 & ( f (v 1 )&u 1 )&a log v 1 = = h h =r v 2 , u 2 + ( f (v 2 )&u 2 ) &r v 1 , u 1 + ( f (v 1 )&u 1 ) >0, = =
\
+ \
+
(3.27)
which implies that r(v 2 , u 2 )>r(v 1 , u 1 ) since 0
\
+ \
+
(3.28)
To prove (3.28), we first observe that there exists a unique state (v, u ) such that (v 1 , u 1 ) and (v, u ) are connected by R 1 , while (v, u ) and (v 2 , u 2 ) are connected by S 2 . In this case, we have s(v 1 , u 1 )=u 1 &a log v 1 =u 2 &a log v 2 =s(v 2 , u 2 ).
(3.29)
190
LUO, NATALINI, AND YANG
Therefore, in view of the sub-characteristic condition and the fact v 1 v due to property (B), we have u &u 1 =a log v &a log v 1 =
|
1
0
|
a d% (v &v 1 ) v 1 +%(v &v 1 )
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
= f (v )& f (v 1 ), i.e., f (v )&u f (v 1 )&u 1 , which, together with (3.29) implies h h s v, u + ( f (v )&u ) s v 1 , u 1 + ( f (v 1 )&u 1 ) . = =
\
+ \
+
(3.30)
On the other hand, we have h h s v 2 , u 2 + ( f (v 2 )&u 2 ) &s v, u + = =
\
+ \
\ f (v)&u)+
h =1 (u 2 &u )& (a log v 2 &a log v )& ( f (v 2 )& f (v )) . =
_
&
(3.31)
f $(v +%(v 2 &v )) } (v 2 &v )= f (v 2 )& f (v ).
(3.32)
Since v 2 v, from the sub-characteristic condition we have a log v 2 &a log v =
| |
1 0
a d% } (v 2 &v ) v +%(v 2 &v )
1
0
Therefore h h s v 2 , u 2 + ( f (v 2 )&u 2 ) s v, u + ( f (v )&u ) , = =
\
+ \
+
(3.33)
since 0
191
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
Since v increases along R 1 and S 2 due to the property (B), we have h h P (v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
=2a(log v 2 &log v 1 )=P(v 1 , u 1 , v 2 , u 2 ). (III) The case when (v 1 , u 1 ) and (v 2 , u 2 ) are connected by S 1 , R 2 is similar to the case when they are connected by R 1 , S 2 . (IV) Finally, let us consider the case when (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 and R 2 . Suppose (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 ((v 1 , u 1 ), (v, u )) and R 2 ((v, u ), (v 2 , u 2 )). Then s(v 1 , u 1 )=u 1 &a log v 1 =u &a log v =s(v, u ),
(3.34)
r(v 2 , u 2 )=u 2 +a log v 2 =u +a log v =r(v, u ),
(3.35)
and u 1 u u 2 ,
v 1 v,
v 2 v.
(3.36)
Now we have u 2 &u 1 =u 2 &u +u &u 1 =a(log v &log v 2 )+a(log v &log v 1 ) =
|
1
0
|
a d% (v &v 2 )+ v 2 +%(v &v 2 )
|
1
0
a d% (v &v 1 ) v 1 +%(v &v 1 )
1
& f $(v 2 +%(v &v 2 )) d%(v &v 2 )+
0
|
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
= f (v 2 )& f (v 1 ). This shows that f (v 1 )&u 1 f (v 2 )&u 2 .
(3.37)
Let us prove now that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=) ( f (v 2 )&u 2 )) cannot be connected by two waves S 1 and S 2 . Otherwise in that case, by arguing as for inequality (3.22), we easily have h
h
\1& = + ( f (v )&u ) \1& = + ( f (v )&u ), 2
2
1
1
which, thanks to (3.37), implies f (v 2 )&u 2 = f (v 1 )&u 1 . It is now clear that in this case (v 1 , u 1 +(h=)( f (v 1 )&u 1 ) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 R 2 , as shown in Figure 2.
192
LUO, NATALINI, AND YANG
FIGURE 2
We need now to consider the following subcases: Subcase 1: (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 R 2 ; Subcase 2: s(v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 S 2 ; Subcase 3: (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 R 2 ; In the Subcase 2 and Subcase 3, since v increases along R 1 S 2 and decreases along S 1 R 2 , we have h h P (v 1 , u 1 + ( f (v 1 )&u 1 ; v 2 , u 2 + ( f (v 2 )&u 2 )) = =
\
+\
&
=2a |log v 2 &log v 1 | 2a( |log v 2 &log v | + |log v &log v 1 | ) =P(v 1 , u 1 ; v 2 , u 2 ), which proves (3.19). It remains to verify the Subcase 1. Assume that the intermediate state is (v^, u^ ), i.e. (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v^, u^ ) are connected by R 1 and (v^, u^ ) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 2 . Therefore h u 1 + ( f (v 1 )&u 1 )&a log v 1 =u^ &a log v, =
(3.38)
h u 2 + ( f (v 2 )&u 2 )+a log v 2 =u^ +a log v, =
(3.39)
v 1 v^, v 2 v^.
(3.40)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
193
h h (u 2 &u 1 )+ ( f (v 2 )& f (v 1 ))+a log v 2 v 1 . = =
(3.41)
Thus
\ +
2a log v^ = 1& Hence
h h P (v 1 , u 1 + ( f (v 1 )&u 1 ); (v 2 , u 2 + ( f (v 2 )&u 2 )) = =
\
&
=2a log v^ &2a log v 2 +2a log v^ &2a log v 1 =2
h
h
_\1& = + (u &u )+ = ( f (v )& f (v ))& . 2
1
2
1
(3.42)
On the other hand, there holds 2a log v =u 2 &u 1 +a log v 2 +a log v 1 . Therefore P(v 1 , u 1 ; v 2 , u 2 )=2a log v &2a log v 1 +2a log v &2a log v 2 =2(u 2 &u 1 ).
