SIAM J MATH ANAL Vol 38, No 5, pp 474 488 c 007 Society for Industrial and Applied Mathematics OPTIMAL TRACING OF VISCOUS SHOCKS IN SOLUTIONS OF VISCOUS CONSERVATION LAWS WEN SHEN AND MEE REA PARK Abstract This paper contains a qualitative study of a scalar conservation law with viscosity: u t + f(u) x = u xx We consider the problem of identifying the location of viscous shocs, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law We introduce a nonlinear functional whose minimizers yield the viscous traveling profiles which optimally fit the given solution We prove that outside an initial time interval and away from times of shoc interactions, our functional remains very small, ie, the solution can be accurately represented by a finite number of viscous traveling waves Key words optimal viscous shoc tracing, viscous conservation laws, viscous traveling shocs, data compression, finite dimensional representation AMS subect classifications 35L60, 35L65, 35L67 DOI 037/0506464 Introduction Consider a scalar conservation law with viscosity () u t + f(u) x = u xx We assume that the flux f is smooth and genuinely nonlinear, so that f (u) κ>0 for every u Our main interest here is how to identify the emergence of viscous shocs in a solution, and how to optimally trace their locations and strengths More generally, one may as the following question Assume that a particular solution u = u(t, x) has already been computed If we are allowed only a finite number of parameters to describe its most relevant features, what is the best way to compress the information? In the literature, the problem of finite dimensional approximation of a dynamical system has been studied mainly by looing at ω-limit sets [T] Several results, valid for evolution equations of parabolic type, provide estimates on the dimension of an attractor Of course, this yields a bound on the number of parameters needed to describe the evolution of the system asymptotically as t + In the present paper, the focus is different Namely, we see a finite dimensional description which is accurate not only in the asymptotic limit as t + but also in the transient regime For solutions to a scalar, viscous conservation law, this transient behavior is actually the most interesting feature that can be observed On the other hand, at least in the case of convex flux, the ω-limit set is rather trivial The asymptotic limit of any solution t u(t, ) can be described in terms of the solution of a Riemann problem, ie, either a single rarefaction or a viscous shoc wave For general theory on hyperbolic conservation laws, we refer to the boos [Sm, B, S] The problem of optimal location of viscous shoc profiles was mentioned also in [W] In this connection, we introduce a scalar functional whose minimizers identify the strengths and locations of viscous shoc profiles present in the solution We also Received by the editors October 3, 005; accepted for publication (in revised form) August, 006; published electronically January, 007 http://wwwsiamorg/ournals/sima/38-5/6464html Department of Mathematics, Penn State University, University Par, State College, PA 680 (shen w@mathpsuedu, par m@mathpsuedu) 474
OPTIMAL TRACING OF VISCOUS SHOCKS 475 prove that outside a set of times with finite measure, at all other times our functional has very small values In other words, the description of the solution profile u(t, ) in terms of finitely many viscous shocs is accurate, for most times t The exceptional set consists of an initial time interval and times at which shoc interactions occur; see Figure decay of positive waves interactions asymptotic limit (Riemann Problem) 0 t Fig The exceptional set of times where the finite dimensional representation is not accurate The main result We consider here the single conservation law with viscosity () u t + f(u) x = u xx We fix M>0 and let F M denote the set of all solutions to the Cauchy problem for () with initial data () u(0,x)=ū(x) satisfying (3) TotVar{ū} M, ū L M We shall assume that the flux f is C and strictly convex, so that f (u) > 0 for all u R In particular, this implies that there exist