Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Digital Object Identifier (DOI) 10.1007/s002110000156Numer. Math. (2000) 86: 377–417 Numerische

Mathematik

Discrete approximations ofBV solutionsto doubly nonlinear degenerate parabolic equations

Steinar Evje, Kenneth Hvistendahl Karlsen

Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen,Norway; (e-mail:steinar.evje; [email protected])

Received November 10, 1998 / Revised version received June 10, 1999 /Published online June 8, 2000 –c© Springer-Verlag 2000

Summary. In this paperwe present and analyse certain discrete approxima-tions of solutions to scalar, doubly nonlinear degenerate, parabolic problemsof the form

∂tu+ ∂xf(u) = ∂xA(b(u)∂xu), u(x, 0) = u0(x),

A(s) =∫ s

0a(ξ) dξ, a(s) ≥ 0, b(s) ≥ 0,

(P)

under the very general structural conditionA(±∞) = ±∞. To mentiononly a few examples: the heat equation, the porous medium equation, thetwo-phaseflowequation, hyperbolic conservation lawsandequationsarisingfrom the theory of non-Newtonian fluids areall special casesof (P). Since thediffusion termsa(s) andb(s) are allowed to degenerate on intervals, shockwaves will in general appear in the solutions of (P). Furthermore, weaksolutions are not uniquely determined by their data. For these reasons wework within the framework of weak solutions that are of bounded variation(in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-calledBV entropyweaksolutions follows fromaworkofYin [18]. Thediscreteapproximationsare shown to converge to the uniqueBV entropy weak solution of (P).

Mathematics Subject Classification (1991):65M12, 35K65, 35L65

The research of the second author has been supported by VISTA, a research cooperationbetween the Norwegian Academy of Science and Letters and Den norske stats oljeselskapa.s. (Statoil).Correspondence to: K. Hvistendahl Karlsen

378 S. Evje, K.H. Karlsen

1 Introduction

In this paper we present and analyse certain finite difference schemes fora class of scalar, doubly nonlinear degenerate, parabolic equations in onespatial dimension. Nonlinear parabolic evolution equations arise in a varietyof applications, ranging frommodels of turbulence, via traffic flow, finanicalmodelling and flow in porous media, to models for various sedimentationprocesses. The problem we study here is of the form

∂tu+ ∂xf(u) = ∂xA(b(u)∂xu), (x, t) ∈ QT = R× 〈0, T 〉 ,u(x, 0) = u0(x), x ∈ R,

(1)

where

A(s) =∫ s

0a(ξ) dξ, a(s) ≥ 0, b(s) ≥ 0.

We assume thata(s), b(s), f(s) andu0(x) are appropriately smooth func-tions. The functionsa(s) andb(s) are allowed to have infinite number ofdegenerate intervals inR. Difficulties arise because of thisdouble degener-acyaswell as thedouble nonlinearityrepresented by the nonlinear functionsa andb. By definingB as

B(u) =∫ u

0b(ξ) dξ,

we may write (1) as

∂tu+ ∂xf(u) = ∂xA (∂xB(u)) .(2)

Examples of such equations include the heat equation, the porous mediumequation and, more generally, convection-diffusion equations of the form

∂tu+ ∂xf(u) = ∂2xB(u).(3)

Included are also hyperbolic conservation laws

∂tu+ ∂xf(u) = 0,(4)

aswell as certain equations arising from the theory of non-Newtonian fluids,

∂tu = ∂x(∂xu

m |∂xum|n−1), n ≥ 1, m ≥ 1,(5)

which corresponds to the caseA(v) = v|v|n−1 andB(u) = um.Kalashnikov [10] has established the existence ofcontinuous solutions

of the Cauchy problem for (2) whenf = 0 under some smoothness andboundedness conditions on the initial datau0 and some structural conditionson a(s) andb(s). In particular, these conditions imply thata(s) andb(s)may have degeneracy at and only at the origins = 0. We also refer to some

Doubly nonlinear degenerate equations 379

recent work by Lu [14] for results concerning the regularity of solutionswhen the equations are degenerate at points at whichu and∂xu vanish.

The more interesting cases are those in whicha(s) andb(s) may haveinfinite or uncountable points of degeneracy. A striking feature of such non-linear strongly degenerate parabolic equations is that the solution will gen-erally develop discontinuities in finite time, even with smooth initial data.This feature can reflect the physical phenomenon of breaking of waves andthe development of shock waves. Consequently, due to the loss of regu-larity, one needs to work with weak solutions. However, for the class ofequations under consideration, weak solutions are in general not uniquelydetermined by their data. Therefore an additional condition, the so-calledentropy condition (see (b) below), is needed to single out the physicallyrelevant weak solution. Hence attention focuses on finding a physically rea-sonable framework which incorporatesdiscontinuous solutionsand at thesame time guarantees uniqueness. The concept of a (weak) solution, whichwe adopt to theCauchy problem (1) in this paper, is that ofBV entropyweaksolutionsas formulated by Yin [18] for the initial-boundary value problem.We shall say thatu(x, t) is aBV entropy weak solution (see Sect. 2 forprecise statements) if:

(a) u(x, t) is inBV (QT ) andB(u) is uniformly Holder continuous onQT .

(b) ∂t|u−c|+∂x[sign(u−c)(f(u)−f(c)−A (∂xB(u))

)] ≤ 0 (weakly).

Letting c → ±∞ in (b), we see that (1) holds in the usual weak sense.Yin [18] has shown well-posedness of the initial-boundary value problemassuming only the (very general) structural condition

A(+∞) = +∞ and A(−∞) = −∞.(6)

The well-posedness for the Cauchy problem (1) in the class of functionssatisfying the conditions (a) and (b) follows by a similar analysis, seeSect. 2.Here we should also note, as pointed out by Yin, that the assumption (6) onA is needed only for the existence result. Under the additional assumptionthatB(s) is strictly increasing, which permitsb(s) to become zero in someset of measure zero,BV solutions are continuous. Esteban and Vazquez[7] studied the occurrence of finite velocity of propagation for the solutionsof the special case (5). In particular, they showed that the interface of theequation is nondecreasing and Lipschitz continuous. Wang and Yin [16]have investigated theproperties of the interfaceof the solution for thegeneralproblem (2) whenf = 0.

Since the diffusion term∂xA(b(u)∂xu) can degenerate both ina andb,different kinds of interactions between nonlinear convection and nonlineardiffusion will take place. The (lack of) smoothness of the solution is a result


of the (lack of) balance between the convective and diffusive fluxes. In thefollowing we will briefly discuss some simple numerical examples whosepurpose is to demonstrate the effect of the degeneracy ina andb on intervals.As long as the diffusion term is nondegenerate (a, b > 0), there is a perfectbalance between the convective and diffusive fluxes and the equation thenhas a classical smooth solution. Thedegeneracywhichmayoccur inaor/andb, implies that there is a loss of regularity in the solution.

First we discuss the effect of degeneracy ina. For this purpose, let usconsider the equation (1) whenb(u) = 1. We then have equations of theform

∂tu+ ∂xf(u) = ∂xA(∂xu).(7)

Let f be the Burgers’ fluxf(s) = s2 andA the continuous function

A(s) =

s+ 4, for s ∈ 〈−∞,−5〉,−1, for s ∈ [−5,−1〉,s, for s ∈ [−1, 1],+1, for s ∈ 〈+1,+5],s− 4, for s ∈ 〈+5,+∞〉.

(8)

HenceA satisfies (6) and degenerates on the two intervals[−5,−1] and[1, 5]. In Fig. 1 (left) we have plotted the solution at timeT = 0.15. Thedegeneracy introduces only a ’mild’ loss of regularity in the solution due tothe fact that the convective and diffusive fluxes will be in balance for largegradients. Hence no jumps will arise in the solution.

Next we consider the general problem (1). Whenb(s) is zero on aninterval, jumps will in general occur in the solution. Letf be the Burgers’flux function as before, whileA is the function given by (8) andb is thecontinuous function given by

b(s) =

0, for s ∈ [0, 0.5〉,2.5s− 1.25, for s ∈ [0.5, 0.6〉,0.25, for s ∈ [0.6, 1].

In Fig. 1 we have plotted the solution of this degenerate parabolic problem(right) at timeT = 0.15. It is instructive to compare this solution with thesolution of the corresponding conservation law (4), see Fig. 1 (middle). Inparticular, we observe that the solution of the degenerate problem has a’new’ increasing jump, despite the fact thatf is convex. In that sense thesolution of the degenerate problem has a more complex structure than thesolution of the conservation law (4), as well as the solution of the problem(7). Moreover, while the speed of the jump of the conservation law solution


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x−axis

u−ax

is

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x−axis

u−ax

is

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x−axis

u−ax

is

Fig. 1. Left: The solution of Burgers’ equation with a diffusion termA which degenerateson intervals. Middle: The solution of the inviscid Burgers equation.Right: The solution ofBurgers’ equation with diffusion termsA andB which degenerate on intervals

is determined solely byf (Rankine-Hugoniot condition), the speeds of thejumps in the solution of the degenerate problem are determined by bothfandA(∂xB(u)), see Sect. 2 for precise statements of the jump conditions.

Convergence of explicit monotone difference schemes has been estab-lished for the special caseA(s) = s, see [8]. To the best of our knowledge,for the general case no convergence results for discrete approximations areavailable. The analysis presented here follows along the lines of [8]. Bothworks were inspired by the theory developed by Crandall and Majda [4].However, due to the double degeneracy as well as the double nonlinearity,the analysis in the present case is significantly more involved than in [4,8]. Compared to [8], new arguments must be used in order to obtain: (i) anappropriate cell entropy inequality, (ii) the required regularity of the discretediffusion term, (iii) identification of the limit to which the discrete diffusionterm converges.