(3.43)
Inequality (3.37) yields h h (u 2 &u 1 )+ ( f (v 2 )& f (v 1 )) (u 2 &u 1 ). = =
\ + 1&
(3.44)
The case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected just by a single wave can be handled easily by a similar argument as above. Thus completes the proof. K
4. GLOBAL EXISTENCE AND ZERO RELAXATION LIMITS In this section, we shall prove Theorems 1.1 and 1.2. Proof of Theorem 1.1. For the approximated solution (v l, u l ), let F n be denote the total wave strength in the time strip 7 n =[(x, t) : &
(4.1)
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LUO, NATALINI, AND YANG
for n=1, 2, ... . It turns out F n F 0 C(TV(u 0 (x))+TV(v 0 (x)),
(4.2)
for n=1, 2, ..., where C is a positive constant independent of =, l and h. Since F n is equivalent to the total variation, we obtain inequality (1.12) for the approximated solutions (v l, u l ). Furthermore, following for instance [14], we can obtain time continuity estimates, again for (v l, u l ). Actually, and for future references, let us discuss in a greater detail this point. First, it is easy to see that, since there is no source term in the equation for v, we promptly obtain that for any bounded interval I R there exists a constant C 2 , which does not depend on =, such that for every s, t 0,
|
|v l (x, t)&v l (x, s)| dx C 2 (|t&s| +h).
(4.3)
I
On the other hand, from (2.5), (2.6) and (4.2), we have that there exists a constant C, which does not depend on =, such that for every s, t 0,
|
|u l (x, t)&u l (x, s)| dx
I n2
C(|t&s| +h)+ : n=n1
|
|u l (x, nh+)&u l (x, nh&)| dx,
I
where s n 1 h
|
|u l (x, nh+ )&u l (x, nh& )| dx
I
h =
|
| f (v l )&u l |(x, nh& ) dx.
I
To estimate this contribution due to the relaxation term, let us set #\ n =
|
| f (v l )&u l | (x, nh\) dx.
I
Now, again from (2.6), h
\ = + ( f (v )&u )((m+: ) l, nh&),
( f (v l )&u l )(x, nh+)= 1&
l
l
n
(4.4)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
195
for (m&1) l
\ + h C=+ 1& \ =+
#& n Ch+ 1&
n&1
#+ 0 .
(4.5)
Thus
|
|u l (x, t)&u l (x, s)| dx C( |t&s| +h)+
I
h n2 : #& = n=n1 n
1 C(|t&s| +h) 1+ e &s= . =
\
+
(4.6)
Now, from standard compactness argument (see [15, 14]) there is a convergent subsequence in L 1loc from the sequence [(v l, u l )]. Let us denote the limit function as (v =, u = )(x, t). By arguing as in [14], it easy to show the consistency of the algorithm, and in particular that (v =, u = )(x, t) is a weak solution of (1.1), (1.2), i.e., for any test function , # C 10([0, )_R), we have,
| | R
(v =, t &u =, x ) dx dt+
0
| | R
+
+
1 =
v 0 (x) ,(x, 0) dx=0,
| | R
+
(4.7)
R
(u =, t + p(v = ) , x ) dx dt+
0
=
|
|
u 0 (x) ,(x, 0) dx R
(u = & f (v = )) , dx dt.
(4.8)
0
Moreover (v =, u = ) satisfies the entropy inequality 1 t '(v =, u = )+ x q(v =, u = )& ' u ( f(v = )&u = ) 0, =
(4.9)
in the sense of distribution, for any smooth entropy flux pairs (', q) with convex '. K Proof of Theorem 1.2. From the estimates (1.11)(1.14), and using the fact that, thanks to (4.6), the constant C =3 can be estimated by C 4 (1+(1=) e &s= ), for some positive constant C 4 , we can extract a subsequence from the sequence (v =, u = ), which converges in C((0, ); L 1loc(R))
196
LUO, NATALINI, AND YANG
as = Ä 0. Let us denote the limit function as (v, u). Let = Ä 0 in (4.8). Hence the uniform bound of (v =, u = ) yields
| | R
+
(u& f (v)) , dx dt=0
(4.10)
0
for any test function , # C 10((0, )_R). Since , is arbitrary, and (v, u) is bounded and has the bounded total variation, we have u(x, t)= f (v(x, t)), a.e.
for
(x, t) # R_(0, ).
(4.11)
Let = Ä 0 in (4.7), and thanks to (4.10), we verify that v is the weak solution of (1.4).
ACKNOWLEDGMENTS The research of the first author was partially supported by the small scale grant 9030642 of City University of Hong Kong. The research of the second author was partially supported by TMR project HCL *ERBFMRXCT960033. The research of the third author was partially supported by Strategic Grant *7000729 of City University of Hong Kong.
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11. J. F. Collet and M. Rascle, Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography, Z. Angew. Math. Phys. 47 (1996), 400409. 12. F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), 22232249. 13. C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys. 46 (1995), S294S307, special issue. Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids. 14. C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31 (1982), 471491. 15. J. Glimm, The interaction of non-linear hyperbolic waves, Comm. Pure Appl. Math. 41 (1988), 569590. 16. F. Golse, P. L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76(1) (1988), 110125. 17. E. Grenier, Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math. 50 (1997), 821865. 18. L. Hsiao, T. Luo, and T. Yang, Global BV solutions of compressible Euler equations with spherical symmetry and damping, J. Differential Equations 146 (1998), 203225. 19. L. Hsiao and P. Marcati, Nonhomogeneous quasilinear hyperbolic system arising in chemical engineering, Ann. Sc. Norm. Super. Pisa 15 (1988), 6597. 20. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235277. 21. M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22(12) (1997), 195233. 22. C. Lattanzio and P. Marcati, The zero relaxation limit for 2_2 hyperbolic systems, Nonlinear Anal. 38 (1999), 375389. 23. C. Lattanzio and P. Marcati, The zero relaxation limit for the hydrodynamic Whitham traffic flow model, J. Differential Equations 141(1) (1997), 150178. 24. T.-P. Liu, Initial-boundary value problems for gas dynamics, Arch. Rational Mech. Anal. 64 (1977), 137168. 25. T.-P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68(2) (1979), 141172. 26. T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153175. 27. T. Luo and R. Natalini, BV solutions and relaxation limit for a model in viscoelasticity, Proc. Royal Soc. Edinb. 128A (1998), 775795. 28. M. Luskin and B. Temple, The existence of a global weak solution to the nonlinear waterhammer problem, Comm. Pure Appl. Math. 35 (1982), 697735. 29. T. Makino, K. Mizohata, and S. Ukai, The global weak solutions of compressible Euler equation with spherical symmetry, Japan J. Ind. Appl. Math. (1992), 431449. 30. I. Muller and T. Ruggeri, ``Extended Thermodynamics,'' Vol. 37, Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1993. 31. R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49(8) (1996), 795823. 32. R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations 148 (1998), 292317. 33. R. Natalini, Recent mathematical results on hyperbolic relaxation problems, in ``Analysis of Systems of Conservation Laws'' (H. Freistuhler, Ed.), Monographs and Surveys in Pure and Applied Mathematics, CRC Press, 1999. 34. T. Nishida, Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Jap. Acad. 44 (1968), 642646.