constants κ, κ, (4) 0 <κ f (u) κ for all u [ M,M] In essence, what we want to show is the following Apart from a small set of times J [0, [, the profile u(t, ) of any solution of () can be accurately described in terms of the superposition of finitely many traveling viscous shocs Indeed, the assumption (4) of genuine nonlinearity implies that all rarefaction waves will decay within an initial time interval Moreover, in regions where the gradient u x is large and negative, viscous shoc profiles will form These can travel for a long time without much changing their shape, except when they interact with each other The set J of exceptional times where our description is not accurate will thus include an initial time interval and also the intervals where wave interactions occur Much of the following analysis aims at maing rigorous the above claims For every u >u + and y R, let ω (u±,y) be the unique viscous shoc profile oining the states u,u +, centered at y This profile can be found as the unique solution to the ODE ω = f(ω) σω [ f(u ) σu ], σ = f(u ) f(u + ) (5) u u +, satisfying the additional conditions (6) ω (y) =0, ω( ) =u, ω(+ ) =u +
476 WEN SHEN AND MEE REA PARK Notice that the last two identities in (6) follow from (5) and the convexity of f Given any solution u F M of the conservation law, for each t>0 we introduce a description based on optimal location of shoc profiles Fix an integer N and let ω i = ω (u± i,yi) be the ith viscous shoc profile we try to fit in We consider the functional J ( ) N u(t), ω,,ω N = u(t, x) ω i (x) ω (x) dx i= R + R u (7) x(t, x) ω (x) dx i= Notice that the first integral measures the distance between u and the traveling viscous shoc ω i, multiplied by a weight function ω which is vanishingly small away from the center of the ith shoc The second integral measures how well the derivative u x is approximated by derivatives of traveling shoc profiles See Figure for an illustration of fitting two viscous shocs in a solution ω u ω y y x Fig Fitting two viscous shocs ω,ω in a solution If we fix a priori the complexity of our description, ie, the integer N, how small can we render the integral J? This problem can be formulated as inf J ( ) (8) u(t), ω,,ω N, ω,,ω N where the infimum is taen over all N-tuples of traveling shoc profiles ω i = ω (u± i,yi), for some states u i >u + i and y i R Notice that if we choose ω i 0 for i =,,N (ie, all traveling waves of zero amplitude), then the first integral in (7) vanishes because trivially ω 0 However, in this case the second integral equals ux (t, ), L which is of order TotVar(u) 3 due to regularization and can be large To estimate the quantity in (8), an intuitive argument goes as follows Set δ = M/N, where M is given in (3) Since the total variation of u(t) is bounded by M, there can be at most N shoc profiles of strength δ Each one of these can be traced accurately In addition, there may be an arbitrarily large number of smaller shocs, say, of strengths σ,, with (9) σ δ, σ M Each shoc which is not traced produces an error in the second integral of (7) of the order (0) ω,x(x) dx = O() σ 3
OPTIMAL TRACING OF VISCOUS SHOCKS 477 Because of (9) we thus expect that the minimum of J is approximately J min O() Mδ = O() M 3 () N The estimate () should indeed hold outside an initial time interval, where positive waves will decay, and away from interaction times Our main results are as follows Theorem Assume f (u) κ>0for every u R Let u F M be a solution of the viscous conservation law (), and fix N Then, for every t>0, the minimization problem (8) has at least one solution Theorem There exist constants α (uniformly valid for all N and u F M ) and β = β N,M (depending only on N and M) such that ( ) () J min u(t) α N for all t [0, [ \I u for an exceptional set I u of times, with meas(i u ) β The remainder of the paper contains a proof of the above two theorems We remar that Theorem states the existence of a minimizer for the scalar function J : R 3N R Since J is continuous and positive, the result would be trivial if J (y) as y However, it is easily seen that this coercivity condition fails The heart of