Inwhat follows,we restrict ourattention to implicit three-point differenceschemes. That is, we consider discretizations of (2) of the following form(see Sect. 3 for more details)

Un+1j − Unj∆t

+D−(h(Un+1

j , Un+1j+1 )−A(D+B(Un+1

j )))

= 0,(9)

whereh denotes amonotone and consistent numerical flux function,∆x,∆tare themeshsizeandD+, D− are theusual forwardandbackwarddifferenceoperators respectively. Extension to generalp-point monotone schemes fol-lows easily. Note here that we choose to discretize the diffusion termwrittenon its conservative form. In [8] we observed that this seems to be essentialin order to ensure that the scheme is consistent with the entropy condition.In this paper we show that (9) satisfies a cell entropy inequality consistentwith the entropy inequality (b). As opposed to [8], this inequality does notdrop out directly from the monotone theory. The key here is the followinginequality which can be established for the discrete diffusion term:

sign(Un+1j−1 − c)A(D+B(Un+1

j−1 )) ≤ sign(Un+1j − c)A(D+B(Un+1

j−1 )).


We establish several regularity estimates for the approximate solutionswhich are sufficient to guarantee convergence (of a subsequence) to a limit.Themain difficulty here is to show that the discrete diffusion term possessesthe regularity properties known to hold forBV entropyweak solutions. Thisis obtained by deriving and carefully analysing a linear difference equationsatisfied by the numerical flux of the difference scheme (9). In addition, itturns out that due to the double nonlinearity the interpolants must be chosencarefully for the construction of approximate solutions. In [8] we could showdirectly from the linear equation satisfied by the numerical flux that thediscrete diffusion termB(Un+1

j ) is Holder continuous with coefficient one-half. This is not possible for the present case due to the double nonlinearitybyA andB.

Finally, in view of the regularity estimates, it is not difficult to showthat the discrete diffusion termA(D+B(Un+1

j )) must converge to a limit

functionA(x, t).However, several argumentsare required to identifyA(x, t)asA(∂xB(u(x, t))). In [8] none of these arguments were necessary sincethe derivatives can be moved to the test function. This again reflects theadditional difficulties introduced by the nonlinear functionA.

As a by-product of our analysis, we establish the existence and regular-ity properties of solutions of the Cauchy problem (1), and in that respectcomplement the work of Yin [18] on the initial-boundary value problem.

We should emphasise that this paper and the companion papers [8,9](on strongly degenerate convection-diffusion equations) are intended as pre-liminary theoretical thrusts at the numerical approximation of non-classicalsolutions of degenerate parabolic equations, and they utilise discrete approx-imations which could be somewhat too ’crude’ for practical applications.Having said this, we are currently looking into the issue of devising higherorder difference schemes for degenerate parabolic equations. Another im-portant issue that is under investigation is the problem of deriving rigorouserror estimates for our schemes. We also mention that our interest in degen-erate parabolic equations ispartially motivated by the recent efforts madein developing mathematical models for the settling and consolidation of aflocculated suspensions in solid-liquid separation vessels (so-called thick-eners). We refer to Burger and Wendland [1] and Concha and Burger [2]for an overview of the activity centring around these sedimentation models,whose main ingredients are degenerate parabolic equations.

Before ending this introduction, we should make some comments con-cerning the structural condition (6) onA. Let us for a moment return toequation (7). Such equations have been studied more recently by Kurganov,Levy, and Rosenau [13,12] under the condition thatA is bounded. In par-ticular, they observe by numerical experiments and analysis how the solu-tions develop infinite spatial derivatives in finite time from smooth initial


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0

0.2

0.4

0.6

0.8

1

x−axis

u−ax

is

Fig. 2. The solution of Burgers equation with a bounded diffusion termA andb(s) = 1

conditions. For an example of this phenomenon, see Fig. 2 where we haveplotted the solution of the problem (7), but now withA bounded. Intu-itively it is obvious what happens. In this case the equation imposes anupper bound on the amount of the diffusive flux while the convective fluxmay be as large as desired.When the fluxes are no longer in balance, smoothupstream-downstream transit becomes impossible andasubshock is formed.The importance of (6) used in this paper, is that under this condition it ispossible to obtain an estimate|∂xB(u(x, t))| ≤ Const from the estimate|A(∂xB(u(x, t))| ≤ Const. This is obviously not true ifA is bounded.

The rest of this paper is organized as follows: In Sect. 2 we give a briefsummary of the theory of doubly nonlinear degenerate parabolic equations.We also recall some classical results needed from the Crandall and Liggetttheory [3]. In Sect. 3 we present and discuss the discrete approximations. InSect. 4 we derive a number of regularity estimates satisfied by the discreteapproximations. In Sect. 5 we exploit these estimates to prove the conver-gence (compactness) of the approximate solutions to the unique solution of(1).

2 Mathematical preliminaries

In this section we recall the known mathematical theory of double non-linear degenerate parabolic equations. To this end, letΩ be an open sub-set ofRd (d > 1). The spaceBV (Ω) of functions of bounded variationconsists of allL1

loc(Ω) functionsu(y) whose first order partial derivatives∂u∂y1, . . . , ∂u∂yd

are represented by (locally) finite Borel measures. The totalvariation |u|BV (Ω) is by definition the sum of the total masses of theseBorel measures. Moreover,BV (Ω) is a Banach space when equipped withthe norm||u||BV (Ω) = ||u||L1(Ω)+|u|BV (Ω). It is well known that the inclu-

sionBV (Ω) ⊂ Ld/(d−1)(Ω) holds ford > 1 and thatBV (Ω) ⊂ L∞(Ω)for d = 1. Furthermore,BV (Ω) is compactly imbedded into the spaceLq(Ω) for 1 ≤ q < d/(d − 1) if Ω is bounded. Finally, we will also


need the Holder spaceC1, 12 (QT ) consisting of bounded functionsz(x, t)onR× [0, T ] which satisfies

|z(y, τ)− z(x, t)| ≤ L(|y − x|+ |τ − t| 12 ), ∀ x, t, y, τ,

for some constantL > 0 (not depending onx, y, t, τ ).In what follows, we shall always assume, if not otherwise stated, that the

structural condition (6) holds. Due to possibly strong degeneracy, we seeksolutions of the Cauchy problem (1) in the following sense.

Definition 2.1 A bounded measurable functionu(x, t) is said to be aBVentropy weak solution of (1) provided the following two requirements hold:

(i) u ∈ BV (QT ) andB(u) ∈ C1, 12 (QT ).

(ii) For all test functionsφ ≥ 0 with support inR× [0, T 〉 and anyc ∈ R,

∫∫QT

(|u− c|∂tφ+ sign(u− c)(f(u)− f(c)−A(∂xB(u))

)∂xφ

)dt dx+

∫R

|u0 − c| dx ≥ 0.(10)

Definition 2.1 is similar to the one used by Yin [18] who studied theinitial-boundary value problem. The uniqueness proof for the Cauchy prob-lem follows from the analysis of the corresponding initial boundary valueproblem. In fact, the Cauchy problem is simpler since theBV solutionsof the boundary value problem must satisfy some extra conditions on theboundary. The following characterization of the set of discontinuity points(jumps) ofu can be proved along the lines of Yin [18].

Theorem 2.2 [18] Let Γu be the set of jumps ofu; ν = (νx, νt) the unitnormal toΓu; u−(x0, t0) and u+(x0, t0) the approximate limits ofu at(x0, t0) ∈ Γu from the sides of the half-planes(t− t0)νt + (x− x0)νx < 0and(t−t0)νt+(x−x0)νx > 0 respectively;ul(x, t) andur(x, t) denote theleft and right approximate limits ofu(·, t) respectively. Letint(α, β) denotethe closed interval bounded byα andβ. Furthermore, define

sign+(s) =

1, if s > 0,0, if s ≤ 0,

and sign−(s) =

0, if s ≥ 0,−1, if s < 0.

Finally, letH1 be the one-dimensional Hausdorffmeasure. ThenH1 - almosteverywhere onΓu

b(u) = 0, ∀u ∈ int(u−, u+) and νx = 0,(11)


(u+ − u−)νt + (f(u+)− f(u−))νx

−(A(∂xB(u))r −A(∂xB(u))l

)|νx| = 0,(12)

|u+ − c|νt + sign(u+ − c)[f(u+)− f(c)−(A(∂xB(u))r sign+ νx

−A(∂xB(u))l sign− νx)]νx

≤ |u− − c|νt + sign(u− − c)[f(u−)

−f(c)− (A(∂xB(u))l sign+ νx

−A(∂xB(u))r sign− νx)]νx.

(13)

By explicitly making use of the above jump conditions, the followingstability result, from which uniqueness follows, can be obtained along thelines of Yin [18].

Theorem 2.3 [18] Letu1 andu2 areBV entropy weak solutions of (1) withinitial functionsu0,1 andu0,2, respectively, contained inL1(R). Then foranyt > 0,∫

R

|u1(x, t)− u2(x, t)| dx ≤∫R

|u0,1(x)− u0,2(x)| dx.

Finally, we note that the jump conditions in Theorem 2.2 can be statedmore instructively as follows.

Corollary 2.4 Assume thatb(u) = 0 for u ∈ [u∗, u∗] for someu∗, u∗ ∈R. Let u be a piecewise smooth solution of (1) and letΓu be a smoothdiscontinuity curve ofu. A jump between two valuesul andur of the solutionu, which we refer to as a shock, can occur only forul, ur ∈ [u∗, u∗]. Thisshock must satisfy the following two conditions:

(i) The shock speeds is given by

s =

[f(ur)−A(∂xB(u))r

]− [f(ul)−A(∂xB(u))l

]ur − ul .(14)

(ii) For all c ∈ int(ul, ur), the following entropy condition holds

[f(ur)−A(∂xB(u))r

]− f(c)ur − c ≤ s ≤

[f(ul)−A(∂xB(u))l

]− f(c)ul − c .

