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Regularity of solutions to the perturbed conservationlawsJerome A. Goldstein a & Yudi Soeharyadi aa Department of Mathematical Science , The University of Memphis , Memphis, TN , 38152Published online: 02 May 2007.

To cite this article: Jerome A. Goldstein & Yudi Soeharyadi (2000) Regularity of solutions to the perturbed conservation laws,Applicable Analysis: An International Journal, 74:1-2, 185-199

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Regularity of Solutions to the Perturbed Conservation Laws Communicated by R.P. Gilbert

JEROME A. GOLDSTEIN and YUDI SOEHARYADI

Department of Mathematical Sciences,

The University of Memphis, Memphis, TN 38152

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L1-norm. A dissipativity condition from the semi- group approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of bounded L1-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions.

AMS: 35L65,47H20

KEY WORDS: Conservation laws, perturbed conservation law, balance law, regularity of solutions, m-dissipative operator, LP-variation.

(Received for Publication June 1999)

1. INTRODUCTION

In this paper we wish to study a kind of regularity for the Cauchy problem

(CP) of a perturbed conservation law in arbitrary space dimension:

u ( x , 0) = uo(x) , for x E Rn

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186 J.A. GOLDSTEIN AND Y. SOEHARYADI

The flux function 4 : R + Rn is an even, continuous function with +(O) = 0

and satisfies a mild regularity near the origin, namely limx-to = 0. 1x1-

(For n = 1, this is no condition at all; for n 2 2, it is a Hijlder condition at

zero, which holds if @(0) exists.) The function g : R --+ R is assumed to be

globally bounded and Lipschitzian, i.e., there are corresponding constants

C1 and C2, positive such that

for x, y E Rn. Further we assume that H : Rn --+ R is continuous, positive

and bounded away from zero. The initial condition uo is in L1(Rn)n L"(Rn).

Here we adopt the notation 4(u), for divd(u). We may think this equation

as a scalar conservation law with a persistent source or sink term, depending

on the sign of g. This equation is also known as a balance law.

This problem evolves from the Cauchy problem for the (unperturbed)

conservation law.

('4 ut + 4(u), = 0, for t > 0 and x E Wn,

u(x ,O)=uO(x) , for x E R n ,

where 4 is described as in the discussion following (1). It is well known

that this equation does not have solutions in the strong sense ( see [4, 91).

Solutions exist only in a weaker sense. Further, if phi is strictly convex, then

nonzero solutions develop discontinuities during their evolution, regardless

of the smoothness of the initial data. Crandall [4] and Benilan [2] however,

showed that as an Abstract Cauchy problem, or (ACP),

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CONSERVATION LAWS 187

where A is the operator defined by Au := -4(u),, the problem (3) is gov-

erned, on L1(Rn), by an L1-contraction semigroup {Tt),

- where u,v E V ( A ) , for t > 0. The corresponding operator A is m-

dissipative, which in term of the resolvent equation means

where p,q E V ( ( I - AA)-I), for any X > 0, and,

which is called the range condition for the (ACP). This establishes the

existence and at the same time, uniqueness of a global mild solution for

the (ACP), for all t > 0. For background on the semigroup theory see,

for example, [I, 3, 81. An important point here is that the description of

D ( A ) is quite subtle and was inspired by Kruzkov's work [9]. The precise

definition of D(A) is not needed here, so we omit specifying it.

For the perturbed problem, Goldstein and Park [6], taking a similar ap-

proach, considered (1) as an (ACP),

with the operator 2 defined as

They proved that the operator is quasi m-dissipative via perturbation the- - ory. That is, A - wI is m-dissipative, for some real number w . Therefore

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188 J.A. GOLDSTEIN AND Y. SOEHARYADI

the (ACP) is also governed by semigroup {'.?$I on L1(Wn) and hence the ex-

istence and uniqueness of a mild solution for the (ACP) for t > 0 is assured.

For earlier work on perturbed conservation laws see Schonbek [13,12], where

she considered a singular perturbation case in one dimension.

