a. Quasidecoupling Transformation . . . . . . . . . . . . . . . . . . . 140 b. Framework for the Uniqueness and Stability of Solutions . . . . . . . 143 c. Framework for the Limiting Behavior of Approximate Solutions . . . . . 149
4. Completely Degenerate Systems . . . . . . . . . . . . . . . . . . . . . 151 a. Propagation of Oscillations and Existence of Generalized Solutions . . . 152 b. Uniqueness and Stability of Generalized Solutions . . . . . . . . . . . . 156
5. Systems with One Contact Field and One Line Field . . . . . . . . . . . 163 a. Uniqueness and Stability of Generalized Solutions . . . . . . . . . . . . 166 b. Systems Arising in Magnetohydrodynamics and Elasticity . . . . . . . . 175 c. A System Arising in Oil Recovery . . . . . . . . . . . . . . . . . . 176
We are concerned with the existence, uniqueness , and quali tative behavior of d i scon t inuous solut ions to non l inea r conservat ion laws. The m a i n objective of this paper is to describe an analyt ical me thod - the quas idecoupl ing me thod - and to s tudy its appl icat ions to these problems for d i scont inuous solut ions to hyperbolic systems of conserva t ion laws with l inearly degenerate characteristic fields. This me thod is developed from the ideas in [4, 5] abou t p ropaga t ion and cancel la t ion of ini t ial oscil lat ions for conservat ion laws and is mot ivated by the t r ans fo rma t ion between Euler ian coordinates and Lagrangian coordinates in f luid dynamics (cf. [7, 56, 49]). The novel feature
132 G.-Q. CUEN
of the method described here is the quasidecoupling of the entropy conser- vative inequalities and, to a lesser extent, the particular forms of entropy con- servative inequalities for conservation laws.
We consider a hyperbolic system of conservation laws
ut + f ( U ) x = 0, -oo < x < oo, (1.1)
where u = u ( x , t ) ~ ~n and f is a smooth nonlinear mapping from R n to R n. The condition of hyperbolicity requires that the Jacobian Vf of f have n real eigenvalues ) . i (u) , 1 <_ i <_ n, and n distinct right eigenvectors r i (u ) , 1 <_ i < n,
Vf(u ) r i (u ) = ~.i(u) r i ( u ) , (1.2)
that is, the Jacobian Vf be diagonalizable for any value of u. An eigenvalue 2i is either genuinely nonlinear or linearly degenerate in the sense of LAX [33] if its derivative in the corresponding eigendirection is either
r i �9 V)~ i :# 0 or ri. V)~i - O. (1.3)
In the classical theory of hyperbolic equations, an efficient method is the diagonalization method for studying the qualitative behavior of smooth solu- tions to quasilinear hyperbolic systems. The systems are reduced to correspond- ing systems of diagonal forms that consist of the transport equations
OtW i + ,~i(W) OxW i = O, 1 <_ i <_ n , (1.4)
where the functions w i ( u ) , 1 <_ i <_ n, are Riemann invariants satisfying
Vwi" V f = i~iVw i . (1.5)
A necessary and sufficient condition for the existence of the Riemann in- variants wi for strictly hyperbolic systems is the well-known Frobenius condi- tion:
li{r i , r~} = 0, for any j , k ~: i , (1.6)
where lg denotes the left eigenvector corresponding to 2~ and {. , .} is the Poisson bracket of vector fields in u-space. For n = 2, the Frobenius condition always holds. It is well known that the Cauchy problem for (1.1) does not, in general, have globally defined smooth solutions because the eigenvalues are nonlinear; hence only discontinuous solutions may exist in the large. For discontinuous solutions, the system (1.1) is not equivalent to the system (1.4) in general, and this method no longer works. In Section 3, we introduce a related method - the quasidecoupling method - and establish a framework for studying the uniqueness and qualitative behavior of discontinuous solu- tions to hyperbolic systems of conservation laws.
The uniqueness problem for conservation laws has been the topic of many papers (see the references and those cited therein). A complete theory of the uniqueness and L: stability of solutions to scalar conservation laws has been established in the space L ~ or L ~ c~ BE We refer the reader to the work of OLEImK [41], VOI?PERT [55], KEYHTZ [29], and KRU2KOV [31] on this problem for a scalar conservation law. For generalized solutions that are piecewise Lipschitz continuous and satisfy certain ordering assumptions, uniqueness
Discontinuous Solutions to Conservation Laws 133
theorems for strictly hyperbolic systems of conservation laws were established by OLEINIK [421, ROZHDESTVENSKII [451, GODUNOV [241, HURD [25, 261, L ~ [35], and DIPERI~A [14] in the space L ~ n BV. These results impose restric- tions on generalized solutions that have not been translated into conditions on the initial data. For an isotachophoresis model, DAFERMOS & GENO [12] established the uniqueness of a BV solution with initial data in BV with small oscillation in which generation points of centered rarefaction waves of the two families are strictly separated. We also refer the reader to DOUGLIS [17], LYAPIDEVSKII [38] and TVEITO & WINTHER [54] for stability results within cer- tain special classes of solutions. In Subsection 3b, we first establish a framework (Hypotheses (A~,2)) for the uniqueness and stability of L ~176 solu- tions of the model Cauchy problem
(ao (w-k ) P(x , t ) ) ,+ ( a o ( w - k ) 2(x, t ) P(x,t))x<=O for any kE(-oo, oo),
wit=0 = Wo(X), (1.7)
where a0(w) satisfies
a0(0) = 0, ao(-W) = ao(w), ao(w) > 0 for w * 0, (1.8)
and theL=vector field ( )~ (x, t)P(x, t), P(x, t) ) , P(x, t) >__ O, P(x, 0) __> P0 > 0, is divergence-free:
divx,t(,;t(x, t)P(x, t), P(x, t)) = 0. (1.9)
Our major finding is that the solution of the Cauchy problem (1.7) is independent of the choice of the L = function P(x, t) > 0 with fixed initial value P(x, 0). This is essential for using the framework to solve the uniqueness problem for conservation laws (see Sections 4-6). We remark that Hypotheses (A1,2) do not require any regularity of L ~176 solutions, in contrast to, for exam- ple, BV regularity that is the basic assumption of all uniqueness theorems established for systems of conservation laws, as in [12, 14, 26, 35, 42, 55].
The study of the limit behavior of dissipative approximate solutions to con- servation laws is another objective of this paper. Essentially, this problem is equivalent to the oscillation problem, that is, to studying the dynamic behavior of initial oscillations of O(1) amplitude with high frequency, as time evolves. More specifically, we study the limit behavior of the solutions corresponding to the oscillatory sequence of large initial data for nonlinear hyperbolic systems of conservation laws. This study can provide insight into how the behavior of the system at the microscopic level affects the behavior of the system at the macroscopic level. Such results are important for understanding both the structure of solutions for the Cauchy problem and the limit behavior of numerical and analytical approximations for the Cauchy problem with oscillatory initial data.
For genuinely nonlinear systems of conservation laws, many compactness theorems have been established in [15, 3, 13, 441 (see also the references cited therein). For 2x2 genuinely nonlinear systems, these theorems indicate that the initial oscillations instantaneously cancel as time evolves. The basic strategy is to reduce the compactness problem of the solution operator into proving
134 G.-Q. C~IrN
that the corresponding Young measures are Dirac masses by solving the Tartar- Murat functional equation for the Young measures. However, for systems with at least a linearly degenerate field, the Tartar-Murat equation cannot describe the complete structure of the corresponding Young measures. This causes an analytical difficulty (see DIPEgNA [16], RASCSE [43] and SEggE [48]). A suitable framework must be introduced to recover the lost information for the Young measures. For a 2 x 2 system with one linearly degenerate field, a rigorous framework is successfully established in [3, 4], which indicates that the initial oscillations propagate along the linearly degenerate field and cancel along the genuinely nonlinear field. We also refer the reader to [48] and [2] for some evidence of large oscillations for strictly hyperbolic and linearly degenerate systems. In Subsection 3c, we develop a more general framework (Hypotheses (Bl,2)), which is shown to be useful for conservation laws in Sections 4-6 .
Section 4 is concerned with n x n systems with fully contact fields, i.e., linearly degenerate characteristic fields, which are described in [50]. In [50], S~RRE established a global existence and uniqueness theorem for smooth solu- tions for the Cauchy problem for such systems by finding a hierarchy of transport equations involving higher derivatives of Riemann invariants wi's and by using the local existence and uniqueness theorem of KA~ [28]. For n = 2, existence theorems of global generalized solutions were established in [47] for BV solutions and in [5] for L ~o solutions. In Section 4 we study well- posedness in the space L ~176 and in the space L ~ n BV, and establish the ex- istence, uniqueness, and stability of generalized solutions. We also study the propagation problem of initial oscillations for such n x n systems by using the quasidecoupling method described in Section 3.
We remark that the basic assumption of all existing uniqueness theorems for systems of conservation laws, as in [14, 12, 35, 42], is that the solutions lie in the space BV n L ~~ In Section 4, our uniqueness theorems are estab- lished in the space L ~176 which is much broader than B V n L ~.
In Section 5 we establish uniqueness and stability theorems for generalized solutions to systems with one contact field and one line field, identified by TEM~'Ln [52], with the aid of the quasidecoupling method. Several important models arising in the fields of elasticity theory (cf. [30, 1]), enhanced oil recovery (cf. [27, 53]), and magnetohydrodynamics (cf. [20, 1]) provide ex- amples of such systems. In this connection we note that for such systems ex- istence theorems of generalized solutions have been established in [53, 4], the propagation and cancellation of initial oscillations have been identified in [4, 5] and well-posedness is obtained for a polymer flooding model within a special class of solutions in [54].
Section 6 describes an example of how to reduce the study of uniqueness and qualitative behavior of solutions for large systems with linearly degenerate fields to corresponding problems for much smaller systems by using the quasidecoupling method. This example is the system of electromagnetic plane waves, derived from the Maxwell equations by SERGE [49], who succeeded in showing global existence of L = solutions by related technique. This example indicates that the quasidecoupling method will prove useful for dealing with
Discontinuous Solutions to Conservation Laws 135
large systems of conservation laws with linearly degenerate fields arising in magnetohydrodynamics, electromagnetic theory, and elasticity theory.
In Section 7 I remark on further applications of the quasidecoupling method to some important problems and methods for conservation laws.
2. Entropy, Solutions, and the Young Measure
We begin with several remarks concerning entropy, solutions for conserva- tion laws, and the Young measure, which motivate part of the subsequent development. We recall that u is called a weak solution of (1.1) with initial data uo(x) if the integral relation
II (u4~t + f ( u ) Ox) dx dt + o( 4~(x, 0) Uo(X ) dx = 0 (2.1) 1 7 T - o o
holds for any function q~ ~ C~(HT), where HT = (--co, co)X [0, T]. One of the main features of conservation laws is that uniqueness is lost
within the broader class of weak solutions; many weak solutions may share the same initial data. Thus the problem arises of identifying an appropriate class of weak solutions (i.e., admissibility criteria) to single out physically rele- vant solutions.