the proof consists in showing that, if {X (m) } m is a minimizing sequence with X (m), then a second minimizing sequence X (m) can be defined (in terms of X (m) ) whose elements remain uniformly bounded The proof of Theorem involves a deeper argument With a solution of the viscous equation () we associate a curve γ moving in the plane By results in [BB, BB, BB], the total area swept by this curve in its motion is a priori bounded in terms of a monotone decreasing area functional Q(u) We then show that at every time t where the rate of decrease d dtq(u(t)) is sufficiently small, the inequality () holds We remar that in (8), the integer N is fixed Of course, one could let N vary and loo at the minimization problem (3) min N 0 { inf ɛn + J ( ) } u(t), ω,,ω N ω,,ω N Here the first term penalizes the complexity of the description, adding a cost for each new viscous profile The small constant ɛ>0 acts as a threshold parameter Small viscous shoc waves, whose strength ω x L is of order <ɛ, will not be traced From Theorem it immediately follows that the problem (3) also admits a global minimizer This can be interpreted as an optimal description of the solution profile u(t, ) as superposition of traveling viscous shocs 3 Proof of Theorem Step At any fixed time t>0, the solution u(t, ) of the viscous conservation law () is a C function with bounded total variation We shall prove, more generally, that the functional J (u; ω,,ω N ) admits a global minimum for every C function u : R R with bounded variation Step Recall that ω = ω (u ±,y) Observing that the traveling wave profiles ω as well as their derivatives ω,x depend continuously on the scalar parameters u, u+, y ( =,,N), we have to prove that the continuous scalar function J (u; ) :R 3N R admits a global minimum
478 WEN SHEN AND MEE REA PARK Since J 0, this function has a nonnegative infimum J min We can thus construct a minimizing sequence in R 3N, converging to J min,say, ( ) } {X (m) (3) = y (m),u (m)+,u (m),, y (m) ; m N,u(m)+ N,u (m) N By possibly taing a subsequence, we can assume that each component of the vector X (m) R 3N either converges to a finite limit or else diverges to ± Step 3 If sup m { y (m) + u (m)+ + u (m) } < for each =,,N, then the entire minimizing sequence {X (m) } m is bounded in R 3N By our previous assumption, it converges to some limit X = ( ȳ, ū +, ū,, ȳ N, ū + N, ū N) By continuity, we thus have J (u; X) =J min, proving the existence of a minimizer Step 4 In general, however, one cannot guarantee the minimizing sequence to be bounded, because the function J (u ; ) is not coercive on R 3N We shall thus adopt an alternative strategy Assume that for some index, lim Consider the new sequence { y (m) + u (m)+ X (m) = (ỹ (m), ũ (m)+, ũ (m) + u (m),, ỹ (m) N } =, ũ(m)+ N ), ũ (m) N, obtained by setting the parameters of the th traveling profile to zero More precisely, for every m weset ( ) ( ) ỹ (m) i, ũ (m)+ i, ũ (m) i = y (m) i,u (m)+ i,u (m) i if i, ( ) ỹ (m), ũ (m)+, ũ (m) =(0, 0, 0) We claim that (3) lim sup J (u; X(m) ) lim J (u; X(m) ) If the original sequence had unbounded components, say, for {i,i,,i } {,,, N}, the above construction yields a new minimizing sequence having unbounded components By induction, in a finite number of steps we obtain a minimizing sequence where all components are bounded Hence, by Step 3, a global minimizer exists Step 5 It now remains to show that (3) holds Equivalently, for every ε>0we will prove that (33) lim sup J (u; X(m) ) lim J (u; X(m) )+ε We shall consider different cases
Case Assume that, as m, OPTIMAL TRACING OF VISCOUS SHOCKS 479 (34),x L 0 By the assumption f (u) κ>0, the strict convexity of the flux function implies (35),x L 0 In this case, observing that 0 because all viscous shoc profiles are decreasing, we have the estimate u x dx i = u x i u x i dx u x dx u x L,x dx,x L Here we used the elementary inequality (a b c) (a b) ac, valid whenever b and c have the same sign Therefore, (36) lim u x i = dx lim sup u x i dx, provided that (34) holds Clearly, (36) implies (3) Notice that the condition (34) is certainly satisfied if u (m)+ and u (m) remain uniformly bounded and u (m)+ u (m) 0 Case Assume that (37) lim inf,x L This breas down into three different subcases Case a We have the limits y (m) the ideas, assume y (m) a viscous shoc profile connecting u = δ > 0 while u (m)+ u +, u(m) u Tofix + Observe that in this case δ = ω,x L, where ω is with u + Given ε>0, choose L so large that ( L u x dx ) / < ε δ