(15)


Proof. For u ∈ L∞(QT ) ∩ BV (QT ) it can be shown that the followingrelation betweenu+,u−,ur andul holdsH1 almost everywhere onΓ ∗

u =(x, t) ∈ Γu : νx = 0:

u+(x, t) = ur(x, t) sign+ νx − ul(x, t) sign− νx ,u−(x, t) = ul(x, t) sign+ νx − ur(x, t) sign+ νx .(16)

These identities are non-trivial and we refer to [17] for a proof. Since|νx| =(sign+ νx + sign− νx)νx, (12) can be written as

(u+ − u−)νt +(f(u+)− f(u−)

)νx −

(wru sign+ νx − wl

u sign− νx)νx

+(wlu sign+ νx − wr

u sign− νx)νx = 0,(17)

wherewru = A(∂xB(u))r andwl

u = A(∂xB(u))l. For c ∈ int(u−, u+) =int(ul, ur) (by (16)) we have the relationsign(u+−c) = − sign(u−−c). Inlight of this and (17), wenowuse (13) andperform the following calculation:

sign(u+ − c)[(u+ − c)νt +(f(u+)− f(c)) νx(

wru sign+ νx − wul sign− νx

)νx

]≤ − sign(u+ − c)[(u− − c)νt + (f(u−)

−f(c))νx −(wlu sign+ νx − wr

u sign− νx)νx

]= − sign(u+ − c)[(u− − u+)νt + (f(u−)

−f(u+))νx +(wru sign+ νx − wl

u sign− νx)νx

−(wlu sign+ νx − wr

u sign− νx)νx

]− sign(u+ − c)[(u+ − c)νt + (f(u+)

−f(c))νx −(wru sign+ νx − wl

u sign− νx)νx

+(wlu sign+ νx − wr

u sign− νx)νx

−(wlu sign+ νx − wr

u sign− νx)νx

]= − sign(u+ − c)[(u+ − c)νt + (f(u+)

−f(c))νx −(wru sign+ νx − wl

u sign− νx)νx

].

Hence

sign(u+ − c)[(u+ − c)νt +(f(u+)− f(c)) νx

−(wru sign+ νx − wl

u sign− νx)νx

] ≤ 0.


Dividing by |u+ − c| yields

νt +(f(u+)− f(c))− (

wru sign+ νx − wl

u sign− νx)

u+ − c νx ≤ 0

or [f(u+)− (wr

u sign+ νx − wlu sign− νx)

]− f(c)u+ − c νx ≤ −νt.(18)

Similarly, we can show that

−νt ≤[f(u−)− (wl

u sign+ νx − wru sign− νx)

]− f(c)u− − c νx.(19)

Combining (18) and (19) we have forc ∈ int(ul, ur) that[f(u+)− (wr

u sign+ νx − wlu sign− νx)

]− f(c)u+ − c νx

≤ −νt ≤[f(u−)− (wl

u sign+ νx − wru sign− νx)

]− f(c)u− − c νx.

(20)

Invoking (16) it is not difficult to see that (12) can be written on the form

(ur − ul)νt +(f(ur)− f(ul)

)νx −

(wru − wl

u

)νx = 0.

Let s = − νtνx, then (14) follows. Finally, in view of (16), we see that (20) is

equivalent to (15). Hence the proof is completed.The jump conditions (14) and (15) represent generalizations respectively

of the Rankine-Hugoniot condition andOleinik’s entropy condition for con-servation laws. The geometric interpretation of (14) and (15) is as follows:

Corollary 2.5 Let(ul, ur)bea jumpwhich satisfies the jumpcondition (14).Then the entropy condition (15) holds if and only if

(i) in caseur < ul:The graph ofy = f(u) over

[ur, ul

]lies below or equals the chord

connecting the point(ur, f(ur)−A(∂xB(u))r) to (ul, f(ul)−A(∂xB(u))l);

(ii) in caseul < ur:The graph ofy = f(u) over

[ul, ur

]lies above or equals the chord

connecting the point(ul, f(ul)−A(∂xB(u))l) to (ur, f(ur)−A(∂xB(u))r).


We close this section by briefly recalling a few key results from theCrandall and Liggett theory, since it will be used later in the discussionof properties of the difference schemes. IfX is a Banach space, a dualitymappingJ : X → X∗ has the properties that for allx ∈ X, ‖J(x)‖X∗ =‖x‖X andJ(x)(x) = ‖x‖2X . A possiblymulti-valuedoperatorA, definedonsome subsetD(A) ofX, is said to be accretive if for every pair of elements(x,A(x)) and(y,A(y)) in the graph ofA, and for every duality mappingJ onX,

J(x− y) (A(x)−A(y)) ≥ 0.

If, in addition, for all positiveλ,I+λA is a surjection, thenA ism-accretive.Let(Ω, dµ) be ameasure space. Then recall that, since the dual ofL1(Ω)

is L∞(Ω), any duality mappingJ in L1(Ω) is of the formJ(u)(v) =∫Ω J(u)(x)v(x) dµ, where

J(u)(x) = ‖u‖L1(Ω)

1, if u(x) > 0,−1, if u(x) < 0,α(x), if u(x) = 0,

whereα(x) is any measurable function with|α(x)| ≤ 1 for almost everyx ∈ Ω. Later we shall rely heavily on the following well-known results (see[3,5,15]) about m-accretive operators onX = L1(Ω):

Theorem 2.6 Let (Ω, dµ) be a measure space. Suppose that the nonlinearand possibly multi-valued operatorA : L1(Ω) → L1(Ω) is m-accretive.Then for anyλ > 0 and anyu ∈ L1(Ω) the equation

T (u) + λA(T (u)) = u

has a unique solutionT (u). Furthermore, suppose thatA satisfies∫Ω A(u)

dµ = 0 and commutes with translations. ThenT : L1(Ω) → L1(Ω)possesses the following properties:

(1)∫Ω T (u) dµ =

∫Ω u dµ,

(2) ‖T (u)− T (v)‖L1(Ω) ≤ ‖u− v‖L1(Ω),

(3) ‖T (u)‖BV (Ω) ≤ ‖u‖BV (Ω),

(4) u ≤ v a.e. implies thatT (u) ≤ T (v) a.e.,

(5) ‖T (u)‖L∞(Ω) ≤ ‖u‖L∞(Ω).


3 The discrete approximations

Selecting mesh sizes∆x > 0,∆t > 0, the value of our difference approxi-mation at(xj , tn) = (j∆x, n∆t) will be denoted byUnj . Capital lettersU ,V etc. will denote functions on the lattice∆ = j∆x : j ∈ Z. The valueof U at(xj , tn) will be writtenUnj . ThusU

n is a function on∆ with valuesUnj . The following notations will be used on occasions:

λ =∆t

∆x, µ =

∆t

∆x2 ,

∆−Uj = Uj − Uj−1, D− =1∆x∆−,

∆+Uj = Uj+1 − Uj , D+ =1∆x∆+.

For later use, we introduce the following two constants

a∞ = supminu0≤ξ≤maxu0

|a(ξ)| <∞, b∞ = supminu0≤ξ≤maxu0

|b(ξ)| <∞.

To approximate (1) we consider three-point implicit difference schemes ofthe form

Un+1j −Un

j

∆t +D−(h(Un+1

j , Un+1j+1 )−A(D+B(Un+1

j )))

= 0,

(j, n) ∈ Z× 0, . . . , N − 1,

U0j =

1∆x

∫ (j+1)∆x

j∆xu0(x) dx, j ∈ Z.

(21)

Weassume that the numerical fluxh(u, v) satisfies the consistency condition

h(u, u) = f(u),(22)

and the monotonicity conditions

∂uh(u, v) ≥ 0, ∂vh(u, v) ≤ 0 .(23)

We will see later that (23) ensures that the solution operator of (21)is monotone. An example of a scheme which satisfies these conditions isprovided by a variant of the Engquist-Osher scheme where the numericalflux h(u, v) is given by

h(u, v) = f+(u) + f−(v) ,

where

f+(u) = f(0)+∫ u

0max

(f ′(s), 0

)ds, f−(u) =

∫ u

0min

(f ′(s), 0

)ds.


For another example, assume thatβ, γ are strictly increasing and non-decreasing respectively, and consider the numerical fluxh given by

h(u, v) =f(u) + f(v)

2− ∆x

2λβ(γ(v)− γ(u)

∆x

).

This corresponds to a central (space) differencing of

∂tu+ ∂xf(u) = ∂xA(∂xB(u)) + ε∂xβ(∂xγ(u)),

whereε is chosen as∆x2

2∆t . Notice that this scheme ismonotone provided that±λf ′(u)+β′(v)γ′(u) ≥ 0 for all u, v. When the problem is nondegenerate(a, b > 0) we can use the numerical flux given by

h(u, v) =f(u) + f(v)

2,

which corresponds to a central (space) differencing of (1). In this case themonotonicity assumptions read:

1∆xa(r4)b(r3) + ∂uh(u, v)|(r1,r2) ≥ 0,

1∆xa(r4)b(r3)− ∂vh(u, v)|(r1,r2) ≥ 0,

(24)

wherer1, r2, r3, r4 are arbitrary numbers inR. It follows that (24) is satisfiedprovided∆x|f ′| ≤ 2a∞b∞.

4 Regularity estimates

In this sectionweestablish the regularityestimateswhichwill beneeded laterfor showing convergence of the discrete approximations. In the followingwe treat the case whereu0 has compact support andf,A,B are locallyC1. Then at the end of Sect. 5 we briefly discuss the general case whereu0 is not necessarily compactly supported andf,A,B are locally Lipschitzcontinuous. If not otherwise stated, we will always assume, without loss ofgenerality, thatf(0) = 0. The function space that containsu0 will be takenas

B(f,A,B) =z ∈ L1(R) ∩BV (R) : |f(z)−A(∂xB(z)|BV (R) <∞

.