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35. F. Poupaud, M. Rascle, and J. P. Vila, Global solutions to the isothermal EulerPoisson system with arbitrarily large data, J. Differential Equations 123 (1995), 93121. 36. M. Renardy, W. Hrusa, and J. Nohel, ``Mathematical Problems in Viscoelasticity,'' Vol. 35, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Essex, England, 1987. 37. D. Serre, ``Systemes de lois de conservation,'' Diderot editeur, 1996. 38. D. Serre, Relaxation semi-lineaire et cinetique des systemes de lois de conservation, Ann. Inst. H. Poincare Anal. Non Lineaire, in press. 39. I. Suliciu, On modelling phase transitions by means of rate-type constitutive equations, Shock wave structure, Internat. J. Engng. Sci. 28(8) (1990), 829841. 40. L. Tartar, ``Compensated Compactness and Applications to Partial Differential Equations,'' Vol. 39 of IV, pp. 136210, Research Notes in Math., 1979. 41. A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), 136161. 42. A. E. Tzavaras, ``Viscosity and Relaxation Approximations for Hyperbolic Systems of Conservation Laws,'' Technical report, Center of Mathematical Sciences, Univ. WisconsinMadison, 1998. 43. W. G. Vincenti and C. H. Kruger, ``Introduction to Physical Gas Dynamics,'' Wiley, New York, 1965. 44. W. C. Wang and Z. Xin, Asymptotic limit of the initial boundary value problems for conservation laws with relaxational extensions, Comm. Pure Appl. Math. 51 (1998), 505535. 45. J. Whitham, ``Linear and Nonlinear Waves,'' Wiley, New York, 1974. 46. L. Ying and C. Wang, Global solutions of the Cauchy problem for a nonhomogeneous quasilinear hyperbolic system, Comm. Pure and Appl. Math. 33 (1980), 579597. 47. W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations 155 (1999), 89132. 48. W. A. Yong, A difference scheme for a stiff system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 14031414.
Global BV Solutions to a p-System with Relaxation Tao Luo Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong and Institute of Mathematics, Academia Sinica, Beijing, 100080 People's Republic of China
Roberto Natalini Istituto per le Applicazioni del Calcolo ``M. Picone'', Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I-00161 Roma, Italia
and Tong Yang Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received September 23, 1998; revised May 3, 1999
In this paper, we study the global existence of BV solutions to a p-system with a relaxation source term using the Glimm's scheme. The pressure p is given by a #-law with #=1. By a suitable choice of the measure for the strength of the shock waves, we show that the total strength of the waves and the total variation of the solutions are uniformly bounded with respect to the relaxation parameter. Furthermore, when the relaxation parameter tends to zero, we show that the sequence of BV solutions eventually converges to a weak solution to the equilibrium equation. 2000 Academic Press
1. INTRODUCTION Let us consider the following p-system with relaxation v t &u x =0, 1 u t + p(v) x = ( f (v)&u), = 174 0022-039600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
(1.1)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
175
for (x, t) # R_(0, + ), p$(v)<0 and where =>0 is a relaxation parameter. We are interested in the Cauchy problem for (1.1) with initial data v(x, 0)=v 0 (x),
u(x, 0)=u 0 (x)
x # R.
(1.2)
According to the discussion in [26, 45] and in order to have stable wave patterns, we assume the sub-characteristic condition, which in the present case reads | f $(v)| - & p$(v),
(1.3)
for all v under consideration. Under this condition we expect that, as = tends to zero, the solutions to problem (1.1)(1.2) are well described by the equilibrium equation given by the scalar conservation law v t & f (v) x =0.
(1.4)
To put this problem in the right perspective, let us recall that relaxation phenomena arise in a number of different contexts. A general survey of the recent issues in the theory of relaxation hyperbolic problems can be found in [33]. Here we just mention the models for gases not in thermodynamic equilibrium and extended thermodynamics [8, 43, 30], the kinetic theories [7], and more general wave models, including chromatography, viscoelasticity with vanishing memory, phase transitions with small transition time, river flows and traffic flows [36, 45, 39, 42]. More recently the study of various classes of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws [20, 12, 3, 4]. The basic linear theory and results for local smooth solutions can be found in [45, 47, 17]. The nonlinear theory essentially starts with the fundamental paper by T.-P. Liu [26] for the general 2_2 system t v+ x h(v, u)=0, 1 t u+ x g(v, u)= ( f (v)&u). =
(1.5)
Here v, u are both scalars and we assume that the system is strictly hyperbolic with two real eigenvalues * 1 <* 2 . The (formal) reduced equation is given by the scalar law t v+ x f *(v)=0,
(1.6)
where f *(v)=h(v, f (v)). The equilibrium speed is **(v)=h v (v, f (v))+ f $(v) h u (v, f (v)). In that paper nonlinear stability for (smooth) expansion
176
LUO, NATALINI, AND YANG
and traveling waves was established. The main stability criterion was the sub-characteristic conditiona sort of causality principle: the speed ** of the equilibrium equation has to be included between the two speeds * 1 and * 2 of the full relaxing system. In the present case this yields condition (1.3). However, it is well known that in the generic situation solutions become discontinuous in finite time. The rigorous investigation of the relaxation limit for weak solutions was first studied by T.-P. Liu and his collaborators in [10] and [9], using the methods of compensated compactness [40]. Other results by the same methods have been given in [23, 22, 11, 42]. For 2_2 strictly hyperbolic systems and under the sub-characteristic condition, uniformly bounded sequences of solutions to relaxation systems converge almost everywhere, as = Ä 0 +, to some weak equilibrium solutions of the related scalar conservation law. More recently, D. Serre [38] proved, still by using the methods of compensated compactness, the convergence of the (semi-linear) JinXin's relaxation approximation [20] and some other discrete kinetic approximations to (one dimensional) genuinely nonlinear hyperbolic systems of conservation laws having a positively invariant domain. Let us recall that a different approach was considered in [5], where the averaging lemmas of the kinetic theory [16] have been used to investigate a special class of hyperbolic systems with relaxation having an equivalent kinetic formulation. Unfortunately the techniques involved with the above results seem to be not effective to deal with more general and more realistic situations. Actually, even without the singular perturbation term, the theory for global weak solutions to n_n hyperbolic systems of conservation laws (n 2) is mainly developed for BV solutions [15, 6], but for the moment very few results have been obtained in this framework for relaxation problems. Actually, all the existing results which use strong convergence techniques (i.e.: the FrechetKolmogorov's compactness theorem), have been obtained only when the limit equation is scalar and the approximating system is semi-linear or weakly coupled. In particular let us recall the results on the convergence of the JinXin's relaxation approximation to conservation laws and related models, see [41, 21, 31, 32, 44]. As far as we know, before the present paper, uniform BV estimates for strongly coupled systems with relaxation were proved only for the simple model in viscoelasticity considered in [27, 48]. Notice that, for systems with relaxation source term, including system (1.1), even the global existence of BV solutions for a fixed value of = needs to be established. In order to explain this point let us consider a quasilinear n_n system with a source term t U+ x F (U)=G(x, U),