In this connection we recall that a function r/: R n ~ ~ is an entropy for (1.1) with entropy flux q : ~n ~ [R if all smooth solutions satisfy an additional conservation law of the form
rl(u)t + q(U)x = O, that is,
Vq = Vr/Vf. (2.2)
Taking the inner product of (2.2) with the right eigenvectors ri of Vf pro- duces the characteristic form
(2i V r / - Vq). ri = 0, (2.3)
2i rlwi = qwi, (2.4)
provided that the system (1.1) is endowed with globally defined Riemann in- variants ,,). For n = 2, the linear hyperbolic system (2.2) is equivalent to (2.4), while the system (2.4) can be reduced to a linear second-order hyperbolic equation for which the Goursat problem is well posed in the coordinates of Riemann invariants. The entropy space is infinite-dimensional and is represented by two families o~(wl) and fl(w2) of functions of one variable, provided 21 < 2 2. For n ____ 3, the system (2.2) is overdetermined and this generally prevents the existence of nontrivial entropy. FRIEDRICHS & LAX [21] observed that most of the conservative systems that result from continuum mechanics (in one and several dimensions) are endowed with a globally de- fined, strictly convex entropy. A description of systems endowed with a rich family of entropies has been provided by SEI~RE [50]. The algebraic charac-
for distinct i, j , and k. For the class of systems (1.1) endowed with a strictly convex entropy v/,
LAX [32] postulated the following entropy criterion: A weak solution u is ad- missible if
vl(u) t + q ( u ) x <= O, (2.6)
in the sense of distributions. The entropy inequality (2.6) is a proper statement of the second law of thermodynamics for the equations of fluid dynamics: The entropy increases along thermodynamic processes. It is easy to check that, on discontinuities of u with speed s, left state u_, and right state u+, (2.6) reduces to the local condition
s ( ~ ( u + ) - ~ ( u _ ) ) - ( q ( u + ) - q ( u _ ) ) >_ O . (2.7)
Note that the inequality (2.7) imposes a restriction on each point of discon- tinuity, a fact that suggests that admissible solutions of (1.1) may be deter- mined by means of shock criteria. LAX [321 proposed the Lax shock criterion for systems in which each of the eigenvalues is either genuinely nonlinear or linearly degenerate:
h i (u+) <-_ s <= hi(u_) . (2.8)
This criterion governs the number and type of characteristics impinging on a shock wave. For more general systems, a comprehensive shock criterion has been proposed by LIU [361:
s ( r + ; u _ ) < = s ( r ; u _ ) , 0 _ < r _ < r +, (2.9)
where s ( r ; u_) denotes the speed of the shock that joins u ( r ; u_) with u_ determined by the Rankine-Hugoniot condition. On the other hand, it is known that the second law of thermodynamics is not powerful enough to rule out all undesirable thermodynamic processes. DAFF~RMOS [8] proposed the en- tropy rate criterion, which recognizes as admissible those solutions that max- imize the rate of entropy production.
DAFERMOS [9, 10] analyzed these criteria and showed that they are all equivalent when applied to weak solutions with nondegenerate shock speed; for shocks of moderate strength, the shock criterion (2.9) and the entropy rate criterion are equivalent and generally more discriminating than either the Lax shock criterion (2.8) or the entropy criterion (2.6) or (2.7). For shocks of great strength, the relationship among these criteria is generally complicated; some peculiar phenomena have been discussed in [4, 37, 57, 58].
In view of the existence theory of conservation laws (cf. G~IMM [22]), it is natural to consider solutions in the space L ~176 or L ~ n BV. Here BV(g2) denotes the space of functions that have locally bounded total variation in the sense of Cesari; that is, a real-valued function u = u ( y ) c : R n defined on a region g2 C ~m is an element of B V ( Q ) if it is locally integrable and if
Discontinuous Solutions to Conservation Laws 137
its gradient is a locally finite Borel measure, namely,
u gO dy = - I O dl~ ~2 Y2
for all ~ E C ~ ( 9 ) , where /~ is a locally finite Borel measure. On the other hand, u E BV(g?) if and only if the local total variation in each x i is locally integrable in x[ = (xl . . . . . x i - 1 , Xi+l . . . . , x,,) :
S TVu(x/ , . ) dx; < ~ , !x i I<-M
for all M < cr It is known (cf. VO~'PZRT [55]) that u induces a parti t ion of ~m into the union of three pairwise-disjoint subsets: the set of points of Lebesgue approximate continuity of u, the set F (u ) of points of Lebesgue ap- proximate jump discontinuity of u contained in the countable union of Lipschitz arcs, and the set of irregular points whose ( m - 1)-dimensional Hausdor f f measure H(m-a) is zero.
In the context of generalized solutions u E BV, the set F ( u ) is referred to as the set of shock waves and contact discontinuities of the solution u. For simplicity, we assume that, for every fixed { ~ [0, ~) , u (., {) is a function of bounded variation on ( - c r cr with one-sided limits u ( ~ • {) for every 2E ( - c r ~ ) , and u is equal to the symmetric mean u = ~. Here
= lim q/m * u m .-..r cx~
is defined Ha-almost everywhere and coincides with u H2-almost everywhere, where {q/m} is a sequence of radially symmetric functions approximating the ~-function. Consider the B-measure defined by
Bu = r l (u ) , + q ( U ) x , (2.10)
where (r/, q) is a C 2 entropy-entropy flux pair of (1.1). Using the generalized Green theorem for measures [19], we can express the B-measure by
B u ( E ) = B , ( E n F ( u ) ) = ~ (nt[t/] + nx[q]) dHa. (2.11) Enf'(u)
Here n = (nx, nt) denotes the unit normal to E r i E ( u ) in the sense of FEDERER [19], and the bracket denotes the jump of the enclosed quantity across E n F ( u ) in the direction of n:
[y/] = [~/] (P) = I1{ I ,u (P)} - q [ l _ , , u ( P ) } ,
where l + , , u ( P ) denote the approximate limits of u at P E E n F ( u ) with respect to the half-planes [ Q : < Q - P , + n >=> 0}.
We normalize the direction of n by requiring that n~ < 0, and refer to
u_ =- l , u and u+ =- l _ n u
as the left and right approximate limits of u at P, respectively. Then the bracket denotes the jump from left to right across a shock or a contact discontinuity. We note that the set of irregular points and points on which nx = 0 for
13 8 G.-Q. CrIEN
u ~ BV(fl~2+) has one-dimensional Hausdorff measure zero. Therefore, we can write (2.11) as
Bu(E) = ~ (s[r/] - [q]) dt, (2.12) Ec~F(u)
where s - - n t ( P ) / n x ( P ) denotes the speed of propagation of F ( u ) n E at P and dt - - n t d i l l .
The Young measure provides an appropriate tool for studying oscillations in solutions of (1.1), that is, the behavior o f oscillations of perturbed solutions u e as the perturbation parameter e goes to zero. We recall from TARTAR [51] and MURAT [39, 40] that, for an arbitrary sequence of measurable maps
U e : ~ m - - - ~ n, I l u ~ l l ~ _ < M < o o ,
that converges in the weak-star topology of L ~~ to a function u,
w*-lim u e = u, e~0
there exists a subsequence (still labeled) u ~ and a family of Young measures
Vy ~ Prob(~n) , supp Vy C {2 :[2 1 _<M}, y E l P , m ,
such that for all continuous functions g,
w*-lim g ( u e ( y ) ) = I g(2) dvy = (Vy, g ( 2 ) ) , e~O ~n
for almost all points y ~ ~ . In particular, u ~ converges strongly to u if and only if the Young measure Vy at almost all points y is equal to a Dirac mass concentrated at u(y ) . Then, we have the following theorem.
w*-lim 8~0
and
Theorem 2.1. Let f2 C R % be a bounded open set and u C : f 2 ~ ~n be measurable functions satisfying
(i) ]l u~ IIL~ =< M. (ii) For continuous function pairs (rli(u), qi(u)) (i = 1, 2),
Orli(U~) + Oqi(ue~) is compact in Hlolc(s (2.13) 3t Ox
Then there exists a subsequence (still labeled) u e and a family of Young measures {Vx, t: R2+ ~ Prob(Rn)}, supp Vx, t C {)~ :[2[ _< m}, such that for all continuous functions g,
g (ue ( x , t)) = ~ g ( 2 ) dvx, t - (Vx, t, g ( 2 ) ) , (2.14) ~+
Next, we consider the system (1.1) endowed with globally defined Riemann invariant coordinates and satisfying the formula (2.5). Suppose that one of its eigenvalues 2i is linearly degenerate in the sense of LAx [33]:
Owi2i -- O, i.e., 2 i = 2i (w[) , (2.16)
Discontinuous Solutions to Conservation Laws 139
where w[ = ( w 1 . . . . . Wi_l, Wi+ 1 . . . . . W n ) ~ R n-1. Then we deduce from (2.4) that there exists a family of entropy-entropy flux pairs (rli, qi) corresponding to the linearly degenerate field which are of the form
rli = Nia(wi), qi = Ni)~i(w') a(wi) , (2.17)
where a is any continuous function of the scalar variable w i, and
Owj)~i N i = exp 2j dwj , j * i, (2.18)
is independent of j ( . i ) because of (2.5). If one can choose a(wi) such that the associated r/i is convex, then
Ot(Nia(wi)) + Ox(2iNia(wi) ) <__ O, (2.19)
in the sense of distributions. Generally, there always exists an ao(wi) >= 0 such that
Ot(Niao(wi)) + Ox(2iNiao(wi)) = 0, (2.20)
which is derived from one equation of the system (1.1). Using the notation of the Young measures Vx, t ~ P r o b ( N n) for the
dissipative approximate solutions, we can obtain the associated average ine- quality of (2.19) and equality of (2.20). In particular, if the Young measures vx, t have the following property: for any continuous function a(wi), there ex- ists an L = function ~i(x, t) such that
(Vx, t, )~i(W[) a (wi ) N i) : ~,i(x, t) (Vx, t a (wi ) Ni ) a.e., (2.21)
then Ot(Vx, t, N i a ( w i ) ) + Ox(~. i (x , t ) (Vx, ,Nia(wi) ) ) <=0, (2.22)
in the sense of distributions. Finally, we remark that if So = {(x, t) :Ni(x, t) = 0} * 4, for nonstrictly
hyperbolic cases, we can assert that the inequality (2.19) holds in the stronger, generalized sense of distributions, that is, for any 4 , 6 C l o ( H r - S o ) n BV(Hr), 4, _-> 0,
Similarly we can also assume that the inequality (2.22), and the equalities (2.20) and (2.23) hold in the same sense.
3. Method of Quasidecoupling
In this section we describe the method of quasidecoupling for studying the existence, uniqueness and qualitative behavior of solutions to systems of con- servation laws with linearly degenerate fields.
140 G.-Q. CHEN
a. Quasidecoupling Transformation
Let w : ~2+ __, R be a bounded and measurable function satisfying the ine- quality
Ot(a(w;x, t) P(x, t ) ) +ax (a (w;x , t ) 2 ( x , t ) P ( x , t ) ) <_60 (3.1)
and the initial condition
w(x, O) = Wo(X) (3.2)
in the generalized sense of distributions; that is, the inequality
l~ a(w;x, t) P(x, t) (~t+ ,~(x, t) ~x) ~ d t tit
+ ~a(wo(x) ;x ,O)P(x ,O)C)(x ,O)dx>=O (3.3) - - o o
holds for any nonnegative function ckE C I ( F I r - { P ( x , t ) = 0 } ) c~BV(IIr). Here the function a(w; x, t) is continuous with respect to w and uniformly bounded with respect to (x, t), and (2(x, t)P(x, t), P(x, t)), P(x, t) >_ O, is a bounded and measurable vector field whose divergence vanishes in the sense of distributions
divx, t(2(x, t)P(x, t), P(x, t)) = 0, (3.4)
o
P(x, O) dx= I P(x, O) d x = +0o. (3.5) 0 - - o o
Then we have the following theorem.
T h e o r e m 3.1. There exists a globally defined coordinate transformation
T: (x, t) ~ (p, t) - (Tt(x), t)
such that, in the new coordinate system (p, t) = T(x, t),
Ota(w(T-l(p, t ) ) ; T-m(p, t)) < O, w[t= 0 = wo(Tol(p)), (3.6)
in the sense of distributions, that is,
I~ a(w(T- l (P, t ) ) ; T-l(p, t)) ~v(p, t)tdp dt lit
+ ~ a(wo(To-l(p));T-l(p,O)) y/(p,O) dp>=O, (3.7)
for all nonnegative functions ~u~ C0~(//7-), where Tol(p) is the inverse map of Tt(x)lt= o = p ( x , O) = f~ P(~, 0) d~, uniquely determined by P(x, 0).
Remark. Under the transformation T, the divergence form of the vector field
(a(w;x, t) 2(x, t) P(x, t), a(w;x, t) P(x, t))
Discontinuous Solutions to Conservation Laws 141
is transformed to a complete single differential form Ota(w(T- l (p , t ) ) ; T - l(p, t)), and the inequality in (3.6) is formally independent of the vector fields (2 P, P) in the new coordinate system (p, t). For these reasons, we call the transformation T a "quasidecoupling transformation" corresponding to the vector field (2 P, P).
I_emma 3.1. Let the L ~ vector field (Q(x, t), P(x, t)) , divergence-free in the sense of distributions in HT, and let
j P ( x , 0) d x = +c~, -c~__<c~<fl_< +c~.