(25)

Convergence inL1loc of a subsequence of the familyu∆ of approximate

solutions generated from (21) is obtained by establishing three estimates forUnj :


(a) a uniformL∞ bound,(b) a uniform (total) variation bound,(c) L1 Lipschitz continuity in the time variable,

and two estimates for the discrete total flux termh(Un+1j , Un+1

j+1 )− A(D+

B(Un+1j )):

(d) a uniformL∞ bound,(e) a uniform (total) variation bound.

The estimates (d) and (e) play a main role in that we utilize estimate (e)to obtain estimate (c), while (d) is used to obtain the Holder continuity intime and space of the discrete diffusion termB(Unj ).

For later use, recall that theL∞(Z)norm, theL1(Z)normand theBV (Z)semi-norm of a lattice functionU is defined respectively as∥∥U∥∥

L∞(Z) = supj∈Z

∣∣Uj∣∣,∥∥U∥∥

L1(Z) =∑j∈Z

∣∣Uj∣∣,∣∣U ∣∣

BV (Z) =∑j∈Z

∣∣Uj − Uj−1∣∣ ≡ ∆x∥∥D−U

∥∥L1(Z).

If not specified,i, j will always denote integers fromZ;m,n, l integers from0, . . . , N; x, y, c real numbers fromR andt, τ real numbers from[0, T ].Throughout this paperC will denote a positive constant, not necessarilythe same at different occurences, which is independent of the discretizationparameters involved.

The following lemma deals with the question of existence, uniquenessand properties of the solution of the (nonlinear) system (21).

Lemma 4.1 If (23) is satisfied, then for anyU there is auniqueU∗ satisfyingthe following equation

U∗j − Uj∆t

+D−(h(U∗

j , U∗j+1)−A(D+B(U∗

j )))

= 0, j ∈ Z.(26)

Furthermore, the solutionU∗ of (26) possesses the following properties:

(a) Uj ≤ Vj ∀j ∈ Z implies thatU∗j ≤ V ∗

j ∀j ∈ Z,(b)

∥∥U∗∥∥L∞(Z) ≤

∥∥U∥∥L∞(Z),

(c)∥∥U∗ − V ∗∥∥

L1(Z) ≤∥∥U − V ∥∥

L1(Z),

(d)∣∣U∗∣∣

BV (Z) ≤∣∣U ∣∣

BV (Z).


Proof. As an aid in the analysis we shall view the equation (21) in termsof an m-accretive operator and an associated contraction solution operator,i.e., we shall use the Crandall and Liggett theory [3]. A similar treatmentof implicit difference schemes for conservation laws has been given earlierby Lucier [15] and for strongly degenerate convection-diffusion equationsin [9].

For a fixedn, let us now rewrite the difference equation (21) as (supress-ing the∆x dependence)

Un+1j +∆tA(Un+1; j) = Unj , j ∈ Z,(27)

where the operatorA : L1(Z)→ L1(Z) is defined by

A(U ; j) = D−(h(Uj , Uj+1)−A (D+B(Uj))

).

We first show that the operatorA is accretive. To this end, it is sufficient toestablish that for anyU, V with U − V ∈ L1(Z),∑

j∈Z

sign(Uj − Vj)(A(U ; j)−A(V ; j)

) ≥ 0.

As a first step to achieve this goal, we perform the following calculation∑j∈Z


)=

∑j∈Z

sign(Uj − Vj)(A(U ; j)−A(V ; j)− c(Uj − Vj))

+c∑j∈Z

∣∣Uj − Vj∣∣,≥ −

∑j∈Z

∣∣cWj −(A(U ; j)−A(V ; j)

)∣∣ + c∑j∈Z

∣∣Wj

∣∣,(28)

whereWj denotesUj − Vj andc = c(∆x) > 0 is a number chosen so that

c ≥ 1∆x

(max(u,v)

∂uh(u, v)−min(u,v)

∂vh(u, v))

+2∆x2a∞b∞.(29)

Next, we observe that

A(U ; j)−A(V ; j)

=1∆x

(hu(αj , Uj+1)Wj + hv(Vj , αj+1)Wj+1

−hu(αj−1, Uj)Wj−1 − hv(Vj−1, αj)Wj

)− 1∆x2

(a(γj)

(b(βj+1)Wj+1 − b(βj)Wj

)−a(γj−1)

(b(βj)Wj − b(βj−1)Wj−1

)),

(30)


whereαj , αj , βj ∈ int(Uj , Vj) andγj ∈ int(D+B(Uj), D+B(Vj)). In-serting this into inequality (28) yields the desired result:∑

j∈Z


)

≥ c∑j∈Z

∣∣Wj

∣∣−∑j∈Z

∣∣∣[ 1∆xhu(αj−1, Uj)

+1∆x2a(γj−1)b(βj−1)

]Wj−1

+[c− 1

∆x(hu(αj , Uj+1)− hv(Vj−1, αj))

− 1∆x2 b(βj)(a(γj) + a(γj−1)

])Wj

+[ 1∆x2a(γj)b(βj+1)− 1

∆xhv(Vj , αj+1)

]Wj+1

∣∣∣≥ c

∑j∈Z

∣∣Wj

∣∣−∑j∈Z

[ 1∆xhu(αj−1, Uj)

+1∆x2a(γj−1)b(βj−1)

]∣∣Wj−1∣∣

−∑j∈Z

[c− 1

∆x(hu(αj , Uj+1)− hv(Vj−1, αj))

− 1∆x2 b(βj) (a(γj) + a(γj−1))

]∣∣Wj

∣∣−

∑j∈Z

[ 1∆x2a(γj)b(βj+1)

− 1∆xhv(Vj , αj+1)

]∣∣Wj+1∣∣ ≡ 0,

due to the monotonicity conditions (23) and the choice ofc given by (29).From (30) we observe that the operatorA is Lipschitz continuous,∥∥A(U)−A(V )

∥∥L1(Z) ≤

( 2∆xL(h) +

4∆x2L(A,B)

)∥∥U − V ∥∥L1(Z),

whereL(h) = max |hu| + max |hv| andL(A,B) = a∞b∞. This impliesthatA is not only accretive but also m-accretive, see [6]. We can now in-voke Theorem 2.6 to conclude the existence of a unique monotone solutionoperatorS associated with (21) such that

U∗j = S(U ; j),

which proves the first part of the lemma. Since∑

j∈ZA(U ; j) = 0 andA

commutes with translations, the second part of the lemma also follows fromTheorem 2.6.


As a direct consequence of Lemma 4.1 the following lemma is estab-lished.

Lemma 4.2 We have∥∥Un+1∥∥L∞(Z) ≤

∥∥U0∥∥L∞(Z) ,

∣∣Un+1∣∣BV (Z) ≤

∣∣U0∣∣BV (Z).(31)

Next we establish a regularity property for the total fluxh(Un+1j , Un+1

j+1 )

− A(D+B(Un+1

j )). As mentioned, this regularity property is of funda-

mental importance when proving convergence of the scheme (21). Let usfirst indicate how this regularity estimate can be derived at the continuouslevel in the case of classical solutions. To this end, consider the uniformlyparabolic equation

∂tu+ ∂xf(u) = ∂xA (∂xB(u)) , A′, B′ > 0,(32)

and recall that this equation has a unique classical solutionu. By differenti-ating (32) with respect tot and subsequently integrating with respect tox,we find that the quantity

v(x, t) =∫ x

−∞∂tu(ξ, t)dξ

satisfies the linear, variable coefficients, uniformly parabolic equation

∂tv + f ′(u)∂xv = a(∂xB(u))∂x(b(u)∂xv).(33)

From the maximum principle for this equation it follows that

‖v(·, t)‖L∞(R) ≤ ‖v0‖L∞(R) .

From (32) and the definition ofv we see thatv = −f(u) + A(∂xB(u)),which implies that

‖A (∂xB(u(·, t)))‖L∞(R) ≤ C,whereC = 2 max |f |+‖A (∂xB(u(·, 0)))‖L∞(R). This ismerely formalismsince thesolutionof (1) ingeneral onlyexists inaweaksense.However, thesecalculations clearlymotivate thenext lemma,whosecontent is auniformL∞bound as well as aBV bound for the discrete total fluxh(Un+1

j , Un+1j+1 )−

A(D+B(Un+1j )).

Lemma 4.3 We have∥∥h(Un+1j , Un+1

j+1 )−A(D+B(Un+1j ))

∥∥L∞(Z)

≤ ∥∥h(U0j , U

0j+1)−A(D+B(U0

j ))∥∥L∞(Z),

(34)


∣∣h(Un+1j , Un+1

j+1 )−A(D+B(Un+1j ))

∣∣BV (Z)

≤ ∣∣h(U0j , U

0j+1)−A(D+B(U0

j ))∣∣BV (Z).

(35)

Proof. To prove these regularity properties for the approximate solutions,we introduce two auxiliary sequencesWn

j andV nj given by

Wn+1j =

Un+1j − Unj∆t

, V n+1j = ∆x

j∑k=−∞

Wn+1k .

Using the finite difference scheme (21) we observe that

Wn+1k ∆x = −∆−

(h(Un+1

k , Un+1k+1 )−A(D+B(Un+1

k ))).(36)

Summing over allk = −∞, . . . , j and having in mind thatUnk = 0 forsufficiently largek andh(0, 0) = f(0) = 0, we get

V n+1j = −(

h(Un+1j , Un+1

j+1 )−A(D+B(Un+1j ))

).(37)

From this relation it is clear that it is sufficient to establishL∞ andBVestimates forV n. As a first step toward that end, we derive an equationfor the auxiliary sequenceV nj . For this purpose, consider the differenceequation given by (21) and subtract the corresponding equation at timetn.Then we obtain

Wn+1k ∆x =Wn

k ∆x−∆−(h(Un+1

k , Un+1k+1 )− h(Unk , Unk+1)

)+∆−

(A(D+B(Un+1

k ))−A(D+B(Unk ))).