We wish to impose some regularity on the initial data of the perturbed

(ACP) (7) and investigate the lifespan of the mild solution of this (ACP)

with additional regularity, in relation to the nature of this initial data.

Recall that according to the celebrated theorem of Glimm on the existence

of solutions to systems of conservation laws, if one starts with initial data

which is small in magnitude and in total variation, then the solutions will

continue to have bounded total variation. We will draw a parallel; in Section

2, we define regularity in terms of variation measured in L1-norm. We will

show that if we start with initial data which has, not necessarily small,

bounded L1- variation, then the solution also has bounded L1-variation. As

a corollary we obtain a Banach space of bounded L1-variation functions,

which is invariant under the semigroup {'.?$I. Via the work of OuYoung [ll]

on the periodic version of the (CP) ( I ) , using the same analysis we show

that the regularity and invariance results are true for the periodic problem,

provided we are more restrictive on the flux function 4. We will show this

in Section 3. In the Appendix we offer a generalization of the function space

defined in Section 2, namely the space of bounded LP-variation functions,

1 5 p 5 m. Normed appropriately this construction gives rise to Banach

spaces. For p = rn this function space is nothing but the space of Lipschitz

functions.

2. PROPERTIES OF SOLUTIONS AND THE SPACE ~i~( ' ) ( lW")

Let u1 denote the translate of u E L/o,(Wn) by a fixed vector 1, i.e.,

u,(x) := U ( X + E ) ,

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CONSERVATION LAWS

for every x E Rn. For every constant K > 0 we define the set,

C i ' := {f I f E L1(Rn), J l fi - fill 5 K I 1 1 , for "11 1 E Rn).

This set consists of functions which have a specific bound, h', for variation

measured in the L1-norm. The Frechet - Kolmogorov theorem implies that

c!) is compact in L1(Rn), for each h' > 0. ( see [5, pp. 292, 3871).

Supposed u solves the resolvent problem in the perturbed conservation

law problem, i.e., for a given h E ~ ( x ) ,

Similarly, the translate ul of u satisfies

Then this pair of equations can be rewritten as

where now we can consider the right hand side of (8) and (9) as the given

data in the resolvent equation for the (unperturbed) conservation law. The

dissipativity condition (5) implies

Additionally let us assume that g E Lm(R)n Lip(R) and H(x) > a > 0. We

impose some L1-variation bound to the initial data. We have the following

results:

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190 J.A. GOLDSTEIN AND Y. SOEHARYADI

LEMMA 1. Let the assumptions above hold. Fix h' > 0 and KO > 0. If

h E c;) and H E c!; then u E c:), where

provided X < I l d l ~ l P '

Proof. We continue developing the estimate in (10) and use the definitions

of c!) and c!:. Thus

As H E c!,), we have I( H - Hl 11, 2 KO 1 1 1, and therefore

provided that the denominator in the right hand side of (11) is positive. 0

For fixed t and n, we set X = i. Also we set h' and lie positive. We iterate

the process in the Lemma 1 to obtain the result as follows.

LEMMA 2. With the assumptions as in the Lemma 1, we have

where

for any la which makes iy < 1 . Here we set a = &%, and 7 = %.

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CONSERVATION LAWS

Proof. Lemma 1 shows that

where Q, as in the proof of Lemma 1, is

Suppose after r iterations (r < n) we have (I - ;L)-' h 6 c::, where

( -)-lrt') h E c?', where according to Then the next iteration yields I - - A Q

Lemma 1,

Plugging Q, into the last equation yields

This concludes the induction. 0

The main result deals with certain situations in which for fixed values of

A' and I&, the mild solution of (1) has L1-variation within a given bound,

i.e., the lifespan of solution in c:), for a given Q. The theorem gives a lower

bound for maximal lifespan of the mild solution with such regularity.