480 WEN SHEN AND MEE REA PARK We then have the estimate u x dx i = u x L dx u x,x dx u x i L u x ( ) L dx u x dx sup,x i (x) x<l ( / u x dx) Observing that lim sup x<l L,x (x) =0,,x L lim,x = δ L,x dx from the above estimate we deduce lim inf u x dx lim inf i= u x dx ε δ i δ This clearly implies (33) Case b Assume that both sequences u (m)+ and u (m) diverge to + The case where they both tend to is entirely similar We then have lim inf u,x dx ( ) lim inf u(x) (x),x = x R L Hence the original sequence was not minimizing This contradiction shows that this case cannot happen Case c Assume that the strength of the th traveling wave becomes arbitrarily large as m, so that u (m) u (m)+ In this case, it is easy to chec that J (u ; X (m) ) Indeed, let K = u L We then have lim inf u,x dx lim inf,x dx (x) >K+ Obviously, this integral diverges to infinity Indeed, let s consider the case when lim u(m) =+ For the case when lim u (m)+ = it is entirely similar We have lim inf,x dx lim inf,x dx (x) >K+ lim inf min,x (x)>k+ (x)>k+,x (x)>k+ dx =
OPTIMAL TRACING OF VISCOUS SHOCKS 48 This proves that the original sequence was not minimizing We again conclude that this case cannot happen This completes the proof of Theorem 4 Proof of Theorem We shall rewrite the parabolic equation () using a different set of variables: (4) v = f(u) u x, τ = t, η = u This change of variable was first introduced in [BB] and then used in later papers [BB, BB] For each fixed time t>0, the solution of () () is smooth The map (4) x γ t (x) = ( u(t, x),v(t, x) ) parameterizes a curve γ t in the u-v plane To see how this curve evolves in time, from (3) one obtains (43) v t + f (u)v x = v xx On regions where u x 0 we can now use (τ,η) as independent variables, instead of (t, x) From (4) and (43) we obtain u x = f(u) v, v η = v x u x, v ηη = v xx u x v x u 3 u xx, x Therefore v τ = v t u t u x v x =(v xx f (u)v x ) v x u x (u xx f (u)u x )=v xx u xx u x v x (44) v τ =(u x ) v ηη = ( v f(η) ) vηη In particular, the curve γ = γ(τ,η) = (η, v(τ,η)) evolves in the direction of the curvature and its total length is monotone decreasing in time Another functional which is monotonically decreasing in time is the area functional (45) Q(γ) = η< η γ η (η) γ η ( η) dηd η, defined as the double integral of a wedge product In terms of the original coordinates u, x, wehave Q(u) = u x ( x) [f (u(x)) u x (x) u xx (x)] (46) x< x u x (x) [f (u( x)) u x ( x) u xx ( x)] dxd x All these calculations, (4) (46), can be found in [BB, BB] As proved in [BB], the decrease of the functional Q controls the area swept by the curve γ in its motion By parabolic regularization estimates, at time t = we now have (47) Q ( u() ) C
48 WEN SHEN AND MEE REA PARK for some constant C, uniformly valid for all solutions u F M Therefore (v f(η) ) vηη dηdτ v τ (τ,η) dηdτ { d dt Q( u(t) ) } (48) dt Q ( u() ) C As a consequence, for any given ε>0, there exists a set of times I u [, [ with (49) meas(i u ) C /ε such that (40) (v(t, η) f(η) ) v ηη(t, η) dη ε for all t,t / I u In addition, the assumption (4) of genuine nonlinearity yields the well-nown decay estimate u x (t, x) (κt), hence (4) <u x (t, x) ε, x R, for all t (κε) To achieve a proof of Theorem, it now suffices to show that at every time t where (40) (4) hold with some ε > 0 sufficiently small, the profile of u(t, ) can be suitably approximated by a finite superposition of viscous shoc profiles, and () holds As before, set δ = M/N We can single out finitely many disoint intervals I =[a,b ], =,,ν, such that (4) (43) min u x (t, x) 7δ x I for all, u x (t, x) u x (t, a )=u x (t, b )= δ 3 for all x I, u x (t, x) > 7δ for all x/ I I ν The images of these intervals through the mapping x γ(x) are graphs of functions v = v () (η), say, with