Again we sum over allk = −∞, . . . , j, yielding

V n+1j = V nj −

(h(Un+1

j , Un+1j+1 )− h(Unj , Unj+1)

)+

(A(D+B(Un+1

j ))−A(D+B(Unj ))).

(38)

We now rewrite the two last terms. To this end, we first we observe that

∆xUn+1j − Unj∆t

= ∆xWnj = ∆x

j∑k=−∞

Wnk

−∆xj−1∑

k=−∞Wnk = V nj − V nj−1.(39)

Then we have


h(Un+1j , Un+1

j+1

)− h (

Unj , Unj+1

)=

(h

(Un+1j , Un+1

j+1

)− h(Un+1

j , Unj+1))

+(h(Un+1

j , Unj+1)− h(Unj , Unj+1))

= ∂vh(Un+1j , α

n+ 12

j+1 )(Un+1j+1 − Unj+1

)+∂uh(α

n+ 12

j , Unj+1)(Un+1j − Unj

)= λ

(∂vh(Un+1

j , αn+ 1

2j+1 )

[V n+1j+1 − V n+1

j

]+ ∂uh

(αn+ 1

2j , Unj+1

) [V n+1j − V n+1

j−1

])= ∆t

(hn+1v,j D+V

n+1j + hn+1

u,j D+Vn+1j−1

),

(40)

where

hn+1v,j = ∂vh(Un+1

j , αn+ 1

2j+1 ), hn+1

u,j = ∂uh(αn+ 1

2j , Unj+1),

αn+ 1

2j , α

n+ 12

j ∈ int(Unj , Un+1j ).

(41)

Similary, we rewrite the last term of (38).

A(D+B(Un+1j ))−A(D+B(Unj ))

= a(γn+ 1

2j )D+

(B(Un+1

j )−B(Unj ))

= a(γn+ 1

2j )D+

(b(β

n+ 12

j )[Un+1j − Unj ]

)(42)

= ∆t · a(γn+ 12

j )D+(b(β

n+ 12

j )D−V n+1j

)= ∆t · an+1

j D+(bn+1j D−V n+1

j

),

where

an+1j = a(γ

n+ 12

j ), γn+ 1

2j ∈ int(D+B(Unj ), D+B(Un+1

j )),

bn+1j = b(β

n+ 12

j ), βn+ 1

2j ∈ int(Unj , Un+1

j ).(43)

From (38), (40) and (42) we obtain the following linear difference equa-tion for V nj :

V n+1j − V nj∆t

+(hn+1u,j D+V

n+1j−1 + hn+1

v,j D+Vn+1j

)= an+1

j D+

(bn+1j D−V n+1

j

).

(44)


This equation can be written as

cn+1j V n+1

j−1 + dn+1j V n+1

j + en+1j V n+1

j+1 = V nj ,(45)

where

cn+1j = −

[λhn+1

u,j + µan+1j bn+1

j

],

dn+1j =

[1 + λ

(hn+1u,j − hn+1

v,j

)+ µan+1

j

(bn+1j+1 + bn+1

j

)],

en+1j = −

[µan+1

j bn+1j+1 − λhn+1

v,j

].

By the monotonicity assumption (23), we have

cn+1j + dn+1

j + en+1j = 1, cn+1

j , en+1j ≤ 0, dn+1

j ≥ 0.(46)

Thanks to (46), the linear system (45) is strictly diagonal dominant. Con-sequently, there exists a unique solutionV n+1. Furthermore, this solutionsatisfies a maximum principle:

cn+1j |V n+1

j−1 |+ dn+1j |V n+1

j |+ en+1j |V n+1

j+1 |≤ |V nj | =⇒

∥∥V n+1∥∥L∞(Z) ≤ ‖V n‖L∞(Z) .

In view of (37) we can now conclude that (34) is satisfied. Next we provethat the solution of (44) has bounded variation onZ. Introduce the quantityZnj = V nj − V nj−1 and observe that

Zn+1j − Znj∆t

+D−(hn+1u,j Z

n+1j + hn+1

v,j Zn+1j+1

)= D−

(an+1j D+

(bn+1j Zn+1

j

)).

Similarly to (45), we can write this equation as

cn+1j Zn+1

j−1 + dn+1j Zn+1

j + en+1j Zn+1

j+1 = Znj ,(47)

where

cn+1j = −

[λhn+1

u,j−1 + µan+1j−1 b

n+1j−1

],

dn+1j =

[1 + λ

(hn+1u,j − hn+1

v,j−1

)+ µbn+1

j

(an+1j−1 + an+1

j

)],

en+1j = −

[µan+1

j bn+1j+1 − λhn+1

v,j

].

Again, due to the monotonicity assumption, we see that

cn+1j+1 + dn+1

j + en+1j−1 = 1, cn+1

j , en+1j ≤ 0, dn+1

j ≥ 0.(48)


Therefore, from (47), it follows that

cn+1j |Zn+1

j−1 |+ dn+1j |Zn+1

j |+ en+1j |Zn+1

j+1 |≤ |Znj | =⇒

∑j∈Z

|Zn+1j | ≤

∑j∈Z

|Znj |,

which immediately implies (35). This concludes the proof of the lemma.

An immediate consequence of (35), and (21) is that the discrete approx-imations (21) areL1 Lipschitz continuous in time, and thus contained inBV (QT ).

Lemma 4.4 We have∥∥Um − Un∥∥L1(Z)

≤ ∣∣h(U0j , U

0j+1)−A

(D+B(U0

j )) ∣∣BV (Z)

∆t

∆x|m− n|.

(49)

Proof. Suppose thatm > n. Using (21), we readily calculate that

∑j∈Z

∣∣Umj − Unj ∣∣ ≤ ∆tm−1∑l=n

∑j∈Z

∣∣∣D−(h(U l+1

j , U l+1j+1)−A

(D+B(U l+1

j )))∣∣∣

≤ ∣∣h(U0j , U

0j+1)−A

(D+B(U0

j ))∣∣BV (Z)

∆t

∆x(m− n),

where theBV estimate (35) has been used. This concludes the proof of thelemma. Lemma 4.5 If (23) is satisfied, then the following cell entropy inequalityholds ∣∣Un+1

j − c∣∣− ∣∣Uj − c∣∣ +∆tD−(h

(Un+1j ∨ c, Un+1

j+1 ∨ c)

− h(Un+1j ∧ c, Un+1

j+1 ∧ c))

(50)

− ∆tD−(sign(Un+1

j − c)A(D+B(Un+1

j )))≤ 0.

Proof. The arguments are as follows. First, observe that

h(Un+1j ∨ c, Un+1

j+1 ∨ c)− h(Un+1j ∧ c, Un+1

j+1 ∧ c)≤ sign(Un+1

j − c)(h(Un+1j , Un+1

j+1 )− h(c, c)),(51)

−(h(Un+1

j−1 ∨ c, Un+1j ∨ c)− h(Un+1

j−1 ∧ c, Un+1j ∧ c)

)≤ sign(Un+1

j − c)(h(c, c)− h(Un+1j−1 , U

n+1j )

).

(52)


These two inequalities follow from themonotonicity ofh. Due to the similar-ity,weonly show the first inequality (51). Theproof is baseduponexaminingseveral cases depending on whetherUn+1

j+1 is larger or smaller thanUn+1j .

If c /∈ int(Un+1j , Un+1

j+1 ), then the left hand side of (51) is equal to the righthand side.

Next, assume thatc ∈ int(Un+1j , Un+1

j+1 ) andUn+1j ≤ Un+1

j+1 . Then

h(Un+1j ∨ c, Un+1

j+1 ∨ c)− h(Un+1j ∧ c, Un+1

j+1 ∧ c)= h(c, Un+1

j+1 )− h(Un+1j , c)

= h(c, c)− h(Un+1j , Un+1

j+1 ) +[h(c, Un+1

j+1 )− h(c, c)]+

[h(Un+1

j , Un+1j+1 )− h(Un+1

j , c)]

= sign(Un+1j − c)(h(Un+1

j , Un+1j+1 )− h(c, c)) +Qn+1

j ,

where

Qn+1j =

[h(c, Un+1

j+1 )− h(c, c)] +[h(Un+1

j , Un+1j+1 )− h(Un+1

j , c)]

= hn+1v,j (Un+1

j+1 − c) + hn+1v,j (Un+1

j+1 − c),

and

hn+1v,j = ∂vh(c, αn+1

j+1 ), hn+1v,j = ∂vh(Un+1

j , αn+1j+1 ),

αn+1j , αn+1

j ∈ int(c, Un+1j ).

Due to the monotonicity assumption (23) and the fact thatc ≤ Un+1j+1 , we

conclude thatQn+1j ≤ 0 and the desired inequality is obtained. Similarly,

we can show that this inequality holds whenUn+1j ≥ Un+1

j+1 .For the discrete diffusion term we have the following inequality.

sign(Un+1j−1 − c)A

(D+B(Un+1

j−1 ))

≤ sign(Un+1j − c)A

(D+B(Un+1

j−1 )).

(53)

In order to see this, consider the relation

sign(Un+1j−1 − c

)A

(D+B

(Un+1j−1

))= sign

(Un+1j − c

)A

(D+B

(Un+1j−1

))+Rn+1

j ,

where


Rn+1j =

(sign

(Un+1j−1 − c

)− sign

(Un+1j − c

))A

(D+B

(Un+1j−1

))=

(sign

(Un+1j−1 − c

)− sign

(Un+1j − c

))×

(Un+1j − Un+1

j−1

)an+1j− 1

2b

(βn+1j− 1

2

)and

an+1j− 1

2=

∫ 1

0a(ξD+B(Un+1

j−1 ))dξ ≥ 0, b(βn+1j− 1

2) ≥ 0,

βn+1j− 1

2∈ int(Un+1

j−1 , Un+1j ).