Then, for almost all t E (0, T),
j P(x, t) dx = +c~. Ol
P(x, t) >_ O, be
Proof. The proof is simple. As in [56], we choose two sequences of non- negative smooth functions Ce(r) and ~ ( x ) with compact support for any t E (0, r ) ,
qSe(r) = f l , rE [0, t], {. 0, r E [ t + e, T],
such that
~'~(x) = f l , x~[c~,/~], 0, xE ( - ~ , , ~ - ~1 u [fl + c~, + ~ ) ,
j [0~(r) l d r = I ~,~(x)l d x = 2 . 0 - - ~
Since div(Q(x, t), P(x, t)) = 0, we have
j~ (O~(r) tp'~(x) P(x, t) +4)e( z) ~ , (x) Q(x, t) ) dx dt + ~ q/~(x) P(x, O) dx = O. FI T - ~
Let a--, O. We obtain
~a(x) (n(x , t) - P ( x , 0)) dx _-< 2IIQIIL~ t,
for any Lebesgue point t of all locally integrable functions t ~ J ~,a(x)x P(x, t) dx. Let fi --, 0. For almost all t ~ (0, T), we have that
f ( P ( x , t) - P ( x , 0)) dx __< 211QIk~ t.
Therefore
J P(x , t) d~ = j e ( x , o) dx = + ~ . OL OL
142 G.-Q. CHEN
Proof of Theorem 3.1. Notice that the vector field (~(x, t)P(x, t), P(x, t)) is divergence-flee. We introduce a coordinate transformation
determined by
T: (x, t) ~ (p, t),
x
Op = P(x, t), Op _ ).(x, t) P(x, t), p(x, O) = ~ P(~, O) d~. (3.8) Ox Ot o
Then the transformation T is well defined, and there exists a globally defined Lipschitz solution (p, t) = T(x, t) in the space of distributions. From (3.5) and Lemma 3.1, we have
; o P(x, t) dx = ~ P(x, t) dr= + ~ .
0 - - o a
Then T is proper and onto in N2. If P(x, t) > 0, then T is a bi-Lipschitz homeomorphism with Jacobian
P(x, t ) > 0. Then we have from (3.3) that, for any nonnegative function ~u ~ Cy (Hr),
jj a(w;x, t) P(x, t) (q/t + )~(x, t) ~x) drdt tit
+ ~ a(wo(x);x, O) P(x, O) q/(x, O) dr __> O. (3.9) - - o o
Making the coordinate transformation (3.8) for the integrals in (3.9) we im- mediately obtain
~ a(w(T- l (p , t ) ) ; T-l(p, t)) ~u(p, t)tdp dt HT
+ ~ a(wo(Tol(p)); r - l (p , 0)) ~u(p, O) dp >= O. - - o o
This leads to (3.7).
If the set {(x, t) :P(x, t) = 0} :~ 0, we can verify the above process of proof under the assumption that (3.1) holds in the generalized sense of distributions and prove that the inequalities (3.1) and (3.6) are equivalent by arguments similar to those of WAGNER [56]. We omit the details.
A useful fact is the following corollary.
Corollary 3.1. Let T be the quasidecoupling transformation (3.8). Then for any g E L I(I-IT),
g(p,t) dp= ~ g(T(x , t ) ) P(x , t ) dx - - o a - - o a
holds for almost all t ~ (0, T).
Discontinuous Solutions to Conservation Laws 143
b. Framework for the Uniqueness and Stability of Solutions
Suppose that L ~ functions w(x, t) and z(x, t) satisfy the following hypo- theses:
AI: w(x, t) is solution of the Cauchy problem
Ot(ao(w-k ) PI(X, t)) + O x ( a o ( w - k ) )Ll(X , t) Pl(X, t)) <=0
w(x, 0) = w0(x),
Vk~ ( -oo , ~ ) ,
(3.10)
in the generalized sense of distributions in H T. A2: z(x, t) is the solution of the Cauchy problem
Here ao is a nonnegative and even Lipschitz function satisfying ao(0) = 0, ao(w) > 0 for w * 0, and vector fields (Pi(x, t), 2i(x, t) Pi(x, t)), Pi(x, t) >= O, i = 1, 2, are two divergence-free fields in (x, t)-space and satisfy
Pl(x, 0) ~ P o > 0, P2(x, 0) =>Po> 0.
Then we have the following theorem.
Theorem 3.2. Suppose that L ~ functions w(x, t) and z(x, t) satisfy Hypotheses A 1 and A 2 in l l T. Then
w(x, t) = wo(Td-l(p(x, t ))) , z(x, t) = zo(To-l(q(x, t))) a.e. t ~ [O, T) ,
_-< tie, IlL-rje21rL- l ao(w0(x) - z0(x)) ~ , Ix I <=Ko(N + Lt)
for almost all tE [0, T), where (p, t) = T(x, t) - (Tt(x), t) and (q, t) = T(x, t)=- ( ~ ( x ) , t) are the quasidecoupling transformations corresponding to the vector fields (2i Pi, Pi), i = 1, 2, respectively, L = max2 = 1,2, (suprz r 12i (x, t) I ), and K 0 = maxi=l, 2 []PiJ[z~/Po .
We first introduce a lemma from [5].
Lemma3.2. Suppose that functions u(x, t) ~L~([O, T ] ; L l ( - c ~ , c~)) and Uo(X) ~L ~ satisfy
jj u(x, t) ~,'(t) & dt + ~,(0) ~ Uo(X) dx >= o H T -o~
for any nonnegative function q/(t) ~ C1[0, T). Then limt__,o ~ ~_~ u(x, t) dx exists and satisfies
u(x, t) dx <= lim ~ u(x, t) dx <= ~ Uo(x) dx a.e. t~ [O, T) . - - o o t ~ O - - o o - - c o
Proof of Theorem 3.2. Using the quasidecoupling transformation (3.8) for (3.10) and (3.11), respectively, we obtain from Theorem 3.1 that the function w ( T - l ( p , t)) satisfies
Ota(w(T- l (p , t)) - k) <= O, wit=0 = wo(ToI (p ) ) , (3.15)
in the sense of distributions in the (p, t) = T(x, t) coordinates for any kr R 1, and the function z ( 57 -1 (q, t)) satisfies
Ota(z (T - l (q , t)) - l) < O, zlt=0 = zo(To1(q) ) , (3.16)
in the sense of distribution in the (q, t) = T(x, t) coordinates for any I~ R 1. First we show that (3.12) holds. Suppose that q/(p, t ;p ' )>=0 is a
C~(Hr x [ R 1) function and, for any fixed p '~ R l ( p ( R 1 ) , that q/(p, t ;p ' ) Co~(Hr). Now we choose k = wo(To1(p') ) and the test function q/(., . ; p ' ) for (3.15). We obtain
~l ao (w(T- l (P , t) ) - wo(Tol(p ' ) )) gtt(p, t; p ') dp dt Hr (3.17)
+ ~ ao (wo(T( l (p ) ) - wo(To-l(p'))) ~(p, 0 ; p ' ) dp >= O. - - o o
Discontinuous Solutions to Conservation Laws 145
Integrate (3.17) over p '~ R 1. We have
o( ~l ao(w(T- l (P , t)) - wo(ro l (p ' ) ) ) q/t(p, t ;p ' ) dp dtdp' -~ rtr (3.18)
z(x , t ) =zo(To l (q (x , t ) ) ) a.e. ( ( x , t )~Hr :Pz (x , t ) > 0 1 .
This proves (3.12). Notice that, for fixed t, both p(x, t) and q(x, t) are monotonically increas-
ing functions of x. If x lies in [ - N , N ] , then both intervals [p( -N, t)/Po, p(N,t)/Po] and [q(-N,t) /Po, q(N,t)/Pol lie in [ -Ko(N+Lt) ,Ko(N+Lt)] , where L = maxi=l,2 suPHT[)~i(X, t)] and Ko = maxi=l,2 lIP/HL=/Po �9 Therefore, for any nonnegative function ~( t ) r C~[0, T) and any positive constant N,
T
I f ao(w(x, t) - z ( L - ~ , ( x ) , t)) &(x , t) P2(~-~7",(x), t) ~,(t) ~ d t 0 Ixl<__U
T
= I I ao(wo(Tol(Tt(x)))-zo(To l(Tt(x)))) P1 (x,t) P2(Tt-lTt(x),t) q/(t)dxdt 0 ]xl<=N
Finally, we can obtain from the boundedness of the domain of dependence of p - q that, for almost all tE [0, T),
j Ip(x, t) - q(x, t)] dx ]x]~N
5- c f j Ipo(x) - qo(x) l dx I Ixl<=u+zt
t ds 1 + j j [,~(q(x, s), s; wo(q(x, s)))- ,~(q(x, s), s; %(q(x, s)))qdx o Ixl<-N
t
<=Cj J ] 2 ( x , s ; w o ( x ) ) - 2 ( x , s ; % ( x ) ) l d x d s j [po(x)-qo(x)ldx o ixl<=~+L s Ixl<=X+Lt
__< c j (Iw0(x) - %(x) l + Ipo(x) - q0(x) l) dx. Ix I<=N+Lt
This completes the proof.
A direct corollary follows.
Corollary 3.2. Suppose that there exists an L o~ function )~ (y, t) such that
21(x, t) = )~(p(x, t), t ) , 22(x, t) = )~(q(x, t), t ) , (3.32)
and suppose that the function ao(w) = Iw] m, m > O. Let Pl(x, O) = P2(x, O) in the Framework (A). Then weighted Lm-stability with weight (P1P2)(x, t) for functions w(x, t) and z(x, t) holds, that is, for almost all tE [0, T),
j Iw(x, t) - z ( x , t) lme (x, t) e2(x, t) ]x]<_N
__< Ilex ]lL~o lip2 Ik~ j [Wo(X)-Zo(X) lmdx , (3.33) Ix l -<K0(N+LO
where L = maxi=l,2 supgrl)~i(x, t) l.
Furthermore, we also obtain a uniqueness theorem from Theorem 3.2 and Theorem 3.3 :
Theorem 3.4. (Uniqueness). Suppose that the functions w(x, t) and z (x, t) satisfy Hypotheses A1, A2 and conditions (3.27), (3.32), and suppose a o ( 0 ) = 0 and ao(w) >O for w t- O. Then Wo(X) = Zo(X) leads to
w ( x , t ) = z ( x , t ) a.e. on I l r - { ( x , t ) : P l ( x , t ) = P 2 ( x , t ) =0}.
Discontinuous Solutions to Conservation Laws 149
Remark. Theorem 3.4 shows that the solution of the Cauchy problem
O , ( a o ( w - k ) P ( x , t ) ) + O x ( a o ( w - k ) 2 ( x , t ) P ( x , t ) ) < O Vkfi ( - c o , co),
w(x, O) = Wo(X),
with a0(0) = 0, ao(w) > 0 for w :~ 0, is independent of the choice of the function P(x, t) > 0 with the same initial data P(x, 0). This finding is essen- tial for solving the uniqueness and stability problems for solutions to systems of conservation laws in Sections 4 - 6 .
c. Framework for the Limiting Behavior of Approximate Solutions
We establish a framework for treating the limiting behavior of approximate solutions to conservation laws. We recall from Section 2 that with each point (x, t )~ H r is associated a Young measure over the range space [R m, which corresponds to a sequence of functions w e(x, t) and describes the oscillatory behavior of w 8 at (x, t). For this reason, we pose our hypotheses on a family of Radon measures.
Consider a family of Radon m e a s u r e s {`Ux, t : H T -+ Prob(R)}, supp `ux, t CC ( - c o , co), and a family of initial Radon measures [`ux,0} that satisfy the following hypotheses: BI: There exists a continuous state variable set A such that, for any a():) EA,
in the generalized sense of distributions in H r . B2: The set A is sufficiently large that, for any function of bounded varia- tion g (2 ) ,
S a (2 ) dg(,~) <__ 0 (3.35)
implies that g(~.) is a monotonically decreasing function. Here the vector field (2(x, t )P (x , t), P(x, t )) , P(x, t) >= O, is divergence-
free in the (x, t)-space:
divx, t (2 (x, t )P (x , t), P(x, t)) = 0, (3.36)
0
j P(x, O) dx = P(x, O) dx = co. (3.37) --oo 0
Then we have the following theorem.