η [b,a ] = [ u(b ),u(a ) ] ; see Figure 3 For each we now choose a point x [a,b ] such that (44) m = ux (t, x ) = min a x b u x (t, x) and call γ the segment in the u-v plane with endpoints on the graph of the function f, tangent to the graph of the function v () at the point c = u(t, x ), as in Figure 4 Let u + <u be the points where γ intersects the graph of f, and call ω the unique viscous traveling wave profile satisfying ω( ) =u, ω( ) =u+, ω = f(ω) σ ω [ f(u ) σ u ], σ = f(u ) f(u+ ) u = f ( u(t, x ) ), u+ (45) ω(x )=u(t, x ), f (ω(x )) = σ It is important to notice that, by the previous construction, the image of the one-to-one map ( x ω (x),f ( ω (x) ) ) ω,x (x)
OPTIMAL TRACING OF VISCOUS SHOCKS 483 f γ v () v () u a b a b x b b a a η Fig 3 Example of a solution of the viscous conservation law and the corresponding curve in the v η plane f v () γ u + a b c u η Fig 4 Fitting in a viscous shoc ω, illustrated in the v η plane is precisely the segment γ Moreover, the tangency condition and the maximality condition (44) imply that, at x = x, u x (t, x )=ω,x (x ), u xx (t, x )=ω,xx (x )=0 Geometrically, this means that both u(t, ) and ω ( ) have an inflection point at x = x We now recall that, by (40), a ( v () (η) f(η)) v ηη () (46) (t, η) dη ε b Restricted to the region where u x δ 3, the previous inequality implies the ey estimate (47) ν v ηη () (t, η) dη ε δ 6 = {v () (η) f(η) δ 3 } Since ε>0 can be chosen arbitrarily small, according to (47) every function v () is almost affine, hence its graph is very well approximated by the tangent line γ Reverting to the original variables t, x, this in turn implies that u(t, ) is closely approximated by the corresponding traveling profile ω on the appropriate interval x [a,b ]
484 WEN SHEN AND MEE REA PARK Lemma 4 Assume that the flux function satisfies (48) Then for every ε following: 0 <κ f (u) κ for all u R > 0 there exists ε>0 small enough so that (47) implies the (49) u ω L ([a,b ]) ε, u ω H ([a,b ]) ε for all Moreover, (40) (4) (4) m u(t, a ) u(t, b ) κ, sup ω,x (x) 3δ 3, x/ [a,b ] R\[a,b ] ω,x (x) dx 6δ κ Proof By choosing ε>0 sufficiently small, we can assume that the C distance v () γ C (43) ([b ]),a is as small as we lie By (4), when x [a,b ]wehaveu x δ 3 The map x u(x) is thus invertible on each interval [a,b ] The two norms in (49) can both be estimated in terms of the distance (43) We now prove (40) Using (43) and recalling (4), by taing ε>0 sufficiently small we can assume that γ (a ) f(a ) v () (a ) f(a )+ v () γ C 0 3δ 3, γ (b ) f(b ) v () (b ) f(b )+ v () γ C 0 3δ 3 The inequality (40) now follows from a simple geometrical inequality (see Figure 5(a)) If f <κ and γ is a linear function such that γ(a ) f(a ) 3δ 3, γ(b ) f(b ) 3δ 3, γ(c) f(c) =m, for some points b <c<a, then a b (m 3δ 3 ) κ Since we are assuming δ<, m 7δ > (3δ 3 ), from the previous inequality we deduce a b m κ, proving (40) The inequality (4) follows from sup ω,x (x) = max { ω,x (a ), ω,x (b ) } δ 3 + v () γ C 3δ 3 x/ [a,b ] ([b,a ])
OPTIMAL TRACING OF VISCOUS SHOCKS 485 f f m 3 3 c c b a b a (a) (b) Fig 5 Some geometrical properties of convex functions To prove (4), call α,β the points where the line γ intersects the graph of f, as in Figure 5(b) Then ω,x (x) dx =(α a )+(b β ) R\[a,b ] Consider the function g(u) = γ (u) f(u) Clearly, we have g(c ) = max g(u) u 7δ, and c is the midpoint of the interval [β,α ] Assuming f κ, then Recalling that a c 4(m 3δ 3 ) κ we conclude that g(a ) 3δ 3, g(b ) 3δ 3, g = f κ, a g (a )= g (u) du κ(a c ) κ(m 3δ 3 ) c Recalling that m 7δ and δ<, we obtain α a 3δ 3 κ(m 3δ 3 ) 6δ3 3δ 8κδ κ The estimate for b β is totally similar Together, these yield (4) The proof of the lemma is completed As approximations to u(t, ) we now choose the N traveling profiles ω in the above list, corresponding to the N largest values of m,say,m m m N Notice that (40) implies Hence (44) N mn κ M, m N κ u x (t, x) κ ( M N ), x / ( ) M N N [a,b ] =