Now we observe thatRn+1j = 0 unlessc is betweenUn+1

j−1 andUn+1j . If c is

in this interval, it is easy to check thatRn+1j is nonpositive. Invoking (51),

(52), (53) and (21) we obtain

|Un+1j − c|+∆tD−

(h(Un+1

j ∨ c, Un+1j+1 ∨ c)− h(Un+1

j ∧ c, Un+1j+1 ∧ c)

)−∆tD−

(sign(Un+1

j − c)A(D+B(Un+1j ))

)≤ |Un+1

j − c|+ λ sign(Un+1j − c)(h(Un+1

j , Un+1j+1 )− h(c, c))

+λ sign(Un+1j − c)

(h(c, c)− h(Un+1

j−1 , Un+1j )

)−λ sign(Un+1

j − c)A(D+B(Un+1

j ))

+λ sign(Un+1j − c)A

(D+B(Un+1

j−1 ))

= sign(Un+1j − c)(Un+1

j − c+∆tD−(h(Un+1

j , Un+1j+1 )

−A(D+B(Un+1j ))

))= sign(Un+1

j − c)(Unj − c)≤ |Unj − c|.

Hence the proof is complete.Remark. The estimates of Lemmas 4.2, 4.3, and 4.4 have been obtainedwithout making use of the structural assumption (6) onA. From these esti-mates it is not difficult to show that there is a subsequence of the approximatesolutions which converges to a limit functionu. However, we do not haveestimates on the diffusion term which ensure thatA(D+B(Unj )) convergesin some appropriate sense to the diffusion termA(∂xB(u)).

In the following we will discuss continuity properties of the discretediffusion termB(Unj ). From (34) and the assumption thatu0 is containedin B(f,A,B) it follows that∥∥A(D+B(Unj ))

∥∥L∞(Z) ≤ C,(54)


whereC is a constant independent of∆. An immediate consequence of (54)and the assumption (6) is the following lemma.

Lemma 4.6 We have

∥∥D+B(Unj )∥∥L∞(Z) ≤ C.(55)

Remark.Theassumption (6) cannot be removed in establishing convergenceto theBV entropy weak solution in the sense of Definition 2.1. In otherwords, the problem may not haveBV entropy weak solutions if (6) is notassumed. Recall the example withA unbounded from Sect. 1 (Fig. 2). HereB(s) = s, but clearlyD+B(Unj ) = D+U

nj is not uniformly bounded

because of the appearance of a discontinuity. Hence this problem cannothave a solution in the class given by Definition 2.1.

Knowing that the discrete diffusion termB(Unj ) is Lipschitz contin-uous in the space variable, the question arises how to obtain informationabout the regularity in the time variable. One strategy would be to continueworking with the linear equation forv = f(u)−A (∂xB(u)) and try to de-rive a result concerning the continuity ofv with respect to the time variablefrom the known modulus of continuity in space. This technique, introducedby Kruzkov [11], was used for the simple degenerate case [8], i.e. whenA(s) = s. To illustrate some of the added difficulties introduced by the dou-ble nonlinearity, let us see why this technique does not work in the generalcase. To this end, letφ(x) be a test function onR and multiply (33) byφand integrate overR. Then we have

∫R

φ(x)∂tv dx = −∫R

f ′(x, t)∂xvφ(x) dx

+∫R

a(x, t)∂x (b(x, t)∂xv)φ(x) dx,(56)

wheref ′(x, t),a(x, t),b(x, t)denotef ′(u(x, t)), a(∂xB(u(x, t))),b(u(x, t))respectively. The first term on the right hand side of (56) is bounded sincev is of bounded variation. For the case whenA(s) = s, that isa(x, t) = 1,the second term is bounded since one derivative can be moved over to thetest functionφ. However, in the general casea(x, t) = a(∂xB(u(x, t))) isnot constant and therefore it is not possible to bound this term. Hence wehave to choose another approach to this problem. We will employ a discreteversion of a technique used by Yin [18] which combines the scheme (21)and the estimate (34). For this purpose, defineu∆ as the interpolant of the


discrete valuesUnj given by

u∆(x, t) =

Unj +Un

j+1−Unj

∆x (x− xj) +Un+1

j+1 −Unj+1

∆t (t− tn),(x, t) ∈ TL

j,n,

Unj +Un+1

j+1 −Un+1j

∆x (x− xj) +Un+1

j −Unj

∆t (t− tn),(x, t) ∈ TU

j,n.

(57)

Here TLj,n denotes the triangle with vertices(xj , tn), (xj+1,

tn), and(xj+1, tn+1) while TU

j,n denotes the triangle with vertices(xj , tn),

(xj , tn+1), and(xj+1, tn+1). Let

Rnj = [xj , xj+1]×[tn, tn+1]

and note thatRnj = TLj,n ∪ TU

j,n. Later we will use the notationRx,t inorder to denote a rectangleRnj , not necessarily unique, which contains thepoint (x, t). In particular, we note thatu∆ is continuous everywhere anddifferentiable almost everywhere inQT .

Lemma 4.7 We have∣∣B(Umi )−B(Unj )∣∣ ≤ C

(|xi − xj |+ √|tm − tn|+∆x).(58)

Proof. We have that∣∣B(Umi )−B(Unj )∣∣ ≤ ∣∣B(Umi )−B(Uni )

∣∣+

∣∣B(Uni )−B(Unj )∣∣ =: I1 + I2.

ClearlyI2 = O (|xi − xj |) by using (55). Nowwe focus on how to estimateI1. Consider the interval[xi, xi + α], whereα will be specified later. Thenfor somex∗ ∈ [xi, xi + α] (that also will be specified later) we have

I1 = |B(u∆(xi, tm))−B(u∆(xi, tn))|≤ |B(u∆(xi, tm))−B(u∆(x∗, tm))|

+|B(u∆(x∗, tm))−B(u∆(x∗, tn))|+|B(u∆(x∗, tn))−B(u∆(xi, tn))|

≤ 2C (|xi − x∗|+∆x) + |B(u∆(x∗, tm))−B(u∆(x∗, tn))|≤ 2C (α+∆x) + |B(u∆(x∗, tm))−B(u∆(x∗, tn))|,

(59)

where the estimates of the first and third terms of the second line followfrom the monotonicity ofB(s). Next we describe how|B(u∆(x∗, tn)) −


B(u∆(x∗, tm))| can be estimated. For this purpose, we introduce the quan-tity

Q(x) =

x∫−∞

(u∆(ξ, tm)− u∆(ξ, tn)

)dξ.

Sinceu∆ is continuous,Q(x) is differentiable everywhere. Hence, there isa numberx∗ in [xi, xi + α] such that

Q′(x∗)α = Q(xi + α)−Q(xi) =∫ xi+α

xi

(u∆(ξ, tm)− u∆(ξ, tn)

)dξ.

We then have the following relation

|B(u∆(x∗, tn))−B(u∆(x∗, tm))| ≤ b∞|u∆(x∗, tn)− u∆(x∗, tm)|= b∞|Q′(x∗)| = b∞

α·∣∣∣ ∫ xi+α

xi

(u∆(ξ, tm)− u∆(ξ, tn)) dξ∣∣∣.(60)

Sinceu∆ is differentiable in time almost everywhere onQT we have

∫ xi+α

xi

(u∆(ξ, tm)− u∆(ξ, tn)

)dξ

=∫ xi+α

xi

∫ tm

tn∂tu∆ dt dx =

∫ xj

xi

∫ tm

tn∂tu∆ dt dx(61)

+∫ xi+α

xj

∫ tm

tn∂tu∆ dt dx =: J1 + J2,

wherej is the integer such that0 < (xi + α)− xj < ∆x. Now, in view of(59), (60) and (61) we want to show that|J1|, |J2| ≤ α2 and then chooseαequal to

√|m− n|∆t. We have

J1 =∫ xj

xi

∫ tm

tn∂tu∆ dt dx

=j−1∑k=i

m−1∑l=n

(∫∫TU

k,l

∂tu∆ dt dx+∫∫TL

k,l

∂tu∆ dt dx)

=12∆x∆t

j−1∑k=i

m−1∑l=n

(U l+1k − U lk∆t

+U l+1k+1 − U lk+1

∆t

).


Using the finite difference scheme (21) and estimate (34) of Lemma 4.3, weobtain the following estimate

|J1| = 12∆x∆t

∣∣∣j−1∑k=i

m−1∑l=n

(U l+1k − U lk∆t

+U l+1k+1 − U lk+1

∆t

)∣∣∣≤ 4 · 1

2∆t|m− n|∥∥h(U0

j , U0j+1)−A

(D+B(U0

j ))∥∥

L∞(Z)

= 2C0|m− n|∆t = 2C0α2,

where

C0 =∥∥h(U0

j , U0j+1)−A

(D+B(U0

j ))∥∥

L∞(Z)(62)

and we have setα equal to√|m− n|∆t. Repeating the arguments forJ2

we also deduce that|J2| ≤ 2C0α2. From (60) and (61) we now conclude

that

|B(u∆(x∗, tn))−B(u∆(x∗, tm))| ≤ 4C0α

and hence, from (59), we obtain

I1 = |B(u∆(xi, tm))−B(u∆(xi, tn))|= O

(α+∆x

)= O

(√|m− n|∆t+∆x).Now the proof of (58) is completed.

5 Convergence results

Wewill now employ the regularity properties established in Sect. 5 to provethat the approximate solutions generated by (21) converge to the solution of(1) in the sense of Definition 2.1. We start by showing that a subsequence ofthe family of approximate solutions converges to a functionu and that thislimit inherits the properties of the approximate solutions (see Lemma 5.2).Finally, using the cell entropy inequality of Lemma 4.5 and the propertiesof the interpolant, we show that this limit satisfies the entropy inequalityof Definition 2.1. The arguments needed to prove this turn out to be ratherinvolved due to the double nonlinearity of the problem. In particular, wewillsee that the convergence to the desired solution requires certain propertiesof the linear interpolants.