(1.7)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
177
with U # R n, under the condition that matrix DF (u) has real, distinct eigenvalues. In general, the presence of a source term distorts the pattern of elementary wave interactions and induces amplification of waves. As a result of this process the total variation of the solution will generally increases with time and thus eventually leave the range of applicability of the Glimm's scheme. However, for source terms coming from geometrical or physical effects, we can expect the global existence of BV solutions, as for example for the quite different case considered in [25], where global existence of BV solutions to the Cauchy problem for (1.7) was established when the L 1 norms of G(x, U) and G U (x, U) are small and the source term have compact support in the x variable. Up to now the natural tool for more general source terms, including the relaxation case, has been the fractional step version of the Glimm's scheme, as used by C. Dafermos and L. Hsiao in [14]. Notice that, to prevent instability, one has to assume some dissipativity conditions on the source term G. In Theorem 2 of [14] the condition of total dissipativity of the source term was taken in account to prove the existence of global solutions, provided that the initial data are confined in a small neighborhood of an equilibrium state U and have small total variation. In that case the total variation of U decays to U exponentially fast as t Ä , see also [2]. Unfortunately it is easy to verify that for system (1.1), this dissipativity condition is not verified (even under the sub-characteristic condition, see [33]) and we cannot apply these results. Actually, as far as one is concerned with global existence of solutions for a fixed value of =, it is possible in principle to argue as in [13], by adopting a clever change of variable, at least for the case of the p-system (1.1) with f #0. Unfortunately, as noticed in [33], the estimates which are obtained by these methods are in general not uniform in the relaxation parameter =. The aim of the present paper is to present a direct approach by studying the wave interaction for system (1.1) when the pressure p is given by the relation p(v)=
a2 , v
(1.8)
for some a>0 and every v>0. What we show is that, although the relaxation term is only partially dissipative, the special structure of the system and the source term enable us to obtain the global existence of the BV solutions by using the fractional step version of Glimm's scheme and under the sub-characteristics condition (1.3), which now reads a | f $(v)| , v
(1.9)
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LUO, NATALINI, AND YANG
for all v>0 under consideration. Since the total variation of the solutions thus obtained is uniformly bounded with respect to the parameter =, the convergence to a BV solution of the equilibrium equation (1.4) follows immediately. Precisely, we shall prove the following results. Theorem 1.1.
Suppose that the initial data u 0 (x), v 0 (x) satisfy |u 0 (x)| M 1 ,
0<$ v 0 (x) M 1 ,
and TVv 0 (x)+TVu 0 (x) M 2 for some positive constants M 1 , M 2 , $ independent on =. There are some positive constants M 1 , $, which do not depend on =, such that, if sup
v | f $(v)| a,
(1.10)
1 $ v M
then there exists a global weak solution (v =, u = ) # L ([0, + ), BV(R)) & Lip([0, + ), L 1loc(R)) to the Cauchy problem (1.1), (1.2), with p given by (1.8). Moreover, the following estimates hold for any (x, t) # R_[0, + ): 0<$ v = (x, t) M 1 ,
|u = (x, t)| M 1 ;
(1.11)
there exists a positive constant C 1 , which does not depend on =, such that TV x v = (x, t)+TV x u = (x, t) C 1 (TVv 0 (x)+TVu 0 (x));
(1.12)
for any bounded real interval I R, there are two positive constants C 2 and C =3 , the first one not depending on =, such that, for every s, t 0,
|
|v = (x, t)&v = (x, s)| dx C 2 |t&s|
(1.13)
|
|u = (x, t)&u = (x, s)| dx C =3 |t&s|.
(1.14)
I
and
I
Finally, for any C 2 entropy flux pairs (', q) of system (1.1), with ' convex, this solution verifies the entropy inequality 1 t '(v =, u = )+ x q(v =, u = )& ' u ( f(v = )&u = ) 0, = in the sense of distribution.
(1.15)
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Theorem 1.2. Let (v =, v = ) be the sequence of solutions to problem (1.1), (1.2) given by Theorem 1.1. Then there exists a subsequence [(v =, u = )] which converges almost everywhere to a pair (v, u) as = Ä 0. Moreover, the limit functions satisfy u(x, t)= f (v(x, t)) and v is a weak solution to the Cauchy problem (1.4) with initial datum v 0 (x). A more detailed discussion would be necessary to check entropy inequalities for the limit solution v given by Theorem 1.2, as pointed out in [22]. The main difficulty here is to find appropriate extensions to the family ' k (v)= |v&k| of Kruz kov-type entropies for (1.4). In particular, one needs to show that, in a suitable fixed domain containing the equilibrium curve, for any k there exists an entropy ' k (v, u) for system (1.1) such that ' k (v, f (v))= |v&k| and which is strictly convex and strictly dissipative, namely ' kuu(v, u)>0. Actually this fact is not covered by the results of [9]. In [22] entropy inequalities are proven for an infinite family of smooth entropy functions which uniformly approach the Kruz kov's entropy functions. Nevertheless it is still not clear whether these inequalities imply in the limit the ``true'' Kruz kov's entropy inequalities t |v&k| & x (sign(v&k)( f (v)& f (k)) 0 , for any k # [&M, M] and in the sense of distributions. Let us explain better the main tools used in the present paper. Our analysis depends on the following observations. First of all we recall that, as first noticed by T. Nishida [34], the essential property of the case #=1 is that the wave curves for shock waves of the homogeneous part of system (1.1) are independent on the starting point and they are obtained one from the another by translations and symmetries. Therefore, we can define a sort of ``distance'' in the set of the states, which is not symmetric but satisfies the triangle inequality, and it is possible to avoid the analysis of the quadratic term in the Glimm's functional. This remark was used in many papers to deal with global weak solutions in presence of a source term, see for instance [46, 28, 19, 35]. On the other hand the sub-characteristics condition (1.9) guarantees that the total wave strength of the approximate solutions will not grow when the source term is added into the approximate solutions, if we measure the strength of waves in a convenient way. Actually the choice of the measure of the wave strength is somewhat subtle here. In [34] and [29], the wave strength of 1-shock (respectively 2-shock) was measured by the absolute value of the variation of rthe right Riemann invariant (respectively sthe left Riemann invariant). However, in our case, following the ideas of [24, 28], it is better to measure the wave strength as the absolute value of the variation of 2a log v, just to reveal the dissipative effect of the relaxation term. As we show in Lemma 3.5 below,
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LUO, NATALINI, AND YANG
with this choice of the wave strength, the relaxation term will not increase the total strength of the waves in the approximate solutions. The paper will be organized as follows. In Section 2 we briefly review the construction of the approximate solutions by using Glimm's scheme. The main wave interaction estimates are given in Section 3. Finally, in Section 4 we complete the proof of Theorems 1.1 and 1.2. It is easy to see that our results could be generalized to other systems sharing the Nishida's properties, as discussed in [37], Ch. 5. Finally, let us mention that after the completion of our work we received the preprint [1], where similar results have been obtained by using a modified version of the Bressan's front tracking algorithm.