Theorem 3.5. Suppose that a family of Radon measures {,Ux. t : H T -,, Prob (R)} with supp `ux, t CC ( - c o , co) and a family of initial Radon measures {`ux, O}, satisfy
150 G.-Q. C~IEN
hypotheses B 1 and B 2. Then
I.tz, t<ltr~(p(x,t)),o a.e. in {(x, t) e H r : P ( x , t ) >0} , (3.38)
where p : H T ~ ~1 is determined by the quasidecoupling transformation, that is,
Remark 1. Suppose that the conclusion in B 2 is that g()~) is a monotonically increasing function. Then the inequality (3.38) of Theorem 3.5 is changed into
~lx, t >~ [ lT6_l(p(x , t ) ) , 0 a . e . in {(x, t) r t) >0} . (3.40)
Remark 2. The formulas (3.38) or (3.40) express that, along the curve dx
C: - - = )L (x, t) , (3.41) dt
the value of the measures Px, t is smaller or larger than the value of the initial measures /Ix,0. Formula (3.38) indicates that the Radon measures Px, t are Dirac measures, provided that #x,0 are Dirac measures, which shows con- vergence of the corresponding approximate solutions in the strong topology {(x, t ) r t ) > 01. Formula (3.40) indicates that the Radon measures l~x,t are not Dirac measures, provided that/~x,0 are not Dirac measures. Thus the initial oscillations propagate along the curve (3.41). This framework is useful for studying the limiting behavior of approximate solutions to conserva- tion laws, especially the dynamic behavior of the initial oscillations (see Sec- tions 4 - 6).
Proof of Theorem 3.5. First we have from (3.34) that
~I (Px, t, a(2)) e (x , t) {~bt+ ~(x, t) (~} dx dt I I T
oo
+ ~ (Px, O, a(2)) P(x, 0) 0(x, 0) dx __> 0 (3.42) - - c o
for any a ~ A and any nonnegative function ~b ~ B V ( H T ) n C I ( H r - {(x, t ) : P ( x , t ) = O } ) . Notice that the vector field ( 2 ( x , t ) P ( x , t ) , P ( x , t ) ) is divergence-free in the (x, t)-space and that
o
f P(x, O) dx = P(x, O) dx = +co. - - ~ 0
Making the quasidecoupling transformation T: (x, t) ~ (p, t) in the integral (3.42) we conclude from Lemma 3.1 that the transformation is proper and on- to in H T. We obtain from (3.42) that
T
~ (lUr_l(p,t), a(2) ) 4~(P) q/( t ) dp dt 0 --co
+ ~ ( 0 ) ~ (Prolr V a e A , - - c o
for any ~(p) ( C0~(-co, ~ ) , ~( t) E C110, T), ~(p) ~,(t) __> 0.
Discontinuous Solutions to Conservation Laws 151
Using Lemma 3.2, we have
(~lT_l(p,t) , a(A~)) 6(P) dp <= ~ (l~Tr a()~)) 6(P) dp, - - o o - - o o
that is,
~ (flT_l(p,t) -- /2T(X(p),0, a (2 ) ) 6 (P) dp <= 0 - - o o
for any nonnegative 6~ C ~ ( - ~ , oo). Using Corollary 3.1, we have
(~x,t--flTo~(p(x,t)),o,a()~))~O a.e. in { (x , t )~ lTT:P(x , t ) >O}. (3.43)
Notice that A is sufficiently large that the assumption (B2) holds. Thus we obtain from (3.43) that
flx, t ~ flT~(p(x,t)),O a .e . in {(x, t) :P(x, t) > 0}.
Therefore, we arrive at (3.38).
4. Completely Degenerate Systems
In this section we consider how the quasidecoupling method can be applied to the Cauchy problem for completely degenerate systems, which are n• systems with n independent contact fields, that is, linearly degenerate fields (cf. SERRE [ 5 0 ] ) :
for distinct i, j , and k, where 2 i, 1 _< i _< n, are eigenvalues of the Jacobian Vf(u), and wg, 1 <_i<_n, are corresponding Riemann invariants. For simplicity, we assume that
�9 ~1 <~ ~2 <~ ' ' " <~ An, h i * 0, 1 _< i _< n, (4.4)
and the Riemann invariants w i, 1 <_ i <_ n, are smooth functions of u. The entropy-entropy flux fields for such systems have the following general
a ( w ) = ( a l ( w l ) . . . . . am(Win)) , where
(4.6)
Owjl~i Ni = exp ~ dwj (4.7)
is independent of j , j . i, because of the equality (2.5) for all distinct i, j , and k, and
Ne(u) __> c0 > 0, (4.8)
on the parallelepiped K = H i [inf w ~ sup w ~ provided (4.4) holds. A simple 2 • example of such systems is (see [52])
0 U 2 t U 3 -~- U 1 U 2 x
SERRE [50] proved the global existence and uniqueness of a smooth solution for the Cauchy problem (4.1) by using a hierarchy of transport equations in- volving higher derivatives of wi, and the local existence and uniqueness theorem of KATO [28].
Theorem 4.1. Assume that 2i, 1 <_ i <_ n, are smooth functions of w = (wl . . . . . wn). Then there exists a unique smooth solution u(x, t) of the Cauchy
problem (4.1) having the following regularity: wE c k ( R 2 ; K) =- {wE c k ( ~ 2) : wEK} if and only if wOE ck(R2+; K).
This result shows that the solution for the Cauchy problem (4.1) is as regular as the initial data because the systems are degenerate. With the aid of the quasidecoupling method, we now study the well-posedness of general- ized solutions for the Cauchy problem with the initial data woEK and Wo E BV(R2+; K) : We treat existence, uniqueness, stability, and the dynamic behavior of the initial oscillations for the generalized solutions.
Definit ion 4.1. A bounded measurable vector function u (x, t) E R n is called a generalized solution of the Cauchy problem (4.1) in H r if the equality
~ {tla(U(X, t)) c) t + qa(U(X, t)) ~x} dx dt + ~ tla(Uo(X)) th(x, 0) dx = 0 (4.9) H r -oo
holds for any continuous vector function a(w) and any smooth function E C~o(HT).
a. Propagation of Oscillations and Existence of Generalized Solutions
We first study the dynamic behavior of the initial oscillations, which are of amplitude O(1) with high frequency, as time evolves. That is, we study the
Discontinuous Solutions to Conservation Laws 153
limit behavior of the smooth solutions u~(x, t) in Theorem 4.1 corresponding to the oscillatory sequence of large initial data:
ut + f(U)x = O,
ult= 0 = u~(x) E ck(~2+; K),
We have the following theorem.
k_>0.
(4.10)
Theorem 4.2. Consider a completely degenerate hyperbolic system of conservation laws satisfying (4.3) and (4.4). Suppose the sequence of the initial data u~(x) E Ck( R% ; K) and the bounded set K are independent of e. Then the corresponding C ~ solutions u~(x, t) have the following limit behavior:
(i) Let one of the corresponding initial Riemann invariants
w~o(x) = wi (uS(x ) )
be a highly oscillatory sequence. Then the initial oscillations propagate along the corresponding linearly degenerate field.
(ii) Let the initial data sequence u6(x) be compact in the strong topology of L ~176 Then there exists a subsequence converging pointwise a.e. to an L ~ generalized solution to the Cauchy problem (4.1) with the Cauchy data Uo(X) = w*-lim uS(x).
Remark. Theorem 4.2 shows that the initial oscillations can propagate along the corresponding linearly degenerate field, which is not affected by other fields. This is consistent with theoretical results of propagation of the initial oscillations for 2 x 2 systems in [4, 5] and with the evidence of oscillatory solu- tions given in [48, 2]. This phenomenon differs from those in [34, 15, 3].
Proof of Theorem 4.2. We first obtain from Theorem 4.1 that
uC(x, t) ~K (K is independent of e), (4.11)
/'/a(Ue(X, t ) ) t "Jr- qa(Ue(X, t ) ) x -~- O, (4.12)
for any continuous vector function a(w). Using Theorem 2.1, we conclude that there exists a family of Young measures v~,t:~. 2 - , Prob(Rn), supp Vx, t C K, uniquely determined by the approximate solutions u~(x, t), satisfying
3t(Vx, t, Niai(wi)) + Ox(Vx, t, fliNia(wi)) = O, 1 <_ i <_ n,
and, for any continuous vector functions a(w) and b(w),
We conclude from (4.14) that there exists an L ~ function 2i(x, t) such that
(Vx, t, ) t iNia i (wi ) ) = ~i (x , t) (Vx, t, N i a i ( w i ) ) (4.15)
154 G.-Q. Crt~N
for any continuous vector function a ( w ) . Therefore, (4.13) can be written in the following form:
Ot(Vx, t, N i a i ( w i ) ) -}- Ox(~ i (x , t) (vx, t, N i a i ( w i ) ) ) :- O, 1 <_ i < n . (4.16)
In particular,
3t(vx, t, N i ) + Ox(~.i(x, t) (Vx , , N i ) ) = O, 1 < i <_ n , (4.17)
(Vx, t, N i ) ~ c O > O.
For fixed i, 1 < i _< n, we define a family of probability measures:
bli, t _ NiVx, t
(Vx, t, N i ) Then i /zx, t satisfies
( i cgt ( Ux, t, a ( w i ) ) ( V x , t, N i ) ) + 3x(QUix, t, a ( w i ) ) 2 i ( x , t) (Vx, t, N i ) ) = 0. (4.18)
We conclude from (4.17)-(4.18) that the family of probability measures satisfies Hypotheses B1 and B2, and that the set A is the space of continuous functions. Therefore we have from Theorem 3.5 that
/.,l/, t i (4.19) ~" ~lT~l(pi(x,t)),O,
where p i ( x , t) is determined by
apAx, t) Ox - (Vx, t, N i ) >= Co > O,
api(x, t) -- ~,i(x, t) (Vx,,, N i ) ,
Ot
p i ( x , 0) = ~ (Vr N i ) d ~. o
If the composite sequence of approximate initial data wf0(x) = w i ( u ~ ( x ) ) is highly oscillatory, then /zx, 0 is not a point measure, that is, /z~, t is not a point measure from (4.19). Therefore Vx, t is not a point measure. This means that the initial oscillations propagate along the "average" eigendirection:
__dx ~-. ~.i(X, t) - (Vx't' ~ iNi ) dt (vx, t, N i )
If the sequence uS(x ) of initial data is compact in the strong topology of L =, then there exists a subsequence (still labeled) u~(x ) such that all com-
E posite sequences Wio converge strongly to Wio(X),
wf0(x) = w i ( u ~ ( x ) ) ~ Wio(X) a.e. 1 < i <_ n ,
and i /tx, 0, 1 < i < n, is a Dirac measure. Therefore i _ _ ltx, t, 1 <_ i <_ n, is a Dirac measure from (4.19). Finally, we conclude that v~,t is a Dirac measure. We know from Section 2 that there exists a subsequence (still labeled) u~(x, t) converging pointwise a.e. and that the limit function is a generalized solution of the corresponding Cauchy problem (4.1) with Cauchy data U o ( X ) = w*-lime_~ 0 u [ ( x ) . This completes the proof of Theorem 4.2.
Discontinuous Solutions to Conservation Laws 155
A direct corollary follows.
Corollary 4.1. For the Cauchy problem (4.1) for completely degenerate systems, the solution operator St:L~~ n) ---,L~ n) with S t (uo)= u( . , t) is not com- pact.
As a byproduct, we have the following theorem.
Theorem 4.3. There exists an L ~ generalized solution u(x, t) E K to the Cauchy problem (4.1) with L ~ Cauchy data Uo(X) ~ K that satisfies
rla(U(X , t)) t + qa(U(X, t)) x ---- 0 (4.20)
for any continuous vector function a (w) in the sense of distributions.
Proof. Note that, for any Uo(X)~L ~~ there exists an approximate sequence u~(x) ~ C~176 K) of Uo(X) such that, for any N > 0,
II U~ -- U 0 I/LP(-N,N) ~ O, ~, ~ O. (4.21)
In fact, we can take
u~(x) = (J~*uo) (x) ,
where J~ is a standard compactly supported nonnegative mollifier:
Then we consider the Cauchy problem (4.1) with Cauchy initial data u~(x). We obtain from Theorem 4.1 that there exists a unique solution u~(x, t) for this Cauchy problem. Using Theorem 4.1, we conclude that there exists a subsequence ue~(x, t) converging a.e. to an L = solution u(x, t) for the Cauchy problem (4.1) satisfying (4.20).
Finally, we establish a regularity theorem for the generalized solutions of the Cauchy problem (4.1) in Theorem 4.3 by using the method of quasi- decoupling.