Recall thatu∆ denotes the interpolant of the discrete valuesUnj givenby (57). Similarly we definew∆ as the interpolant of the discrete values


B(Unj ) given by

w∆(x, t) =

B(Unj ) +B(Unj+1)−B(Unj )

∆x(x− xj)

+B(Un+1

j+1 )−B(Unj+1)∆t

(t− tn),(x, t) ∈ TL

j,n,

B(Unj ) +B(Un+1

j+1 )−B(Un+1j )

∆x(x− xj)

+B(Un+1

j )−B(Unj )∆t

(t− tn),(x, t) ∈ TU

j,n.

(63)

For later use, observe that the following important relations hold

∂xu∆ = D+Unj , ∂xw∆ = D+B(Unj )(64)

on the parallelogramPnj with vertices(xj , tn−1), (xj , tn), (xj+1, tn), and

(xj+1, tn+1), i.e.,Pnj = TU

j,n−1 ∪ TLj,n. Similary,

∂tu∆ =Un+1j − Unj∆t

(65)

on the parallelogramQnj with vertices(xj−1, tn), (xj , tn), (xj , tn+1), and

(xj+1, tn+1), i.e.,Qnj = TL

j−1,n∪TUj,n. Note also that for(x, t) ∈ Rnj neither

w∆ norB(u∆) will introduce new minima or maxima, that is

min(B(Unj ), B(Unj+1), B(Un+1

j ), B(Un+1j+1 )

)≤ w∆, B(u∆)

≤ max(B(Unj ), B(Unj+1), B(Un+1

j ), B(Un+1j+1 )

).

(66)

This follows from the definition ofu∆, w∆ and the fact thatB(s) is mono-tone. The next technical lemma deals with the interpolation error associatedwith the linear interpolant (63) of Holder continuous functions.

Lemma 5.1 Assume thatG(x, t) ∈ C1, 12 (QT ) and letΠ∆G(x, t) denotethe interpolant given by

Π∆G(x, t) =

G(xj , tn) +G(xj+1,tn)−G(xj ,t

n)∆x (x− xj)

+G(xj+1,tn+1)−G(xj+1,t

n)∆t (t− tn),

(x, t) ∈ TLj,n,

G(xj , tn) +G(xj+1,tn+1)−G(xj ,t

n+1)∆x (x− xj)

+G(xj ,tn+1)−G(xj ,t

n)∆t (t− tn),

(x, t) ∈ TUj,n.


Then the following error estimate holds

‖Π∆G−G‖L∞(QT ) ≤ C(∆x+

√∆t

).

Proof. To see this, let(x, t) be an arbitrary point inQT . Then (x, t) iscontained in some rectangleRnj and we have

|Π∆G(x, t)−G(x, t)| ≤ |Π∆G(x, t)−G(xj , tn)|+|G(xj , tn)−G(x, t)|.

(67)

For the first term on the right hand side of (67) we have

|Π∆G(x, t)−G(xj , tn)| ≤

|G(xj+1, tn)−G(xj , tn)|

+|G(xj+1, tn+1)−G(xj+1, t

n)|,(x, t) ∈ TL

j,n,

|G(xj+1, tn+1)−G(xj , tn+1)|

+|G(xj , tn+1)−G(xj , tn)|,

(x, t) ∈ TUj,n.

Therefore, sinceG(x, t) ∈ C1, 12 (QT ), it follows that the first and the secondterm on the right hand side of (67) is of order∆x+

√∆t.

Now we show that the following compactness and convergence resultshold.

Lemma 5.2 There exists a functionu ∈ L∞(QT )∩BV (QT ), withB(u) ∈C1, 12 (QT ), such that

(a) u∆(x, t)→ u(x, t), inL1loc(QT ) and pointwise

a.e. in QT .

(b) w∆(x, t)→ B(u(x, t)), uniformly on compact

sets in QT .

(c) ∂xw∆∗∂xB(u), in L∞

loc(QT ).

(d) A (∂xw∆) ∗A (∂xB(u)) , in L∞

loc(QT ).

(68)

Proof. The functionsu∆(x, t) andw∆(x, t) satisfy the following estimates:

||u∆||L∞(QT ) ≤ C, |u∆|BV (QT ) ≤ C,(69)


and

|w∆(y, s)− w∆(x, t)|≤ C

(|x− y|+ √|t− s|+∆x+

√∆t

), ∀x, y, s, t.

(70)

The first estimate of (69) follows immediately from the definition of thelinear interpolantu∆ and Lemma 4.2. The second estimate of (69) is aconsequence of the following two estimates:∫∫

QT

|∂xu∆| dt dx =∑j,n

∫∫Pn

j

|∂xu∆| dt dx

= ∆x∆t∑j,n

|D+Unj | ≤ T |u0|BV .

Herewehave used (64) and Lemma4.2. Similarly, by using (65) and Lemma4.4 we obtain the estimate∫∫

QT

|∂tu∆| dt dx =∑j,n

∫∫Qn

j

|∂tu∆| dt dx

= ∆x∆t∑j,n

|Un+1j − Unj |∆t

≤ C0 · T,

whereC0 = |h(U0j , U

0j+1) − A(D+B(U0

j ))|BV (Z). The estimate (70) re-quires argument. Let(x, t) and(y, s) be some arbitrary given points andchoose two rectanglesRx,t andRy,s such that(x, t) ∈ Rx,t and(y, s) ∈ Ry,s(they may coinside). Moreover, let(xi, tm) and(xj , tn) denote vertices ofRx,t andRy,s respectively, such that

|xj − xi|+√|tn − tm| ≤ |x− y|+

√|t− s|.

Then we have

|w∆(y, s)− w∆(x, t)| ≤ |w∆(y, s)− w∆(xi, tm)|+ |w∆(xi, tm)

− w∆(xj , tn)|+ |w∆(xj , tn)− w∆(x, t)|

=: E1 + E2 + E3.

Clearly, by (58),

E2 = |B(Umi )−B(Unj )| ≤ C(|xi − xj |+ √

|m− n|∆t+∆x)≤ C

(|x− y|+ √|t− s|+∆x).


Now estimate (70) follows since we have, in view of (66), that

E1, E3 ≤ C(∆x+

√∆t

).

By virtue of estimates (69),u∆ is bounded inW 1,1(K) ⊂ BV (K) foreach compact setK. Using thatBV (K) is compactly imbedded inL1(K),it is not difficult to show, passing if necessary to a subsequence, thatu∆converges inL1

loc(QT ) and pointwise almost everywhere inQT to a functionu,

u ∈ L∞(QT ) ∩BV (QT ).

Next we discuss convergence properties of the sequencew∆. By estimate(70) we can repeat the proof of the Ascoli-Arzela theorem to conclude thatthere is a subsequence ofw∆ and a limitw,

w ∈ C1, 12 (QT ),

such that

w∆ → w uniformly on compact sets and pointwise inQT .

By the continuity ofw and the pointwise convergence, we conclude thatw = B(u). To see this, let(x, t) be an arbitrary point such thatu∆(x, t)→u(x, t), i.e.B(u∆(x, t))→ B(u(x, t)). We have

|B(u(x, t))− w(x, t)| ≤ |B(u(x, t))−B(u∆(x, t))|

+ |B(u∆(x, t))− w∆(x, t)|+ |w∆(x, t)− w(x, t)|.Sincew∆(x, t) → w(x, t), we only have to check that|B(u∆(x, t)) −w∆(x, t)|must tend to zero. For this purpose, assume that(x, t) is containedin a rectangleRx,t. Then, in view of (66) we have

|B(u∆(x, t))− w∆(x, t)| ≤ |B(u∆(xj , tn))− w∆(xi, tm)|

= |B(Unj )−B(Umi )| ≤ C(∆x+

√∆t

),

where(xj , tn) and(xi, tm) are appropriate chosen vertices of the rectangleRx,t. Hencew = B(u) almost everywhere inQT . By the continuity ofw,this must hold for all points inQT .

Now we continue by showing the convergence result (c) of (68). From(55) and (64) it follows that

‖∂xw∆‖L∞(QT ) ≤ C.Hence there is a limit functionW such that‖W‖L∞(QT ) ≤ C and, passingif necessary to a subsequence

∂xw∆∗W, in L∞

loc(QT ).


Since∫∫QT

w∆∂xφdt dx→∫∫QT

B(u)∂xφdt dx, φ ∈ C∞0 (QT ),

it is obvious thatW = ∂xB(u) and (c) follows. Finally we show why (d) issatisfied. Due to the fact that

‖A(∂xw∆)‖L∞(QT ) ≤ C,(see (54)) we know there is a functionA(x, t) in L∞(QT ) such that, againpassing if necessary to a subsequence,

A(∂xw∆) ∗A, in L∞(QT ) .

By using a discrete version of the arguments of Yin [18], we now show thatA = A(∂xB(u)). Let Π∆B(u) be the interpolant of the discrete valuesB(u(xj , tn)) defined as in Lemma 5.1. For the moment, assume thatK is a compact subset ofQT of the formK = ∪j,nPnj where(j, n) ∈J1, . . . , J2 × N1, . . . , N2. We then have∫∫

KA(∂xw∆) (∂xw∆ − ∂xB(u)) dt dx

=∫∫KA(∂xw∆) (∂xw∆ − ∂xΠ∆B(u)) dt dx

+∫∫KA(∂xw∆) (∂xΠ∆B(u)− ∂xB(u)) dt dx

=: E1 + E2.(71)

First, we estimateE1 as follows (recall (64)).