2. THE DIFFERENCE SCHEME Let us recall that system (1.1) is strictly hyperbolic, provided that 0
a += . v
(2.1)
The Riemann invariants are given by r=u+a log v,
s=u&a log v.
(2.2)
Let us introduce, following [14], a modified version of the Glimm's scheme. For an equidistributed random sequence : 0 , : 1 , ..., : n , ... in (&1, 1), we selected a space mesh-length l and a time mesh-length h satisfying the following CFL condition lh>a
<\exp \
r 0 &s 0 2a
++ ,
(2.3)
where r 0 =inf x # R r(v 0 (x), u 0 (x)), s 0 =sup x # R s(v 0 (x), u 0 (x)). For any fixed =, we choose h such that 0
(2.4)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
181
Assuming that (v l, u l ) has already been determined on n&1 k=0 7 k , we extend (v l, u l ) to 7 n as the admissible solution to the Cauchy problem v t &u x =0,
(2.5)
u t + p(v) x =0, with the initial data at t=nh as v l (x, h)=v l ((m+: n ) l, nh&), h u l (x, nh)= u l + ( f (v l )&u l ) ((m+: n ) l, nh&), =
\
+
(2.6)
for (m&1) l
3. BOUNDS FOR THE APPROXIMATE SOLUTIONS It is well known (see [34, 28]) that all shock curves for the homogeneous system 2.5 in the (r, s)-plane are independent on their starting points. The 1-shock wave curve S 1 starting from (r R , s R ) can be expressed in the form s R &s= g(r R &r)
for
r r R ,
(3.1)
and the 2-shock wave curve S 2 starting from (r R , s R ) can be expressed in the form r R &r= g(s R &s)
for
s s R ,
(3.2)
where 0
g"(x)>0,
for x>0,
lim g$(x)=1,
xÄ
g$(0)= g"(0)=0. The 1-rarefaction wave curves R 1 can be expressed in the form s&s R =0,
for
r r R ,
(3.3)
and the corresponding expression of the 2-rarefaction wave curves R 2 is r&r R =0,
for
s s R .
(3.4)
Following [34], from (3.1), (3.2) we can obtain the following results, which play a key ro^le in our paper, concerning the geometry of the shock curves, see also the thorough discussion in [37].
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LUO, NATALINI, AND YANG
Theorem 3.1. (A) All the shock curves of equal strength are translates of one another. The 2-shock curve S 2 is the reflection of the 1-shock curve S 1 with respect with the line v=v R . (B) the quantity 2a log v increases along R 1 , S 2 curves and decreases along R 2 , S 1 curves. The Riemann solution of 2.5 with initial data (v 0 (x), u 0 (x))=
{
(v & , u & ), (v + , u + )
x<0 x>0
(3.5)
with v + , v & >0 is the building block of Glimm's scheme. We first give some lemmas for estimating the Riemann invariants in the Riemann solutions. Lemma 3.2. holds
Let (v, u) be the solution of (2.5), (3.5). Then the following
v(x, t)>0, r(x, t)#r(v(x, t), u(x, t)) min(r(v + , u + ), r(v & , u & )),
(3.7)
s(x, t)#s(v(x, t), u(x, t)) max(s(v + , u + ), s(v & , u & )). The above lemma was proved in [34]. Using this result, we can obtain a lower bound estimate for v, which ensures the CFL condition is verified and shows that our scheme is well defined. Lemma 3.3. Under condition (1.9), let [v l, u l ] be the approximate solutions constructed in Section 2, and set r 0 =inf x # R r(v 0 (x), u 0 (x)), s 0 =sup x # R s(v 0 (x), u 0 (x)). Then for t 0, 0
x#R
(3.7)
x#R
and v l (x, t) exp Proof.
\
r 0 &s 0 . 2a
+
(3.8)
Lemma 3.2 implies inf r l (x, t)&sup s l (x, t) x#R
x#R l
inf r (x, (n&1) h+ )&sup s l (x, (n&1) h+ ), x#R
x#R
(3.9)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
183
for (n&1) h t
_\
1&
r l &s l h l h l h r & s + f exp 2= 2= = 2a
+
\ \
++&
_((m+: n ) l, nh&),
(3.10)
and s l (x, nh+)=
_\
1&
r l &s l h l h l h s & r + f exp 2= 2= = 2a
+
\ \
++&
_((m+: n ) l, nh&),
(3.11)
for (m&1) l
h
h
h
r&s
_\1&2=+ r&2= s+ = f \exp \ 2a ++&
and q 2 (r, s)=
h
h
r&s
h
_\1&2=+ s& 2= r+ = f \\ 2a ++&& .
Then we have r&s r&s h h 1 q 1 = 1& exp + f $ exp r 2= 2= a 2a 2a
\
+
\ + \ \ ++
and q 1 h h 1 r&s r&s f $ exp = & exp s 2= 2= a 2a 2a
\ + \ \ ++ .