Theorem 4.4. Consider the generalized solutions of the Cauchy problem (4.1) satis- fying (4.20). I f the initial data Uo(X)~ BV([E1;K), then the corresponding generalized solution u (x, t) is also a function of bounded variation:
u (x, t) E BV( ~2+; K) , (4.22) satisfying (4.20).
Proof. Suppose that wi, 1 < i < n, are Riemann invariants corresponding to i-th characteristic fields, respectively. Choose
P(x, t) = N i ( u ( x , t )) , ).(x, t) = )~i(u(x, t )) , ao(w -- k ) = Iwi - k I
in Hypotheses AI and A 2. We immediately obtain from Theorem 3.2 that, for almost all t~ [0, T) and x~ ( - o o , oo),
w i ( u ( x , t) ) -= w i ( u o ( T ~ l ( p i ( x , t)))), (4.23)
156 G.-Q. CI-IEN
where the Lipschitz function Tdil(p) is the inverse map of Toi: x
x ~ Toi(X ) = i Ni(uo(~)) d~, o
and the Lipschitz functions p~(x, t), 1 < i<_n, are determined by the quasidecoupling transformations corresponding to the vector fields (2i(u) N(u) , Ni(u) ) , respectively. Therefore, we conclude from (4.23) that
w ( u ( x , t)) ~BV(R2; K), 1 <_ i <_ n,
and (4.22) follows.
b. Uniqueness and Stability of Generalized Solutions
We first study the stability of L ~ generalized solutions for the Cauchy problem (4.1) established in Subsection 4a.
Theorem 4.5 (Stability of the L ~ Generalized Solution). Suppose that L ~ func- tions u(x, t) and v(x, t) are generalized solutions of the Cauchy problem (4.1) with initial data Uo(X) and Vo(X), respectively. Then, for almost all t ~ [0, T) and any positive constant m,
w i ( u ( x , t ) ) = w i ( u o ( T ~ l ( p i ( x , t ) ) ) ) , w i ( v ( x , t ) ) = w i ( v o ( T ~ l ( q i ( x , t ) ) ) ) a.e.
(4.24)
S Iwi (u(x , t)) - w,(v(?,71T.(~), t))I m d~ Ixl<=g
<=c I IxI <-_Ko(N § Lt)
I w ( u o ( x ) ) - w (v0( Z0~ 1Toi(X ) ) ) I m d x ,
I w i ( u ( r t 7 1 T t i ( x ) , t ) ) - wi(v(x , t)) lm dx Ixl <-_N
(4.25)
<=c f Ix[ <=Ko(N + Lt)
I w ( u o ( T ~ 1 Toi(X))) - W(Vo(x) ) l m dx,
I IWi( H(x' t ) ) -- w i (v (T t71~ 'o iT~ lZ t i ( x ) , t ) ) I m dx Ixl<=N
<_ c j I w( uo ( x ) ) - w(vo(x)) I'n a~, Ix] "< Ko(N + Lt)
~_~ C i IW(Uo(x)) -- W(Vo(X)) I md'x, Ixl < Ko(N + Lt)
(4.26)
Discontinuous Solutions to Conservation Laws 157
where L = maxlsi___~(su p 12i(u) l, sup 12i(v) l ) ; C is a constant independent of m, L, and N; Ko=maxl<_i<_.(supnNi(u(x, t )) , supr iNi (v(x , t ) ) ) /Co; w = (wl, w2 . . . . . w~) with Riemann invariants wi, 1 <_ i <_n, corresponding to the i-th characteristic field; and pi(x, t) = Tti(x) and qi(x, t) = ~ i (x ) are determined by the quasidecoupling transformations corresponding to the vector fields (2~(u(x, t ) )N~(u(x , t ) ) ,N~(u(x , t ) )) and (2 i (v (x , t ) )N~(v(x , t ) ) ,N~(v(x , t ) )).
and then (4.25)-(4.26) follows for any N > 0. Now we consider the difference between
(pi(x , t ) , t ) =- (p~ (x , t ;w(u (x , t ) ) ) , t )= r~(x, t)
and (qi(x, t), t) - (qi(x, t; w(v(x , t ) ) ) , t) = ~.(x, t) for the systems consisting of two equations (n = 2).
Theorem 4.6. Suppose that L ~ functions u(x, t) and v(x, t) are generalized solutions of the Cauchy problem (4.1) with initial data Uo(X) and Vo(X), respec- tively. Then, for almost all t ~ [0, T),
I I (T t i - l r t i - - I ) (x) l dx Ix]~N
<= C ~ {]W(Uo(X))-W(Vo(X)) I +lW(Uo(X))--W(vo(T~lToi(X)))l}dx, Ix] <=Ko(N + Lt)
i = 1, 2, (4.27)
where L = maxi=l,2 (sup 12i(u) I, sup 12i(v) 1) ; /2 = L + 1 + maxi=l,2 (suP[0,rl x (I Tti-1Tti(q-N) [ - N ) / t ) ; Ko=maxi=l,z(supii(22(u ) - ~ l ( U ) ) -1, suPH (22(V) -- 21(v))-1)/Co; C is a constant independent of m, L, and N; w = (Wl, w2) with Riemann invariants wi, 1 <_ i <_ 2, corresponding to the i-th characteristic field; and pi(x, t) = Ta(x) and qi(x, t) = ~ i (x ) are determined by the quasidecoupling transformations corresponding to the vector fields (~i(22 --~1) -1, (22 - -2 i ) -1) (u(x , t)) and (2i()~ 2 - 21) -1, (22 - 21) -1) (v(x, t)) .
Proof. Using Theorem 4.5, we know that the Lipschitz functions pi(x, t) and qi(x, t), determined by the quasidecoupling transformations, satisfy the following Cauchy problems:
p i ( x , t ) t + 2 i ( w j ( u ( x , t ) ) ) p i ( x , t ) x = O , j . i , (4.28)
X
pi(x, 0) = ~PI (~ , 0) d~, 0
158 G.-Q. C ~ N
qi(x, t)t + ~i(wj(v(x , t) )) qi(x, t)x = O, j e~ i, (4.29)
X
qi(x, 0) = j P2(~, 0) d~. 0
Notice that, under the trans formation T i (x, t) = (Pi (x, t), t ) , wj (u ( T i- 1 (Pi, t) )) (i :#j) satisfies the Cauchy problem
Iwj - h i , + Iwj - ~l,,i = o , (4.30)
Wilt= 0 = w j ( u o ( T ~ l ( p i ) ) ) .
Furthermore, under the transformation ~ (x, t) = (qi (x, t), t ) , wj ( v ( ~ - l ( q i , t ) )) (i :#j) satisfies the Cauchy problem
Iwj - k l , + Iw j - k lq, = o ,
wjlt:o = w+(vo(T(,'(qt) )).
Therefore, we can deduce that, for almost all (x, t )~ H r,
Substitute
pi(x, t)t + )~i(wj(uo(T~l(pi(x , t) - t ) ) ) ) pi(x, t)x = O,
X
pi(x, 0) = I PI(~, 0) d~, 0
(4.31)
wj(u(x , t)) = w j (uo(T~ l (p i ( x , t) - t ) ) ) , (4.32)
wj(v (x , t)) = wj (vo (T~ l (q i ( x , t) - t) ) ) .
(4.32) into (4.28)-(4.29). For almost all te [0, T) we have
j r - i , (4.33)
qi(x, t)t + ) L i ( w j ( v o ( ~ ' ~ l ( q i ( x , t ) - t ) ) ) ) qi(x, t)x = O,
X
qi(x, 0) = ~ P2(~, 01 d~. 0
j :# i , (4.34)
Notice that
Ipi(x, O) -- qi(x, O) I dx <= C f I W(Uo(X)) - W(Vo(X))] dx, Ixl <=N+Lt Ixl <__N+Lt
Iw j (uo (T~ l ( x - t) )) - w j ( v o ( T ~ l ( x - t) ))[ dx Ix I ~ll P~IIL ~N§
~ c S Ix I _-<11PII]L~o N+ (L+l)t
Iwj( .o(To71(x))) - wj(vo(To71(x))) t d~
(4.35)
~ c S Ix l<=Ko(N+ (L+l)t)
Iwj(uo(x)) - wj(vo(T~lToi(X) ))[ dx.
Discontinuous Solutions to Conservation Laws 159
Using Theorem 3.3 and (4.33)-(4.35), we finally obtain
IPi(X, t) -- qi(x, t)[ dx Ixl <=N
[x l<=Ko(N+ (L+l)t)
Thus (4.27) follows. This completes the proof of Theorem 4.6.
Remark. Theorem 4.5 and Theorem 4.6 show that for completely degenerate systems (n = 2), the generalized solution of the Cauchy problem (4.1) is L m- stable in the sense of shift, for any positive constant m.
As a corollary, we obtain the following theorem.
Theorem 4.7 (Uniqueness). The L o~ general&ed solution of the Cauchy problem (4.1) (n = 2) is unique.
Finally, we consider the systems (4.1)-(4.3) consisting of n equations. For simplicity, we assume that
sup 21(w) < inf)L2(w)< sup 22(w) < . . . < sup 2n- l (w) < inf 2n(w), wEK wEk wEK wEK wEK
where K is the parallelepiped K = IIi[inf wi(uo), sup wi(uo)]. Then we have the following stability result.
Theorem 4.8 (Stability of BV Generalized Solutions). Suppose that the vector functions u(x, t) and v(x, t) are generalized solutions of the Cauchy problem (4.1) in H r with initial data Uo(X ) and Vo(X ) in BV(E2+; K). I f TV=_o~ (Uo(X) ) is suffi- ciently small and u 0 - v0 E L 1 ( - o o , c~), then
lu(x,t)-v(x,t)ldx C luo(x)-vo(x)4dx (4.36) - o o - ~
for almost all tE (0, T) , where the constant C = C(TV~_~o(Uo(X) )) with C(s) E C[0, s0) and lims~0 C ( s ) = +oo for some S o > 0 depending only on the parallelepiped K.
Proof. We first prove that the Lipschitz functions pi(x, t) and qi(x, t), 1 _< i _ n, satisfy
sup T l(qi(x,t))l <__c luo(x)- v0(x)l ax, (4.37) / 7 T - - c o
i = 1
where the constant C = C ( T V = _ = ( u o ( x ) ) ) , with C(s) EC[O, so) and with lims_.s0C(s) = +c~ for some s o > 0 only depending on the parallelepiped K; the functions pi(x, t) and qi(x, t) are determined by the quasidecoupling
160 G.-Q. CHEN
transformations corresponding to the vector functions u(x, t) and v(x, t), and satisfy
pi(x, t), + 2i (u (x , t ) )p i ( x , t)~ = O, (4.38) x
pi(x, O) = Toi(X) = J" PI(~, O) d~, o
qi(x, t)t + 2 i (v (x , t)) qi(x, t)x = O,
qi(x, o) = To~(X) - j Pz(~, o) cl~ o
(4.39)
for almost all t fi (0, T). Then
I r0s l (p i ) - T0~l(qi)J, + ,~(u) lr~l(p~) - T&l(qi)l ,
<= C Tg~_=(Uo(X)) sup [T~i(pi(x, t)) - T~l(qi(x, t)) I nr i=1
+ ~ luo(x)-vo(x)t --oo
<= c ~ tuo(x) - vo(x) l d~.
This completes the proof of Theorem 4.8.
As a corollary, we obtain the following uniqueness theorem.
Theorem 4.9 (Uniqueness). The generalized solution of the Cauchy problem (4.1) with the initial data Uo ( X ) ~ BV ( [R I ; K) is unique provided that TV~_oo (u0(x)) is sufficiently small.
5. Systems with One Contact Field and One Line Field
In this section, with the aid of the quasidecoupling method, we discuss systems with one contact field and one line field, for which the shock wave curves and rarefaction wave curves coincide (see TEMPZE [52]). Such systems arise in the fields of elasticity theory (cf. [30, 1]), enhanced oil recovery (cf. [27, 53]), and magnetohydrodynamics (cf. [20, 1]).
164 G.-Q. CI-IEN
We note tha t a 2 x 2 system of conservat ion laws
ut +f(U)x = 0 (5.1)
has one contac t field and one line field if and only if the flux funct ion
f ( u ) = u ~b(u),
for some smoo th funct ion ~ of u, where 0 with 4)u * 0 is bo th the wave speed (i.e., ~b = )L 1 is an integral curve o f R1) and the R iemann invariant for the contac t field. The R iemann invariant for the line field is
U2 W ~ - - ,
Ul
tha t is, w = const , is a line of w = dR2/dtt 1. Using the po la r coordinates (r, 0), we have tha t the eigenvalues o f the
system (5.1) are
hi ~- ~, ~2 ~- (F(~)r,
respectively. The cor responding R iemann invariants are
w = O, z = O(r, O), respectively.