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192 J.A. GOLDSTEM AND Y. SOEHARYADI

(1) T H E O R E M 1. Fix positive numbers K , KO and Q . Assume u0 E CK

and H E c!:. Also assume that the mild solution of ( 1 ) stays in the class

c!' for t E ( 0 , ~ ) . Then we may choose

with a and 7 defined as in Lemma 2.

Proof. By the quasi m-dissipativity of the operator i, the Crandall- Liggett

theorem ( ref. [ I , 8, 31) applies to give

lim I - -A uo = u ( t ) . n--too ( :-I-"

From Lemma 2, we take the limit as n + w , and the Dominated Conver-

gence theorem implies, with h = uo,

where Qoa = limn-+oo Qn. We need to compute Q,. For this purpose let

us fix n and t positive, and notice that

Then we can rewrite Qn as

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Letting n + ca yields

Xotice that Q , depends on t . We require that u ( t ) E c:', therefore we

need Q , < Q. Upon solving the last equation for t we have

hence the lower bound for r .

Let us define the set of functions with bounded L1-variation to be

Normed appropriately, it is a Banach space ( see the Appendix ). Theorem 1 - implies that the semigroup for the perturbed conservation law, {Tt) , leaves

the space Lip(')(Rn) invariant for all time. That is, for any t > 0,

Moreover it gives us the growth bound for L1-variation over all time.

COROLLARY 1. With the assumptions as in the Theorem I , we have

Example. In the absence of perturbation, i.e. g = 0, with uo = h,we have

llu - ~ 1 1 1 1 5 llh - hllll,

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194 J.A. GOLDSTEM AND Y. SOEHARYADI

( see [4]) which implies

for all t > 0 and all A' > 0.

3. THE PERIODIC PROBLEM

We wish to use the same approach as the above for the periodic problem,

(13) g(u)

ut + 4(u), + - = 0, t 2 O,x E Rn, H(x)

U(X, 0) = uo(x), 5 E lRn,

u(x + p, t ) = u(x, t ) , t 2 0, x E lRn.

Here p, a fixed vector in Rn, is the period of the given problem. The func-

tions g and H are assumed to have properties as in the (CP) (1). To apply

the analysis done in the previous sections we need to show that the opera-

tor which arises in (ACP) which corresponds to (13) is quasi m- dissipative.

The corresponding (ACP) is

where Au = -4(u),, and Bu = -a. H(x)

OuYoung [ll] showed that the (unperturbed) periodic problem is gov-

erned by a contraction semigroup, and the corresponding A is m-dissipative,

provided that the flux 4 is in C1(IR,IR). His definition of D(A) was deli-

cate and differed from the definition used by Crandall [4] and Benilan [2].

Basically more regularity was built into D(A), since otherwise the proof of

dissipativity (based on Kruzkov's uniqueness argument [9] as adapted in

[4]) did not carry over. But then the proof of the range condition became

more subtle, see [ l l ] . With this fact in hand, we use perturbation theory to

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CONSERVATION LAWS 195

show that A $ B is quasi m- dissipative. It is sufficient to show that B is

globally Lipschitzian. Let T n denote the torus Rn/({O, pl) x . . . x (0, p,)),

where p = (pl, . . 8 , p,) is the period.

T H E O R E M 2. Let g : !R -t R be (globally) Lipschitzian [and nondecreas-

ing], and let H be continuous and such that H ( x ) 2 a > 0, for x f Tn.

Then the operator B = -g(u)/H(x) is (globally) Lipschitzian [and dissipa-

tive]. The conclusion in the square brackets follows from the hypothesis in

the square brackets. The statement without brackets is also valid.

Proof. Let u and v be in V(B) . Then,

thus B is Lipschitzian. For the dissipativity part, first we let $(u) =

I I u I I l~ ignou, for each u E L1(Tn). Il, is a section of the (multivalued) duality

map of L1(Tn). For u and v in V(B) we have

since g is nondecreasing. Thus we have dissipativity of B . 0

Now all the "computations" we have done in the previous section apply to

this periodic problem. We have the following.