Also, we observe that &1 exp
r&s 1
r&s
\ 2a + a f $ \exp \ 2a ++ 1,
due to the sub-characteristic condition. Thus, q 1 0, r
q 1 0, s
(3.12)
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LUO, NATALINI, AND YANG
if 0 h<=. Similarly, we have q 2 0, r
q 2 0. s
(3.13)
Lemma 3.2, together with (3.10)(3.13), yields inf r l (x, nh+) x#R
\
h 2=
+ h inf + f exp \ = \
1&
inf r l (x, (n&1) h+)&
x#R
x#R
h sup s l (x, (n&1) h+) 2= x # R
r l (x, (n&1) h+)&sup x # R s l (x, (n&1) h+) , 2a (3.14)
++
and sup s l (x, nh+) x#R
\
h h inf r l (x, (n&1) h+) sup r l (x, (n&1) h+)& 2= x # R 2= x # R
+ h inf + f exp \ = \
1&
x#R
r l (x, (n&1) h+)&sup x # R s l (x, (n&1) h+) , 2a (3.15)
++
Hence inf r l (x, nh+)&sup s l (x, nh+) x#R
x#R
inf r l (x, (n&1) h+)&sup s l (x, (n&1) h+). x#R
(3.16)
x#R
Thus inf r l (x, nh+)&sup s l (x, nh+ ) r 0 &s 0 .
x#R
(3.17)
x#R
The conclusion follows from (3.9)(3.17).
K
Let us consider Riemann problem (2.5), (3.5). Suppose the two states (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 and S 2 and f (v 1 )&u 1 > f (v 2 )&u 2 . It is easy to check that two states (v 1 , u 1 +(h=)( f (v 1 &u 1 )) and (v 2 , u 2 +(h=)( f (v 2 &u 2 )) can still be connected by R 1 and S 2 if 0
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
185
(see Lemma 3.5 below). In this case, one can easily verify that, if one uses the absolute value of the variation of the Riemann invariants in the shock wave to measure the shock strength as in [34], the total wave strengths of the waves which connect the states (v 1 , u 1 +(h=)( f (v 1 &u 1 )) and (v 2 , u 2 +(h=)( f (v 2 &u 2 )) will be greater than those of the waves which connect the states (v 1 , u 1 ) and (v 2 , u 2 ). However, if we take, following [24, 28], for the wave strength the quantity |2(r&s)| = |2(2a log v)|, the absolute value of the variation of 2a log v across the wave, we have a regular parameterization of the wave curves. Then we can show (see Lemma 3.5 below) that the total wave strength will not increase after updating the value of u at t=nh on account of the effect of relaxation. In the following we shall use this new definition of the wave strength. For the Riemann problem (2.5), (3.5), we use P(v & , u & , v + , u + ) to denote the summation of the total strength of the solution. The following lemma, which can be found for example in [18, 35], will be often used in our analysis. Lemma 3.4.
Under the assumptions of Lemma 3.9, there holds
P(v 1 , u 1 , v 3 , u 3 ) P(v 1 , u 1 , v 2 , u 2 )+P(v 2 , u 2 , v 3 , u 3 ),
(3.18)
where v 1 , v 2 and v 3 are arbitrary positive constants and u 1 , u 2 and u 3 are arbitrary constants. Let us remark that it is possible to show, see [29], that inequality (3.18) still holds true if one uses the measure of the wave strength as in [34]. However, this fact is no longer true for the following result, actually the main ingredient of our analysis, which shows that the total wave strengths will not increase when we update the values of u in the scheme to take into account the effect of relaxation. Lemma 3.5.
If 0
h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) P(v 1 , u 1 ; v 2 , u 2 ), (3.19) = =
\
+
for any constants u 1 , u 2 and any positive constants v 1 , v 2 . Proof. The proof of this lemma is also based on properties (A) and (B) of Theorem 3.1. For illustration, we consider the following four cases. (I) (v 1 , u 1 ) and (v 2 , u 2 ) are connected by S 1 and S 2 , i.e., there exists a unique state (v, u ) such that the following Rankine-Hugoniot condition and entropy condition hold,
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LUO, NATALINI, AND YANG
p(v )&p(v 1 )
& v&v (v&v ), p(v )& p(v ) u &u =& & v &v (v &v), u &u 1 =
1
(3.20)
1
2
2
2
(3.21)
2
and v v 2 .
v v 1 , First, we show that
f (v 1 )&u 1 f (v 2 )&u 2 .
(3.22)
In fact, due to the fact v v 1 , v v 2 and thanks to (3.20), (3.21) and the sub-characteristic condition (1.9), we have f (v 2 )& f (v 1 ) = f (v 2 )& f (v )+ f (v )& f (v 1 ) =
| |
1
f $(v +%(v 2 &v )) d%(v 2 &v )+ 0
|
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
1
&- & p$(v +%(v 2 &v )) d%(v 2 &v ) 0
+
|
1
- & p$(v 1 +%(v &v 1 )) d%(v &v 1 )
0 1
& p$(v+%(v &v)) d% (v &v) + | p$(v +%(v &v )) d%(v &v ) p(v )& p(v ) =& & v &v (v &v) p(v )& p(v ) + & v &v (v&v ) &
&
|
2
2
0
1
1
1
1
0
2
2
2
1
1
1
=u 2 &u +u &u 1 =u 2 &u 1 , which yields (3.22).