486 WEN SHEN AND MEE REA PARK Using the estimates (49) (4) we now chec that the functional J at (7) is small, as claimed by Theorem The first half of the right-hand side in (7) can be estimated as = R = u(t, x) ω (x) ω,x (x) dx ( = + + x [a,b ] x/ [a,b ] u ω L ([a,b ]) ω,x L (R) ( ) u L (R) + ω L (R) ) u(t, x) ω (x) ω,x (x) dx sup x/ [a,b ] O() Nε + N M 3δ 3 6δ = O() κ N 4 ω,x (x) ω,x (x) dx R\[a,b ] For the second half of the right-hand side of (7), we can use the inequality (a+b) (a + b ), valid for all real numbers of a, b, and we get R u x(t, x) ω,x (x) dx = = b a u x(t, x) ω,x (x) dx + = R\ [a,b ] u x(t, x) ω,x (x) = b u x (t, x) ω,x (x) + ω,x (x) dx a ( + u x (t, x) + ω,x (x) ) dx R\ [a,b ] b b u x (t, x) ω,x (x) dx + ω,x (x) dx a a ( ) + u x (t, x) + ω,x (x) dx R\ [a,b ] I + I + I 3 + I 4, dx where I = u ω H ([a,b ]),
OPTIMAL TRACING OF VISCOUS SHOCKS 487 I = b ω,x (x) dx ω,x (x) dx a R\[a,b ] { } sup ω,x (x) ω,x (x) dx, x/ [a,b ] R\[a,b ] { } I 3 = sup u x (t, x) u x (t, x) dx, I 4 = x/ [a,b ] { } sup x/ [a,b ] ω,x (x) R R\ [a,b ] Using the estimates in Lemma 4 and (44), we get I Nε, I N 3δ 3 6δ, κ I 3 κ M N M, I 4 N3δ 3 N 6δ κ ω,x (x) dx Note that I measures how well the viscous shoc profile matches the solution on the interval [a,b ], and this term is arbitrarily small I measures the H norm of the viscous shoc waves outside the interval [a,b ], and it is of O()/N 4 And I 3 is the sum of all the shoc waves that are not represented This is the largest term here, and is of O()/N Finally, I 4 is similar to I, and is of O()/N 3 In summary, we have R u x(t, x) ω,x (x) = dx O() N Putting these two parts together, we get the desired result 5 Concluding remars For solutions to the conservation law (), the transient behavior is nontrivial and can last an arbitrarily long time This happens because we are considering solutions defined on the whole real line On the other hand, if the equation is restricted to a bounded interval, say, (5) u t + f(u) x = u xx, x ]a, b[, with boundary conditions (5) u(a) =α, u(b) =β, then all solutions would converge at an exponential rate to a unique steady state w( ) Indeed, from basic theory of parabolic equations [H] it follows that there exists a unique function w :[a, b] R which satisfies the two-point boundary value problem (53) f(w) x = w xx, w(a) =α, w(b) =β
488 WEN SHEN AND MEE REA PARK Linearizing (5) around the steady state w, one obtains the existence of some δ>0 such that, for every initial data ū L the corresponding solution of (5) (5) satisfies u(t) w C ([a,b]) Ce δt Here one can choose a constant C uniformly valid on bounded subsets of L After an initial time interval, the long-term behavior of the solution is thus trivial In the case of a bounded domain, the corresponding equation (44) in the (η, v) variables must be supplemented with the boundary conditions v η (α) =v η (β) =0 The unique steady state solution of (53) corresponds to a constant function: v(η) κ for all η [α, β] Observing that β β b a = du = α u x α f(η) v(η) dη, one can uniquely determine the constant κ from the relation b a = β α f(η) κ dη Acnowledgments The authors are grateful to Prof Alberto Bressan at the same department for proposing the problem and for many useful discussions We also want to than one of the referees for carefully reading through the paper and for various remars REFERENCES [BB] S Bianchini and A Bressan, A case study in vanishing viscosity, Discrete Contin Dyn Syst, 7 (00), pp 449 476 [BB] S Bianchini and A Bressan, On a Lyapunov functional relating shortening curves and viscous conservation laws, Nonlinear Anal, 5 (00), pp 649 66 [BB] S Bianchini and A Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann Math, 6 (005), pp 3 34 [B] A Bressan, Hyperbolic Systems of Conservation Laws The One Dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 000 [H] D Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math 840, Springer-Verlag, New Yor, 98 [S] D Serre, Systems of Conservation Laws I, Cambridge University Press, Cambridge, UK, 000 [Sm] J Smoller, Shoc Waves and Reaction-Diffusion Equations, Springer-Verlag, New Yor, 983 [T] R Temam, Infinite Dimensional Dynamical Systems in Mathematics and Physics, Springer- Verlag, New Yor, 988 [W] G Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New Yor, 974