T H E O R E M 3. Let g be bounded and globally Lipschitzian. Let H be con-

tinuous and satisfy H ( x ) > a > 0 , and let the flux function c,b be C1. Let

{T t ) be the semigroup which governs the perturbed periodic problem as in

( 1 3 ) . Then the following hold:

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1 % J.A. GOLDSTEIN AND Y. SOEHARYADI

4. APPENDIX

We can generalize the function space ~ i p ( ' ) by using the LP-norm instead

of using the L1-norm, for 1 5 p 5 oo. Let G be measurable subset of IRn , not necessarily of finite measure. Let u E L;o,(G). As before, let ul denote

the translate of u by a fixed vector 1, i.e.,

for every x in the domain of u. Since G is not necessarily closed under

translation, we define

for u E LP(G), and 1 5 p < w. Here GI = {x + 1 I x E G). In particular,

we call the number

the LP-variation of u. We will use the notation LPVar(u) for this quantity.

Now we can define

the space of bounded LP-variation functions. Notice that for p = a, we

have

lip(")(^) = {f E LM / for some Kf > 0,sup 1 fl - f 12 Kf 1 1 1,

for every 1 E G),

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CONSERVATION LAWS 197

i.e., the usual space of Lipschitz functions on G.

Since LpVar(f) = 0 if and only if f is constant, LPVar merely defines

a seminorm on L i p ( p ) ( ~ ) . To make it into a norm we couple it with the

LP-norm as follows :

for 1 < p < co. With this definition of norm we have the following.

T H E O R E M 4. With this norm, the space Lzp(~)(G) is a Banach space.

Proof. This is well known for p = m, so suppose 1 5 p < cm. We only have

to show completeness. Let {fn) be a Cauchy sequence in L ~ ~ ( P ) ( G ) , and let

c > 0. Choose N > 0 such that

for m, n > AT. Therefore {f") is a Cauchy sequence in LP(G). Let fn + f in LP(G). We will show that LPVar(fn - f ) -+ 0, as n + oo. That is, we

will find M > 0 so that Il(fn- f ) ~ -(fn - f)llp 5 Kn 1 1 / , fo r any 1 E G,

while 0 5 h;, < ~/2 ' /p , for all n > M. We will also show that f E Lip(p)(G).

We choose a subsequence, which for simplicity of notation will also be

denoted by {f"), for which f n -+ f a.e. Choose M > 0 such that ,

for any 1 E G, while 0 5 Kn, 5 c /2 ' /~ , for all n, m > M. Here h',, =

LPVar(fn - f m ) . We further extract a subsequence { f n r ) such that 1 1 f". - f n l j p 5 1/2', for any n > n,. Now we define,

Then the sequence {Fns) is monotone, nondecreasing, and further 1 1 FnsII; 5 11 f n l Ilz+l, therefore by the Lebesgue monotone convergence theorem, Fn* /'

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198 J.A. GOLDSTEIN AND Y. SOEHARYADI

F E LP(G). We conclude that F is finite a.e. Notice that f n 6 = f n l $

xJ~:(f"J+l - f " ) , and thus

We get a domination, ( ( f n 3 - fnr) , - (fn8 - f n r ) ( 5 ( 2 4 + 2 F ) E LP(G).

Dominated convergence implies

as r -+ co. Next, we take the limit as r -+ m on both sides of the inequality

(15) to get

for all 1 E G, while 0 5 Kn, 5 ~ / 2 ~ / p , for any n, > M. Here K,,, = - lim,,, K,,,, . And therefore LPVar( fn - f ) 5 6/2l/p, for n > M.

Lastly, we show that f E Lip(p)(G). Recall that we assume f" -+ f a.e.

And we also have for every n, 1 1 fin - f n l l p 5 Kn I 1 1 , for any 1 E G. Again

by dominated convergence, as n + m ,

showing that f E L i p ( p ) ( ~ ) and therefore fn converges to f in L ip (p ) (~) . 0

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