There are two families o f en t ropy-en t ropy flux pairs (E a, Fa):
Ea = q~oa(w) = ra(O),
and ( Gb, Hb ) :
F a = ~ . l~oa(w) = rO(r, O) a(O), (5.2)
r
Gb • ~0 (22 -- ~1) (0, /1) /]'2( 0, ~) -- /~1(~)
4; 4;
6(r,O)
= r(or(r(~, 0), O) b'(rl) exp - ~1(~ )
4; 4;
H~ = ;~Gb + b(4~(r, 0)),
where a and b are two arb i t ra ry cont inuous functions. (Ea, Fa) is a convex ent ropy pair if and only if a"(O) + a(O) >= O. Fur-
the rmore , if ]rrbr [ ---_ Co > 0 and [1 u I[L~ _--__ C, then there exists a family of C 2 en t ropy pairs (Gbk, Hbk) tha t are convex with respect to 4), with bk(q~) = e ko, k~> l.
One can verify tha t any admissible solut ion to the system (5.1) (see [4]) satisfies
Ea(u)t + Fa(u)x = O, (5.4)
for an arb i t ra ry cont inuous funct ion a(O). Moreover, if the system (5.1) is strictly hyperbol ic on the solut ion d o m a i n under considerat ion, then on the
Discontinuous Solutions to Conservation Laws 165
contact discontinuity C = (x ( t ) , t ) ,
d) (u(x( t ) - O, t)) = (9(u(x( t ) + O, t ) ) ,
(9( Gbk(U(X(t) -t- 0)) -- Gbk(U(X(t ) -- O, t) ))
= Hb~(U(X(t ) + O, t)) -- Hbk(u(x( t ) -- O, t ) ) , (5.5)
and on the second kind of shock wave S = (x ( t ) , t) with slope s,
O ( u ( x ( t ) - o, t ) ) = O ( u ( x ( t ) + o, t ) ) ,
s( Gb~(u(x(t) + O, t) ) -- Gb~(u(x(t) -- O, t) ))
- (Hbk(U(X(t) + O, t)) -- Hbk(U(X(t) -- O, t ) )) >-- 0 (5.6)
for almost all t e (0, co). We consider the Cauchy problem
ut + (u(~(U))x = O, ult= o = Uo(X). (5.7)
Definition 5.1. A BV vector function u(x, t), r ( u ( x , t)) = r0 > 0 is called a generalized solution for the Cauchy problem (5.7) in H r if (a) u(x, t) is a weak solution of the Cauchy problem (5.7). (b) u(x, t) satisfies the following entropy conditions:
(i) For any function a~ Cloc(-co, co) and ~u~ C~(Hr),
~ (ra(O) gt t + r4a(O) ~'x) dx dt + ~ ro(x ) a(Oo(x)) ~(x, 0) dx = 0. (5.8) H T - o o
(ii) On discontinuity wave curves with 0 = const., left state (r_, 0 ) = (r(O_, 0), 0) and right state (r+, 0) = (r(4~+, 0), 0),
s(r+, r_; O) <= s(r, r_ ; 0), for all r between r_ and r+, (5.9)
where r+~(r+, O) - r_da(r_, O)
s(r+, r_; O) = r + - - r _
(iii) On discont inu i ty wave curves wi th ~ (r, O) = const., lef t state ( r_ , 8_) = (r(qS, 0_), 0_) and right state (r+, 0+) = (r(4~, 0+), 0+),
( r ( o , 0_) - r( l , 0_)) ( r ( r 0+) - r( l , 0+)) -> O, (5.10)
where l is any constant and r(l, O) is any branch such that the curve 0(r, 0) = l passes through both (r( l , 0_), 0_) and (r( l , 0+), 0+).
for the constants l and 0 such that r(l, 0) = k, for any constants k and 0. If the system (5.1) is strictly hyperbolic: [r~rl _-> Co > 0, then the condition
(5.10) holds naturally. For some nonstrictly hyperbolic systems, for example, systems arising in magnetohydrodynamics and elasticity (see Subsection 5.b) and a system arising in oil recovery (see Subsection 5.c), condition (5.10), together with (5.8) and (5.9), rules out unstable solutions for the Riemann problem.
Remark 2. If the function ~b satisfies
Ir(r(~)rr[ >-- c O > O, [rOrl >= Co > O,
then the condition (5.9) or (5.11) is equivalent to the Lax shock criterion:
/~2( '+') < S("[- , - - ) < ~ 2 ( - - ) , (5.13)
where ( + , - ) denotes (u+, u_), which is a pair of the right state and left state of the second kind of shock waves. Furthermore, the condition (5.9) is also equivalent to the entropy criterion: for some entropy (Gbo, Hbo) strictly convex with respect to 4~,
Gbo(U)t + Hbo(U)x <= O, (5.14)
in the sense of distributions by using (5.8).
Remark 3. The condition r(u(x , t)) >__ r 0 > 0 in Definition 5.1 can be removed by following the process we shall introduce and by WgGNER'S arguments [56].
We recall that there exist generalized solutions to the Cauchy problem (5.7) for the cases of physical significance (see [4, 53]). Now we study the unique- ness and stability of generalized solutions for the Cauchy problem (5.7) with the aid of the method of quasidecoupling introduced in Section 3.
a. Uniqueness and Stability of Generalized Solutions
Theorem 5.1 (Stability). Suppose that BV functions u(x, t) and v(x, t) are generalized solutions of the Cauchy problem (5.7) in 117. with initial data Uo(X) and Vo(X), respectively. Then, for almost all t ~ [0, T) and for any positive con-
Discontinuous Solutions to Conservation Laws 167
stant m, N,
r O(u(x, t)) = O(uo(T( i (p(x , t ) ) ) ) , O(v(x, t)) = O(vo(To-l(q(x, t ) ) ) ) ,
Using Theorem 3.2 with Hypotheses As and A2, we immediately obtain (5.15).
Remark. This lemma holds for L ~176 weak solutions u(x, t) and v(x, t) of the Cauchy problem (5.7) satisfying (5.8) with initial data Uo(X) and Vo(X), respectively. We did not use the assumption that both u(x, t) and v(x, t) are BV functions.
Denote r l (x, t) = r (u (x , t)) -1 and rz(x, t) = r (v (x , t)) -1. We have the following lemma.
Lemma 5.2. I f O(uo(To l (x ) )) = O(vo(T~-l(x) )) - Oo(x), a.e., then
TI(X , t) = "t-1 (p(x, t), t; r (uo(To-l (p(x , t ) ) ) ) - l ) , (5.19)
rz(X, t) = rz (q(x , t), t; r ( v o ( T o l ( q ( x , t ) ) ) ) - l ) ,
where both ri(y, t; To(Y)), i = 1, 2, are solutions of the Cauchy problem
z t -- ~b('r, Oo(Y))y : 0, tit=0 = to(y) , (5.20)
satisfying the following entropy condition: On discontinuity wave curves with Oo(y) = const., left state r_ = r(~b_, Oo(y) ) and right state r+ = z(q~+, Oo(y) ),
a ( z + , r_ ; Oo(y)) <-_ a ( z , z_; Oo(y)) for any r between 7:_ and z+, (5.21)
and on discontinuity wave curves with 4~ = const., left state r_ = r(q~, 0_) and right state r+ = z(~b, 0+),
(z(~b, 0_) - r( l , 0_)) ( z ( r 0+) - r(l , 0+)) _-__ 0, (5.22)
Discontinuous Solutions to Conservation Laws 169
where l is any constant, r(l, O) is any branch such that the curve 4)(r, 0) = l passes both (r( l , 0_), 0_) and (r( l , 0+), 0+), and
4) ( r+ , Oo(y)) - 4) ( r_ , Oo(y)) a ( r + , r _ ; Oo(y)) =
Z'+-- "C_
Proof . Using the quasidecoupling t ransformat ions (5.17) and (5.18), we can immediately verify that bo th r1 ( T -1 (p, t)) and h (]?-1 (q, t)) are weak solu- tions o f the Cauchy problem (5.20) with the initial data r ( u o ( T o l ( p ) ) ) -1 and r ( v0 ( T0-1 (q) ) ) - 1, respectively.
Now we prove the entropy condi t ion (5.21). For concreteness, we prove only that r l ( x , t ) = r l ( T - l ( p , t ) ) satisfies (5.21). In fact, on the shock wave curves of u(x, t) , i.e., on 0 = O(uo(To-l(p))) = const.,
= r+4)(r+, O(uo(To l (p ) ) ) ) - r_O(r_ ,O(uo(To- l (p ) ) ) )
F+ - - r _
The quasidecoupling t ransformat ion T1 : (x, t) ~ (p, t) t ransforms the shock waves S = (x ( t ; r+, r_) , t) with O(uo(To-l(p))) = const, and slope s ( r+ , r_ ;4 ) (uo (To- 1 ( p ) ) ) ) in the (x , t ) - p l ane to shock waves S ' = (p ( t ; r+ , r_ ) , t) with O(uo(To-l(p))) = const, and slope
Similarly, we can prove that rz(X, t) = r 2 ( T - l ( q , t)) satisfies (5.21). The en- t ropy condi t ion (5.22) can be directly verifed f rom (5.10).
170 G.-Q. CHEN
1.emma 5.3. For any positive constant N1,
j ] z ( Y , t ; z o ( Y ) ) - z ( Y , t ; r o ( Y ) ) l d y < = C ~ Izo(y)-~o(y)ldy, lYl <Nt lY[ <N1 +Ltt
where L1 = suPil~ I <=suPnTl~l,y~RilldPr(r, 00(Y)) I �9
P roof . First we note tha t the ent ropy condi t ion (5.21) is equivalent to the con- di t ion
o ( I z - - - I z + - k [ )
+ ( 6 ( z _ , O) - (h(k, 0)) s i g n ( r _ - k) - (q~(z+, O) - 4 ) ( k , 0)) s ign(z+ - k) < 0
(5.23) for any cons tant k and 0; equivalently,
a(lr(4,-, 0) - z ( l , o) I - l z ( 8 + , 0) - z( / , 0 ) l )
(5.24) for the constants l and 0 such tha t r ( l , 0) = k.
Next we consider the difference between v ( y , t ) - z ( y , t; z0(y)) and ~(y, t) - r (y , t; ~'0(Y)) in the n o r m of L 1. For this purpose , we associate with the pair (z, ~) a R a d o n measure B = B(~,~) def ined on H r as follows: B is the divergence o f the vector field (] r - ~], (qS(z, 00) - 4~(r, 00)) s ign(z - D ) :
B = [z - ~lt - ((4)(z, 00) - q~(f, 0)) s ign(z - ~ ) ) y . (5.25)
We recall tha t with each funct ion r in BV, there is associated a set o f points F ( z ) at which z experiences an approx imate j u m p discontinui ty [55]. We refer to F(z) , F(D, F(c>), F($), and F(Oo) as the j u m p set o f the funct ions z, ~, q~, (~, and 0o, respectively. Note tha t if z(y, t) , ~(y, t) and Oo(y) are cont inuous functions, then
B(~,~) ( H r ) = 0 .
Then we can conclude tha t the measure B(~,~) is concentra ted on F = F(q~) u /'(qS) u F(00) :
B(E) = B(E n F) (5.26)
for any Borel set E C / /T. According to the generalized Green theorem for measures [19], the B-
measure is expressed by
B(~,~)(E) = I ( n t [ I z - r l ] - ny[(q~(r, 00) - ~b(?, 00)) s i g n ( z - ? ) ] ) dill. E n r (5.27)
Here H a denotes the one-d imens iona l H a u s d o r f f measure , n = (ny, nt) denotes the uni t n o r m a l to E n F in the sense of FBDZRER [19], and the bracket denotes the j u m p o f the enclosed quant i ty across E in the direction of n:
Discontinuous Solutions to Conservation Laws 171
where ( re , ~• = (z(+• 0o• z(~• 0o• denote the approximate limits of (z, ~) at P ~ E n F with the half-planes
H • ) = {Q: (Q - P, -4-n>__> 0, n y < 0 } .