(3.23)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
187
FIGURE 1
In view of (3.22), one must consider the following subcases Subcase 1: f (v 2 )&u 2 f (v 1 )&u 1 0; Subcase 2: f (v 2 )&u 2 0 f (v 1 )&u 1 ; Subcase 3: 0 f (v 2 )&u 2 f (v 1 )&u 1 . In the following we just discuss the Subcase 1, since Subcases 2 and 3 can be handled similarly. Assume that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 and S 2 , as in Fig. 1. Since f (v 2 )&u 2 f (v 1 )&u 1 , we can take a point C in the straight line v=v 2 such that |CE| = |AB|. The point C is the point (v 2 , u 2 +(h=)[( f (v 2 )&u 2 )&( f(v 1 )&u 1 )]). It follows from property (A) that h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
h =P v 1 , u 1 ; v 2 , u 2 + [( f (v 2 )&u 2 )&( f (v 1 )&u 1 )] =
\
+
P(v 1 , u 1 ; v 2 , u 2 ). In the case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 and R 2 , since v decreases along S 1 and R 2 , due to property (B), we have
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LUO, NATALINI, AND YANG
h h P v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
=a |log v 2 &log v 1 | a |log v 2 &log v | +a |log v 1 &log v | =P(v 1 , u 1 ; v 2 , u 2 ). The case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 and S 2 can be handled similarly. In the following, we show that that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) cannot be connected by two waves R 1 and R 2 . In fact, in this case h h s v 2 , u 2 + ( f (v 2 )&u 2 ) >s v 1 , u 1 + ( f (v 1 )&u 1 ) , = =
(3.24)
h h r v 2 , u 2 + ( f (v 2 )&u 2 ) >r v 1 , u 1 + ( f (v 1 )&u 1 ) . = =
(3.25)
\
and
+ \
\
+
+ \
+
Suppose (3.24) and (3.25) are true and let us set along this proof, 1 :=(1&(h=)). Assume v 2 v 1 . Then (1.9) yields 1 (s(v 2 , u 2 )&s(v 1 , u 1 )) =1 ((u 2 &u 1 )&(a log v 2 &a log v 1 ))
\
=1 (u 2 &u 1 )&
|
\
1 0
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
h 1 (u 2 &u 1 )& 1& =
=
|
1 0
|
+
1
f $(v 1 +%(v 2 &v 1 ) d%
0
|
1
0
a d% v 1 +%(v 2 &v 1 )
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
+
h =1 (u 2 &u 1 )& a log v 2 &a log v 1 & ( f (v 2 )& f (v 1 ) =
_
&
h h =u 2 + ( f (v 2 )&u 2 )&a log v 2 &u 1 & ( f (v 1 )&u 1 )+a log v 1 = = h h =s v 2 , u 2 + ( f(v 2 )&u 2 ))&s v 1 , u 1 + ( f (v 1 )&u 1 ) >0, = =
\
\
+
(3.26)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
189
which implies that s(v 2 , u 2 )>s(v 1 , u 1 ), since 0
\
=1 (u 2 &u 1 )+
|
\
1 0
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
h 1 (u 2 &u 1 )+ 1& =
>
|
1 0
|
+
1
f $(v 1 +%(v 2 &v 1 ) d%
0
|
1 0
a d% v 1 +%(v 2 &v 1 )
a d% } (v 2 &v 1 ) v 1 +%(v 2 &v 1 )
+
h =1 (u 2 &u 1 )+ a log v 2 &a log v 1 & ( f (v 2 )& f (v 1 ) =
_
&
h h =u 2 + ( f (v 2 )&u 2 )+a log v 2 &u 1 & ( f (v 1 )&u 1 )&a log v 1 = = h h =r v 2 , u 2 + ( f (v 2 )&u 2 ) &r v 1 , u 1 + ( f (v 1 )&u 1 ) >0, = =
\
+ \
+
(3.27)
which implies that r(v 2 , u 2 )>r(v 1 , u 1 ) since 0
\
+ \
+
(3.28)
To prove (3.28), we first observe that there exists a unique state (v, u ) such that (v 1 , u 1 ) and (v, u ) are connected by R 1 , while (v, u ) and (v 2 , u 2 ) are connected by S 2 . In this case, we have s(v 1 , u 1 )=u 1 &a log v 1 =u 2 &a log v 2 =s(v 2 , u 2 ).
(3.29)
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LUO, NATALINI, AND YANG
Therefore, in view of the sub-characteristic condition and the fact v 1 v due to property (B), we have u &u 1 =a log v &a log v 1 =
|
1
0
|
a d% (v &v 1 ) v 1 +%(v &v 1 )
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
= f (v )& f (v 1 ), i.e., f (v )&u f (v 1 )&u 1 , which, together with (3.29) implies h h s v, u + ( f (v )&u ) s v 1 , u 1 + ( f (v 1 )&u 1 ) . = =
\
+ \
+
(3.30)
On the other hand, we have h h s v 2 , u 2 + ( f (v 2 )&u 2 ) &s v, u + = =
\
+ \
\ f (v)&u)+
h =1 (u 2 &u )& (a log v 2 &a log v )& ( f (v 2 )& f (v )) . =
_
&
(3.31)
f $(v +%(v 2 &v )) } (v 2 &v )= f (v 2 )& f (v ).
(3.32)
Since v 2 v, from the sub-characteristic condition we have a log v 2 &a log v =
| |
1 0
a d% } (v 2 &v ) v +%(v 2 &v )
1
0
Therefore h h s v 2 , u 2 + ( f (v 2 )&u 2 ) s v, u + ( f (v )&u ) , = =
\
+ \
+
(3.33)
since 0
191
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
Since v increases along R 1 and S 2 due to the property (B), we have h h P (v 1 , u 1 + ( f (v 1 )&u 1 ); v 2 , u 2 + ( f (v 2 )&u 2 ) = =
\
+
=2a(log v 2 &log v 1 )=P(v 1 , u 1 , v 2 , u 2 ). (III) The case when (v 1 , u 1 ) and (v 2 , u 2 ) are connected by S 1 , R 2 is similar to the case when they are connected by R 1 , S 2 . (IV) Finally, let us consider the case when (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 and R 2 . Suppose (v 1 , u 1 ) and (v 2 , u 2 ) are connected by R 1 ((v 1 , u 1 ), (v, u )) and R 2 ((v, u ), (v 2 , u 2 )). Then s(v 1 , u 1 )=u 1 &a log v 1 =u &a log v =s(v, u ),
(3.34)
r(v 2 , u 2 )=u 2 +a log v 2 =u +a log v =r(v, u ),
(3.35)
and u 1 u u 2 ,
v 1 v,
v 2 v.
(3.36)
Now we have u 2 &u 1 =u 2 &u +u &u 1 =a(log v &log v 2 )+a(log v &log v 1 ) =
|
1
0
|
a d% (v &v 2 )+ v 2 +%(v &v 2 )
|
1
0
a d% (v &v 1 ) v 1 +%(v &v 1 )
1
& f $(v 2 +%(v &v 2 )) d%(v &v 2 )+
0
|
1
f $(v 1 +%(v &v 1 )) d%(v &v 1 )
0
= f (v 2 )& f (v 1 ). This shows that f (v 1 )&u 1 f (v 2 )&u 2 .
(3.37)
Let us prove now that (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=) ( f (v 2 )&u 2 )) cannot be connected by two waves S 1 and S 2 . Otherwise in that case, by arguing as for inequality (3.22), we easily have h
h
\1& = + ( f (v )&u ) \1& = + ( f (v )&u ), 2
2
1
1
which, thanks to (3.37), implies f (v 2 )&u 2 = f (v 1 )&u 1 . It is now clear that in this case (v 1 , u 1 +(h=)( f (v 1 )&u 1 ) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 R 2 , as shown in Figure 2.