We note that the set E0 of irregular points on which n y * 0 for (r, ~) B V ( H r ) is a set of zero one-dimensional Hausdorff measure. Therefore we can write (5.27) as
where a--- - n t ( P ) / n y ( P ) denotes the speed of propagation of F at P E E n E~ c~ F and dt = - n y dH 1 . Then for any Borel set E, the Borel measure
B(~,~) (E) _< 0.
In fact, if E C (F(+) n F ( ~ ) n F ( O o ) ) c, we use (5.23) and (5.28) and im- mediately obtain
B(~,~) (E) __< 0.
If E C F ( + ) c~ (F (0 ) n F ( 0 0 ) ) c, or r ( ~ ) c~ ( r ( + ) c~r(Oo) )c, or F(Oo) n (F(+) n F ( 0 ) ) ~, we use (5.22), (5.23) and (5.28) and also deduce
if E C F (+) n g(q~) n g(Oo) c, (5.24) that
a ( l z ( + _ , Oo) - r ( ~ _ , 0o)l -
+ ( + _ - q~_) s ign( r (+_ ,
- ( + + - ~_) s ign( r (++,
G ( l r ( ~ _ , 0o) - z (++ , 0o)] -
+ (q~_ - ++) sign(r(q~_,
- (q~+ - ++) s ign( r (0+ ,
Therefore,
B(~,b (E) _< 0;
then for any point P E E n E~, we have from
I t ( + + , 0o) - z ( ~ _ , 0o)l)
0o) - r ( ~ _ , 0o))
0o) - r ( ~ _ , 0o)) _--- o ,
I z (++ , 0o) - r ( + + , 0o) 1)
0o) - r(++, 0o))
0o) - r(++, 0o)) _-< O.
~( Iz (+_ , Oo) - z (~_ , Oo)1 - I ~ ( + + , Oo) - z(~+, Oo)1)
+ ( + _ - q~_) s ign( r (+_ , 0o) - r ( 0 _ , 0o))
- ( + + - ~_) s ign( r (++, 0o) - z(~+, 0o)) -< 0,
and, equivalently,
a ( l r _ - r - I - [ r + - ~+]) + ( + ( r _ , 0o) - + ( ? - , 0o)) s i g n ( r _ - ~ _ )
- (+ ( r+ , 0o) - +(~+, 0o)) s i g n ( z + - ~ ) __< 0.
172 G.-Q. CnEN
Thus (5.28) yields B(~,~)(E) __< 0.
Suppose that E C F(qS) n F(Oo) n F ( ~ ) C or E C F ( ~ ) n F(Oo) n F(q~) c. To be specific, say that the former case holds. Then
a - - 0 ,
( ~ _ - q~) sign(r(q~_, 0o-) - r(q~, 0o_))
- (q~+ - q~) s ign( r (~+ , 0o_) - r (~ , 0o-)) _-< 0,
( r ( ~ • o_) - r (~, o_)) (r(~,:~, o+) - r (~, o+)) __> o,
and therefore, a - - 0 ,
(8_ - q~) sign(r(qS_, 0o_) - r(d), 0o_))
- (q~+- ~) s ign( r (~+ , 0o+) - r (~ , 0o+)) _-< 0, or, equivalently,
( 7 = 0 ,
( ~ ( r _ , 0o-) - q~(r-, 0o-)) s i g n ( r _ - ~ _ )
D = [ ( y , s ) l l y [ < = N + L ( T - s ) , t l < = s < = t } , O < t l < t < = T ,
S s = { y l l y [ < _ N + L ( T - s ) } , tl<-S<=t,
where the constant L will be chosen below. It follows from the generalized Green theorem for measures that
B(r,~) (D) = { n t l r - ~ l - ny(4~(v, 0o) -qS(~, 0o)) s i g n ( r - ~)}dH 1 OD
I [nt] r - el - ny(q~(r, 00) - ~b(e, 00) ) s i g n ( r - e)} dill
+ SIr-el - SIr-el , St St 1
where y is the lateral boundary. Note that on ~,
nt ]r - e] - ny(q~(r, 00) - 0 ( r , 0o)) s i g n ( r - ~)
( (qs(r, 0 o ) - q ~ ( e , 0 o ) ) s i g n ( r - e ) ) = [ Z - - e I n t - i ~ - - ~ i ny
>--[r-e l ( L - m a x ( s u p l O ~ ( r ' O ~ ~r = 0 ,
provided that
L = max ( sup lea(r, 0o)[, sup 10~(e, 0o) [~ �9 z %
\ Hr Hr /
Therefore, from Lemma 5.3 we obtain
I It(y, t) - ~(y, t)[ dy <= I It(y, t~) - e(y, tl)[ dy. st s~ 1
Notice that r(x, t) and e(y, t) are BV functions. Letting tl ~ 0, we obtain
]r(y, t) -e(y, t)[ dy<_ I ]to(y) - eo(y)l dy. st so
This completes the proof of Lemma 5.3.
Therefore, we have the following lemma.
Lemma 5.4. The solution of the Cauct~ problem (5.20) with (5.21) and (5.22) is unique. In particular,
r l ( y , t; to(y)) = h(Y, t; to(y)) a.e.
Lemma 5.5. For the generalized solutions u ( x, t) and v ( x, t) with 0 ( Uo ( To -1 ( x ) ) ) = O(vo(Tol(X) )) in Theorem 5.1, the functions r(u(x , t)) and r(v(x, t)) satisfy (5.16).
174 G.-Q. CHEN
Proof. We conclude from Lemma 5.2 and Lemma 5.3 that
r (v(x , t)) = T ( v ( ~ - ] ( q ) , t)) = r (q , t; r(vo(T~-](q) ) ) - l ) ,
and both r (u(x, t)) = r (u(T t -1 (p), t)) and r ( v ( ~ -1Tt(x), t)) = r (v(Tt -1 (p), t)) satisfy the Cauchy problem (5.19) and (5.20) with initial data r (Uo (To -1 (p)))-1 and r ( Vo ( To- 1 (p))) - 1, respectively. Using Lemma 5.3, we obtain
f I r (u(Tt - l (P) , t)) - - r (v (T t - l (p ) , t))[ dp Ip[<-N1
__< c ~ Ir(~0(rol(p))) - i - r (v0 (To~Ip) ) ) - I i @ , ]p[ <<-N 1 +Lit
where L1 = max (supnr ] ( r -2G) (u (x, t) )[, sUPnr / ( r -ZG) (v(x, t) ) [ ) . Therefore, for any positive constant N, we have
i I r ( u ( x, t)) - r ( v ( T t - l T t ( x ) , t)) I dx [x]<=N
<= C ~ Ir(uo(x)) - r (vo(TolTo(x) ))l dx. [xI <=Ko(N +Lt)
I.emma 5.6. I f O ( uo ( Tol (x ) ) ) : - O ( v o ( T o I ( x ) ) ) a.e. , then there exists a constant C > 0 such that for almost all t E [0, T],
] ( T t - l T t - I ) ( x ) [ dx Ixl~N
<= C j (]r(uo(x)) - r ( v o ( x ) ) ] +]r(uo(x)) -r(vo(Td-~To(x)))[) dx Ix ] -<_Ko(N+Lt)
for any constant N > 0, where [, = L + max(suPEO, T] (] ~-1Tt(q_N) ] - N ) / t ) , and L = max(supnr ( ]0(u) ] +] ( r -2G) (u) ]), super (]q~(v) ] +] ( r -2G) (v) l)).
Proof. It suffices to prove that for any constant N > 0, there exists a constant C > 0 such that the Lipschitz functions p (x, t) and q(x, t), determined by (5.17) and (5.18), satisfy
IT(q, t; u o ( T o - l ( q ) ) ) - v(q, t; Vo(Toa(q))) I dr
+ t [r(uo(x)) - r(vo(x))] dr I~1= <N+Lt
=< I (luo( ) - o(x)l + [uo(x) - [xI <-Ko(N +Lt)
This completes the proof of Lemma 5.6.
We can immediately obtain Theorem 5.1 from Lemmas 5.1, 5.4, and 5.6. As a corollary, we have the following theorem.
Theorem 5.2 (Uniqueness). The BV generalized solution of the Cauchy problem (5.7) is unique.
Propagation and cancellation of initial oscillations for such systems (5.1) have been studied by CI-I~N [4, 5].
Now we discuss some applications of Theorems 5.1 and 5.2 to several im- portant systems arising in the fields of elasticity theory, enhanced oil recovery, and magnetohydrodynamics.
b. Systems Arising in Magnetohydrodynamics and Elasticity
Consider 2 • 2 systems with symmetry that are discussed in [4]. We assume that 4~(u) satisfies the following assumptions:
S~: 4~(u) > 0. Sz: (~(u) = const. > 0 is a simple closed curve enclosing the origin u = 0.
This class of systems contains the elastic string model and the magnetohy- drodynamics model [30, 37],
= (~(r) > O, qS"(r) > O.
In particular, the choice of 4 ~ = 6 ( r ) = 1 + f i ( r - 1 ) Z / r , f i > 0 constant models a nonlinear stress-strain relation. On the region K ~ [u:r(u) = 1}, the system is nonstrictly hyperbolic.
Theorem 5.3. If the function (~(u) satisfies
r4~r => Co > 0, r(r4)) rr --> Co > 0,
together with the assumptions S1, $2, and if the initial data Uo(X) fi{u:~b 1 -< ~)(u) ~ 02 < ~}, Oi > 0(0), then there exists a unique BV generalized solution u(x, t) of the Cauchy problem (5.7) satisfying (5.8) and (5.14).
176 G.-Q. C~EN
Proof. Following [4], we first obtain a global BV weak solution u(x, t) satis- fying (5.8). It is easy to check that for any C 2 strictly convex entropy (Gb, Hb) on K, the solution u(x, t) also satisfies (5.14), which is equivalent to (5.9) for such systems. The result follows. We omit the details.
Furthermore, we have the following theorem.
Theorem 5.4. If the function (~(u) = ~(r) satisfies mes[ r : 49"(r) = 0} = 0 and if the initial data Uo(X) E{u:0 < r I < r(u) <= r 2 < oo}, then there exists a unique L ~ solution u(x, t) of the Cauchy problem (5.7) satisfying the following entropy conditions:
(i) For any function aE Cloc(-Co, oo) and for WE CI(I-IT),
(ii) For any constant k E R 1 and any nonnegative function W E C~(17T -- {t = 0}),
II ( [ r - kl Wt'~- (r~b(r) - kdp(k) ) sign(r-k) Wx) dx dt <= O. HT (5.30)
Proof. Following [4], we first obtain a global BV weak solutions u(x, t) satis- fying (5.29), (5.30). Using the framework established in Theorem 3.1 in [5], we conclude that there exists a global L ~176 solution of the Cauchy problem (5.7) satisfying (5.29), (5.30). Then using KRvs theorem, we immediately conclude the uniqueness of r. Finally, using the Theorem 5.1, we obtain the uniqueness of 0. This completes the proof of Theorem 5.4.
c. A System Arising in Oil Recovery
Consider
a(ul'~l ) ~ ( U l , U2) - - , (5.31)
Ul
where u I E [0, i ] , U 2 E [0, Ul] , are respectively the concentrations of water and a polymer. Here a(ul , c) is a smooth function such that a(u 1, c) increases f rom 0 to 1 and has one inflection point for c constant and such that for fixed ua, a(ul , c) decreases as c increases (see [53]). The essential feature of the system is that strict hyperbolicity fails along a certain curve in state space.
TZMeL~ [53] obtained a global weak solution to the system (5.1) and (5.31) with Cauchy data Uo(X)EBV. One can check that TZMP~E'S solution satisfies (5.8).
Theorem 5.5. Suppose that BV functions u(x, t) and v(x, t) are solutions of the Cauchy problem (5.7) satisfying (5.8) in I I T with BV initial data Uo(X) >= Co > 0
Discontinuous Solutions to Conservation Laws 177
i Ix[<-N
and Vo(X), respectively. Then for almost all t~ [0, T),
l IO(u(x, t)) - O(v (~ -~r t ( x ) , t))l dx Ixl<=N
<= c I IO(uo(x)) - O(vo(To~To(x) ) ) ld~, Ixl <=Ko(N + Lt)
IO(u(x, t)) - O(v(Tt-l ToTol Tt(x), t)) l dx
f Ixl=<N
i Jxl<=N
f Ixl < Ko(N + Lt)
I O(uo(x)) - O(~o(x))l ~ ,
I r( u(x, t) ) - r ( v( Tt-l Tt(x), t) ) I dx
--< S Ir(uo(x) ) - r (vo(Tol :ro(X)) )I Ix l<_Ko(N+Lt)
I ( ~ - I T ~ - I) (x) I dx
C ~ (]r(uo(x)) - r(vo(X))l + Ir(uo(x)) - r (vo(roXTo(x) ) )[ ) dx, Ixl <=Ko(N + Lt)
provided that (5.9), (5.10), and the equation 0 (uo(To 1 (x) )) = O(vo(T~ I (x ) ) ) a.e. also hold; here T(x, t) = (Tt(x) , t) and T(x, t) = ( Tt(x), t) are the quaside- coupling transformations corresponding to the vector fields ( ( r r t)), r (u(x, t ) ) ) and ( ( r 6 ) ( v ( x , t)) , r (v (x , t ) ) ) . Furthermore, the BV generalized solution of the Cauchy problem (5.7) satisfying (5.8)-(5.10) is unique in TI T.
In fact, for this system, the condition (5.10) is equivalent to the condition that the left state and the right state of any contact discontinuity lie in the same side of the degenerate curve {(u, v ) : 0 r = 0}, which is used to ensure the uniqueness of solutions for the Riemann problem as in [27] and [53]. It would be intrersting to study the regularity of TEMPle'S solution u (x, t) of the Cauchy problem (5.7) and (5.31) with the Cauchy data Uo(X)EBV, especially with BV regularity. In this connection we refer the reader to the work of TVEITO & WINTHER [54] for a stability result within the solution space
B ~ (ul, u2): ul, eBV, - - e L i p c ~ B . x Ul
6. The System of Electromagnetic Plane Waves
In this section we present an example of how to reduce important problems for large systems with linearly degenerate fields into problems for correspond- ing smaller systems by using the quasidecoupling method. We are concerned with the Maxwell equations
Bt + curl E = 0, Dt - curl H = 0, (6.1)
178 G.-Q. CrIE~
where B, D, E, and H are the magnetic induction, the electric induction, the electric intensity, and the magnetic intensity in N3, respectively. The con- stitutive laws can be of the following form in the electromagnetic field (see COLEMAN & DI5~ [6]):
OW OW E = - - H - . (6.2)
OD ' OB
Here W(B, D) is a convex electromagnetic energy density, i.e., VZW>__ O, satis- fying
Wt + d i v ( E • = 0, (6.3)
for smooth electromagnetic fields. As described in [49], the system is translationally invariant and admits
plane waves depending only on t and one space variable x = x3. Then the components B3 and D3 can be constant, and (6.1) reduces to the four equa- tions
0/11 "]- 0 OU 2 0 a~- Ox (4~u4) = o, at Ox (4~/13) = o,
Ou 3 0 (4)u2) = O,
Ot Ox
O/g 4 0 - - + - - ( 4 m l ) = 0 . Ot Ox
(6.4)
Here we make the assumption of axisymmetry:
W(u) = W(r), r = ]lull,
and use the notation (ul, u2, u3, u4) = (D1, D2, B1, B2),
W ' ( r ) 6 ( r ) - (6.5)
F
The system (6.4) is nonstrictly hyperbolic because two of the four eigenvalues coincide: 21 = 2 2 ~-~(/ ') . The eigenvalues 21 = 2 2 are linearly degenerate, and the other two eigenvalues 23, 24 are genuinely nonlinear.
We first focus on how to reduce the uniqueness problem for the system (6.4) to the uniqueness problem for a 2 x 2 system with the aid of the quasidecoupling method. Introduce a nonlinear transformation u - , (p, o-, o~, 0) ~2+ X [0, 27~]2:
u l = � 8 9 u 2 = � 8 9 (6.6)
u3 = �89 (P sin o~ + a sin 0), u4 = �89 (P sin a - o- sin 0).
Then one can verify (see [49]) that there exists an L ~ global solution u (x, t), (p(u) , a (u) ) >= O, Ilullr= _<_ C(llu01[L~,), of the Cauchy problem of (6.4) sub-
ject to
ult=0 = Uo(X), (6.7)
Discontinuous Solutions to Conservation Laws 179
satisfying, for any continuous functions a(a) and b(O),
(Pa(cO)t + (pdpa(a))x = O, (ab(O))t - (ad~b(O))x = O, (6.8)
2W t + ((p2 - a 2) q52)~ < 0, (6.9)
in the generalized sense of distributions under certain physical conditions on q~ or W:
W'(r) > 0, ( 7 ) - ) ) '
where r = x/-p 2 + a 2.
(rW" (r)'~ ' > 0 , W" (r) > O, k, W'~r ) ] > 0 , V r > O ,
(6.10)
Using Lemma 3.1 and Corollary 3.3, we can immediately conclude the following stability behavior of the solutions.
Theorem 6.1 (Stability). Suppose that u(x, t) and v(x, t) are L ~176 weak solutions satisfying (6.8) of the Cauchy problem (6.4), (6.7) with initial data uo(x ) and Vo(X) in FIT, respectively, and
Co = rain ( i n f P(Uo(X)), infa(uo(x)) , inf P(Vo(X)), i~afa(vo(x))) > O. (6.11)
Then for almost all t E [0, T) and any positive constant m,
[O~(U(X, t)) - o ~ ( v ( T t { 1 T o l T o l l T t i ( x ) , t))[ m I~r =<N
• t) ) p ( v (T t ;1To l To-{1Tti(x), t)) dx
I lpl l~ j [~(uo(x)) - ~(vo(x)) l m dx, Ixl < Ko(N § Lt)
J IO(u(x, t)) - O(v(Tt~lTo2T~lTt2(x), t))[ m (6.12) I~1 _-_<N
where Tl(X, t) = (Ttl(x), t) , Tl(x, t) = ( ~ l ( x ) , t) , T2(x, t) = (Tt2(x), t), and Tz(x, t) -- (f t2(x), t) are the quasidecoupling transformations corresponding to the vector fields ( (pO) (u), p(u)) , ((GO) (v), a(v)) , respectively, and
L --- max @upl0(u(x, t))[,
((pO) (v), p(v)),
supl0 (v(x, t))I'] , rtv g
((GO) (u), G(u)) and
K~ = max (sup I G ( u ) I ' k , err sup I a ( v ) ] ' n r suplp(u)I ' suplp(v)l)/e~
In fact, in Hypotheses A 1 and a 2 we can choose
Go(W) = I~(u) - k l m, ( )~ ie l , e 1) = (p(u) O(u) ,p (u) ) ,
ao(z) = I~(v) -- II m, ()~2P2, Pa) = (p(v) O(v), p(v)) ,
Ao(w) =10(u) - k l m, ()L1P1,P1) = (G(U) O(u), G(u)) ,
ao(Z) = 1 0 ( v ) --II m, ()L2P2, P2) -- (G(v) O(v),G(v)),
for any 0 < m <co and any constants k, l e ( - co , c~). The result (6.12) follows.
Then we have the following theorem.
Theorem 6.2. The L ~ weak solution of the Cauchy problem (6.4), (6.7) with (6.11) satisfying (6.8), (6.9) is unique if and only if the weak solution of the Cauchy problem
Pt + (pO(r))x = O, at - (GO(r))x = 0, (6.14)
(p, G)I,= 0 = (P(Uo(X)), G(U0(X))), (6.15)
satisfying (6.9) is unique.
Proof. Suppose that u(x, t) and v(x, t) are two solutions of the Cauchy prob- lem (6.4), (6.7) with (6.11) satisfying (6.8), (6.9). If the weak solution of the Cauchy problem (6.14), (6.15) satisfying (6.9) is unique, then
where O(x, t) = dp(r(x, t)), r = ~p(u) 2 + G(u) 2 = x/p(v) 2 + G(v) 2.
Discontinuous Solutions to Conservation Laws 181
Notice that all p, fi, q, and ~ are Lipschitz functions. We immediately con- clude that
p(x, t) =/5(x, t), q(x, t) = gl(X, t) a.e.
Then we complete the proof by using Theorem 6.1.
Thus we can transfer the uniqueness problem for (6.4), (6.7) to the unique- ness problem for (6.14), (6.15) with the aid of the quasidecoupling method.
Let PL = PL[HT} be the class of piecewise Lipschitz functions. A function u E PL if for each t in [0, T) there exists a set of isolated points xj = xj(t) such that the restriction u(. , t) of u to each interval (xj, Xj+l) is a Lipschitz function of x; the dependence of the Lipschitz constant on the interval (:9, xj+i) as well as the dependence of the partition points xj on t is ar- bitrary. Note that the system (6.14) is genuinely nonlinear in the sense of LAX [33]. Then combining Theorem 6.2 with DIPERNA'S theorem [14], we can conclude the following corollary.
Corollary 6.1. Suppose that u(x, t) and v(x, t) are L ~176 weak solutions of the Cauchy problem (6.4), (6.7), (6.11) satisfying (6.8), (6.9) in H r such that
(O~(U), 0(U), p(u), if(u)) E (L~176 (BV)2{HT}, (a (v ) , O(v), p(v), a(v) ) ~ (L~)2 • (PL)2{HT},
u(x, O) = v(x, 0), for almost all x e ( - co, co). Then
u(x, t) = v(x, t ) ,
for almost all (x, t) in FIT.
Finally, we obtain the following theorem by using Lemma 3.1 and Theo- rem 3.5, and by following the proof of Theorem 4.2 for completely degenerate systems.
Theorem 6.3. Initial oscillations propagate along linearly degenerate fields for the system (6.4).
A direct corollary follows.
Corollary 6.2. For the Cauchy problem (6.4), (6.7), the solution operator St: L~176 4) ---~L~oc(~ 4) with St(u ~ = u(. , t) is not compact.
We omit the details of the proof of Theorem 6.3.
7. Remarks
The quasidecoupling method and ideas developed in the previous sections can be applied to solve some nonlinear problems for more general systems.
a. Propagation and Cancellation of Initial Oscillations. Using the same arguments as in the proof of Theorem 4.2 and with the aid of Hypotheses B 1
182 G.-Q. CHEN
and B2, we can extend the results virtually without modification to strictly hyperbolic systems (1.1) endowed with (1.6) and (2.5). If there exists a linearly degenerate characteristic field for such systems, then the entropy-entropy fluxes (2.17) of the same form can be obtained, which correspond to this field. In fact, Hypotheses B I and B 2 are useful for solving oscillation problems for systems of conservation laws endowed with the entropy-entropy fluxes of the form (2.17).
b. Generalized Characteristics and the Structure of Solutions. Theorem 3.1 and Hypotheses AI,2, B1, 2 in Section 3 could be applied to acquire under- standing on generalized characteristics and the structure of solutions to systems of conservation laws endowed with the entropy-entropy fluxes of the form (2.17). In particular, they can provide insight into the interaction between large linear waves and large nonlinear waves, and into the large-time asymptotic behavior of solutions for systems of conservation laws with such mixed linear characteristic fields and nonlinear characteristic fields. In this connection, we refer the readers to GLIMM • LAX [23], DAFERMOS [11], and the references cited therein on generalized characteristics and the structure of solutions for conser- vation laws.
c. Uniqueness and Existence of Generalized Solutions. From discussions in the previous sections, it should become evident that Hypotheses A~ and A 2 and ideas developed here could be used to solve uniqueness and existence problems for more general systems by combining them with other techniques.
d. Equivalence of the Euler and Lagrangian Equations of Gas Dynamics for Entropy Solutions. WAGNER [56] verified the equivalence of entropy discon- tinuous solutions between the Eulerian and the Lagrangian equations of gas dynamics. As a corollary of Theorem 3.3, we can conclude the equivalence of the uniqueness of entropy discontinuous solution between the Eulerian and Lagrangian equations of gas dynamics via the uniqueness of the transformation between the Eulerian and Lagrangian coordinates with the same initial data from Theorem 3.3.
Acknowledgements. I thank Professors C. DAFERMOS, L. C. EVANS, T.-P. LIU, D. SERRE, and B. TEMPLE for valuable discussions and remarks. I am grateful to Pro- fessor P. D. LAX, J. GLIMM, and C. MORAWETZ for their encouragement. This research was supported in part by NSF Grant DMS-8505550 from MSRI at Berkeley, by a NSF Grant from the IMA at Minnesota, and by the Argonne-University of Chicago Fellowship funded by the Applied Mathematical Sciences subprogram on the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
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Department of Mathematics The University of Chicago