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LUO, NATALINI, AND YANG
FIGURE 2
We need now to consider the following subcases: Subcase 1: (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 R 2 ; Subcase 2: s(v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 1 S 2 ; Subcase 3: (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by S 1 R 2 ; In the Subcase 2 and Subcase 3, since v increases along R 1 S 2 and decreases along S 1 R 2 , we have h h P (v 1 , u 1 + ( f (v 1 )&u 1 ; v 2 , u 2 + ( f (v 2 )&u 2 )) = =
\
+\
&
=2a |log v 2 &log v 1 | 2a( |log v 2 &log v | + |log v &log v 1 | ) =P(v 1 , u 1 ; v 2 , u 2 ), which proves (3.19). It remains to verify the Subcase 1. Assume that the intermediate state is (v^, u^ ), i.e. (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v^, u^ ) are connected by R 1 and (v^, u^ ) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected by R 2 . Therefore h u 1 + ( f (v 1 )&u 1 )&a log v 1 =u^ &a log v, =
(3.38)
h u 2 + ( f (v 2 )&u 2 )+a log v 2 =u^ +a log v, =
(3.39)
v 1 v^, v 2 v^.
(3.40)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
193
h h (u 2 &u 1 )+ ( f (v 2 )& f (v 1 ))+a log v 2 v 1 . = =
(3.41)
Thus
\ +
2a log v^ = 1& Hence
h h P (v 1 , u 1 + ( f (v 1 )&u 1 ); (v 2 , u 2 + ( f (v 2 )&u 2 )) = =
\
&
=2a log v^ &2a log v 2 +2a log v^ &2a log v 1 =2
h
h
_\1& = + (u &u )+ = ( f (v )& f (v ))& . 2
1
2
1
(3.42)
On the other hand, there holds 2a log v =u 2 &u 1 +a log v 2 +a log v 1 . Therefore P(v 1 , u 1 ; v 2 , u 2 )=2a log v &2a log v 1 +2a log v &2a log v 2 =2(u 2 &u 1 ).
(3.43)
Inequality (3.37) yields h h (u 2 &u 1 )+ ( f (v 2 )& f (v 1 )) (u 2 &u 1 ). = =
\ + 1&
(3.44)
The case when (v 1 , u 1 +(h=)( f (v 1 )&u 1 )) and (v 2 , u 2 +(h=)( f (v 2 )&u 2 )) are connected just by a single wave can be handled easily by a similar argument as above. Thus completes the proof. K
4. GLOBAL EXISTENCE AND ZERO RELAXATION LIMITS In this section, we shall prove Theorems 1.1 and 1.2. Proof of Theorem 1.1. For the approximated solution (v l, u l ), let F n be denote the total wave strength in the time strip 7 n =[(x, t) : &
(4.1)
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LUO, NATALINI, AND YANG
for n=1, 2, ... . It turns out F n F 0 C(TV(u 0 (x))+TV(v 0 (x)),
(4.2)
for n=1, 2, ..., where C is a positive constant independent of =, l and h. Since F n is equivalent to the total variation, we obtain inequality (1.12) for the approximated solutions (v l, u l ). Furthermore, following for instance [14], we can obtain time continuity estimates, again for (v l, u l ). Actually, and for future references, let us discuss in a greater detail this point. First, it is easy to see that, since there is no source term in the equation for v, we promptly obtain that for any bounded interval I R there exists a constant C 2 , which does not depend on =, such that for every s, t 0,
|
|v l (x, t)&v l (x, s)| dx C 2 (|t&s| +h).
(4.3)
I
On the other hand, from (2.5), (2.6) and (4.2), we have that there exists a constant C, which does not depend on =, such that for every s, t 0,
|
|u l (x, t)&u l (x, s)| dx
I n2
C(|t&s| +h)+ : n=n1
|
|u l (x, nh+)&u l (x, nh&)| dx,
I
where s n 1 h
|
|u l (x, nh+ )&u l (x, nh& )| dx
I
h =
|
| f (v l )&u l |(x, nh& ) dx.
I
To estimate this contribution due to the relaxation term, let us set #\ n =
|
| f (v l )&u l | (x, nh\) dx.
I
Now, again from (2.6), h
\ = + ( f (v )&u )((m+: ) l, nh&),
( f (v l )&u l )(x, nh+)= 1&
l
l
n
(4.4)
GLOBAL BV SOLUTIONS TO A p-SYSTEM WITH RELAXATION
195
for (m&1) l
\ + h C=+ 1& \ =+
#& n Ch+ 1&
n&1
#+ 0 .
(4.5)
Thus
|
|u l (x, t)&u l (x, s)| dx C( |t&s| +h)+
I
h n2 : #& = n=n1 n
1 C(|t&s| +h) 1+ e &s= . =
\
+
(4.6)
Now, from standard compactness argument (see [15, 14]) there is a convergent subsequence in L 1loc from the sequence [(v l, u l )]. Let us denote the limit function as (v =, u = )(x, t). By arguing as in [14], it easy to show the consistency of the algorithm, and in particular that (v =, u = )(x, t) is a weak solution of (1.1), (1.2), i.e., for any test function , # C 10([0, )_R), we have,
| | R
(v =, t &u =, x ) dx dt+
0
| | R
+
+
1 =
v 0 (x) ,(x, 0) dx=0,
| | R
+
(4.7)
R
(u =, t + p(v = ) , x ) dx dt+
0
=
|
|
u 0 (x) ,(x, 0) dx R
(u = & f (v = )) , dx dt.
(4.8)
0
Moreover (v =, u = ) satisfies the entropy inequality 1 t '(v =, u = )+ x q(v =, u = )& ' u ( f(v = )&u = ) 0, =
(4.9)
in the sense of distribution, for any smooth entropy flux pairs (', q) with convex '. K Proof of Theorem 1.2. From the estimates (1.11)(1.14), and using the fact that, thanks to (4.6), the constant C =3 can be estimated by C 4 (1+(1=) e &s= ), for some positive constant C 4 , we can extract a subsequence from the sequence (v =, u = ), which converges in C((0, ); L 1loc(R))
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LUO, NATALINI, AND YANG
as = Ä 0. Let us denote the limit function as (v, u). Let = Ä 0 in (4.8). Hence the uniform bound of (v =, u = ) yields
| | R
+
(u& f (v)) , dx dt=0
(4.10)
0
for any test function , # C 10((0, )_R). Since , is arbitrary, and (v, u) is bounded and has the bounded total variation, we have u(x, t)= f (v(x, t)), a.e.
for
(x, t) # R_(0, ).
(4.11)
Let = Ä 0 in (4.7), and thanks to (4.10), we verify that v is the weak solution of (1.4).
ACKNOWLEDGMENTS The research of the first author was partially supported by the small scale grant 9030642 of City University of Hong Kong. The research of the second author was partially supported by TMR project HCL *ERBFMRXCT960033. The research of the third author was partially supported by Strategic Grant *7000729 of City University of Hong Kong.
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