E1 =∫∫KA(∂xw∆)

(∂xw∆ − ∂xΠ∆B(u)

)dt dx

=∑j,n

∫∫Pn

j

A(∂xw∆)(∂xw∆ − ∂xΠ∆B(u)

)dt dx

= ∆x∆t∑j,n

A(D+B(Unj ))[D+B(Unj )−D+B(u(xj , tn))

]= −∆x∆t

∑j,n

D−A(D+B(Unj ))[B(Unj )−B(u(xj , tn))

]

+∆x∆t∑n

(A(D+B(UnJ2

))[B(UnJ2

)−B(u(xJ2 , tn))

]−A(D+B(UnJ1

))[B(UnJ1

)−B(u(xJ1 , tn))

] )


= −∆x∆t∑j,n

(Unj − Un−1j

∆t+D−h(Unj , U

nj+1)

)

× [w∆(xj , tn)−B(u(xj , tn))

]

+∆x∆t∑n

(A(D+B(UnJ2

))[B(UnJ2

)−B(u(xJ2 , tn))

]−A(D+B(UnJ1

))[B(UnJ1

)−B(u(xJ1 , tn))

]),

where we have used the difference scheme (21) for the last equality. Hence,by Lemmas 4.2 and 4.4 and (54),

|E1| ≤ (C0T + T (max |∂uh|+ max |∂vh|)|u0|BV)

×‖w∆ −B(u)‖L∞(K) + C∆x.(72)

In order to estimateE2 letωδ(x) denote a standardmollifier in thex variablewith support in[−δ, δ]. Let Aδ(·, t) = ωδ(·) ∗ A(∂xw∆(·, t)). ForE2 wethen have

E2 =∫∫KA(∂xw∆) (∂xΠ∆B(u)− ∂xB(u)) dt dx

=∫∫KAδ(x, t) (∂xΠ∆B(u)− ∂xB(u)) dt dx

+∫∫K

(A(∂xw∆(x, t))−Aδ(x, t)

)(∂xΠ∆B(u)− ∂xB(u)) dt dx

= −∫∫K∂xA

δ(x, t) (Π∆B(u)−B(u)) dt dx

+∫∫K

(A(∂xw∆(x, t))−Aδ(x, t)

)(∂xΠ∆B(u)− ∂xB(u)) dt dx

=: E2,1 + E2,2.

Clearly, in view of Lemma 5.1,

|E2,1| ≤∫ T

0|Aδ(·, t)|BV (R) dt · ‖Π∆B(u)−B(u)‖L∞(K)

= O(∆x+√∆t),

(73)

due to the Holder continuity ofB(u) and the fact that

|Aδ(·, t)|BV (R) ≤ |A(∂xw∆(·, t))|BV (R)

= |A(D+B(Unj ))|BV (Z) ≤ C, (for some appropriaten),


which is true because of (35). Moreover, we have

|E2,2| ≤ ‖∂xΠ∆B(u)− ∂xB(u)‖L∞(K)

×∥∥∥Aδ(x, t)−A(∂xw(x, t))

∥∥∥L1(K)

= O(δ),(74)

since‖∂xB(u)‖L∞(QT ) ≤ C. From (71), (72), (68)b, (73) and (74) it followsthat

lim∆→0

∫∫KA(∂xw∆)

(∂xw∆ − ∂xB(u)

)dt dx = 0.(75)

Note that for a general compact setK we can splitK into two setsKP and∆K such that

K = KP ∪∆K, KP = ∪j,nPnj , meas(∆K) = O(∆x+∆t).

Hence ∫∫KA(∂xw∆)

(∂xw∆ − ∂xB(u)

)dt dx

=∫∫KP

A(∂xw∆)(∂xw∆ − ∂xB(u)

)dt dx

+∫∫∆K

A(∂xw∆)(∂xw∆ − ∂xB(u)

)dt dx.

In light of the analysis above, the first term tends to zero. Because theintegrand of the last integral is uniformly bounded, it follows that this termis of order∆x+∆t and thus tends to zero. Hence (75) holds for all compactK ⊂ QT . On the other hand, sinceA(∂xB(u)) is in L∞(QT ) we have by(68)c

lim∆→0

∫∫KA (∂xB(u)) (∂xw∆ − ∂xB(u)) dt dx = 0.(76)

From (75) and (76) it follows that

lim∆→0

∫∫Ka∆

(∂xw∆ − ∂xB(u)

)2dt dx

(77)

= lim∆→0

∫∫K

(A(∂xw∆)−A(∂xB(u))

)(∂xw∆ − ∂xB(u)

)dt dx = 0,


where

a∆ = a∆(x, t) =∫ 1

0a (ξ∂xw∆ + (1− ξ)∂xB(u)) dξ

=A(∂xw∆)−A(∂xB(u))∂xw∆ − ∂xB(u)

.

(78)

Using (78) and Holder’s inequality we now deduce

∣∣∣∫∫QT

(A (∂xw∆)−A(∂xB(u))

)φ dt dx

∣∣∣≤ ‖φ‖L∞(QT )

∫∫suppφ

√a∆

∣∣∣√a∆(∂xw∆ − ∂xB(u)

)∣∣∣ dt dx≤ ‖φ‖L∞(QT )

(∫∫suppφ

a∆ dt dx) 1

2

×(∫∫suppφ

a∆(∂xw∆ − ∂xB(u)

)2dt dx

) 12

.

Sincea∆ ≤ a∞ <∞ and by using (77), it now follows that

lim∆→0

∫∫QT

(A(∂xw∆)−A(∂xB(u)))φ dt dx = 0, φ ∈ C∞

0 (QT ).

This concludes the proof of (d) and thus the lemma.The next two technical lemmas will be used in the sequel.

Lemma 5.3 LetΩ ⊂ R2 andgj(x) → g(x) a.e. inΩ. Then there exists a

setF , which is at most countable, such that for anyc ∈ R\F

sign(gj(x)− c)→ sign(g(x)− c), a.e. inΩ.

The proof is elementary and is omitted.

Lemma 5.4 Let u∆ be a piecewise constant interpolant of the discrete datapointsUnj defined such thatu∆

∣∣Pn

j= Unj . Then, passing if necessary to

a subsequence,u∆ → u pointwise a.e. inQT , whereu denotes the limitfunction obtained in Lemma 5.2.


Proof. Clearly we have

∫∫QT

|u∆ − u∆| dt dx =∑j,n

∫∫Pn

j

|u∆ − u∆| dt dx

=∑j,n

∫∫TU

j,n−1

|u∆ − u∆| dt dx+∑j,n

∫∫TL

j,n


=: S1 + S2.

ForS1 we have

S1 =∑j,n

∫∫TU

j,n−1


=∑j,n

∫∫TU

j,n−1

∣∣∣(Unj+1 − Unj )(x− xj∆x

)

+ (Unj − Un−1j )

( t− tn−1

∆t− 1

)∣∣∣ dt dx≤ 1

2∆x∆t

∑j,n

(|Unj+1 − Unj |+ |Unj − Un−1

j |)

≤ T2

(|u0|BV∆x+ C0∆t),

whereC0 is given by (62). Similary, we haveS2 ≤ 12T

(|u0|BV∆x+C0∆t).

Hence

∫∫QT

|u∆ − u∆| dt dx ≤ T(|u0|BV∆x+ C0∆t

),

from which the lemma follows. We continue by showing that the limitu satisfies the integral inequality

(10).

Lemma 5.5 Letφ be a nonnegative test function with compact support onR× [0, T 〉 andc ∈ R. Then the limit functionu(x, t) of Lemma 5.2 satisfiesthe integral inequality (10).

Proof. Letφ be a suitable test function and putφnj = φ(j∆x, n∆t). Multi-plying the cell entropy inequality (50) byφnj∆x, summing over allj andnand applying summation by parts, we get


∆x∆t

N−1∑n=0

∑j∈Z

(|Un+1j − c|

[φn+1j − φnj∆t

]+

(h(Un+1

j ∨ c, Un+1j+1 ∨ c)

− h(Un+1j ∧ c, Un+1

j+1 ∧ c))D+φ

nj

− sign(Un+1j − c)A(D+B(Un+1

j ))D+φnj

)+∆x

∑j

|U0j − c|φ0

j ≥ 0.(79)

For the first term we have

∆x∆tN−1∑n=0

∑j∈Z

|Un+1j − c|

[φn+1j − φnj∆t

]

=∫∫QT

|u∆ − c|∂tφdt dx+O(∆x+∆t).

Using Lemma 4.2 and the fact thath is consistent withf , we can obviouslywrite

∆x∑j∈Z

(h(Un+1

j ∨ c, Un+1j+1 ∨ c)− h(Un+1

j ∧ c, Un+1j+1 ∧ c)

)D+φ

nj

= ∆x∑j∈Z

sign(Un+1j − c)(f(Un+1

j )− f(c))D+φnj +O(∆x).

Hence we have for the second term of (79)

∆x∆tN−1∑n=0

∑j∈Z

(h(Un+1

j ∨ c, Un+1j+1 ∨ c)− h(Un+1

j ∧ c, Un+1j+1 ∧ c)

)D+φ

nj

=∫∫QT

sign(u∆ − c) (f(u∆)− f(c)) ∂xφdt dx+O(∆x+∆t),

For the discrete diffusion term of (79) we now have

∆x∆t

N−1∑n=0

∑j∈Z

sign(Un+1j − c)A(D+B(Un+1

j ))D+φnj

=N−1∑n=0

∑j∈Z

∫∫Pn+1

j

sign(u∆ − c)A(∂xw∆)∂xφdt dx+O(∆x+∆t)

=∫∫QT

sign(u∆ − c)A(∂xw∆)∂xφdt dx+O(∆x+∆t).


Hence, we can replace (79) by∫∫QT

(|u∆ − c|∂tφ+ sign(u∆ − c) (f(u∆)− f(c)) ∂xφ

− sign(u∆ − c)A (∂xw∆) ∂xφ)dt dx

+∫

R

|u0 − c|φ(x, 0) dx ≥ −C(∆x+∆t

).

Using (68) and Lemma 5.3 and 5.4 we conclude thatu satisfies (10) foralmost allc ∈ R. To complete the proof, note thatA(∂xB(u)) = 0 a.e. inEc = (x, t) ∈ QT : u(x, t) = c for any constantc. Therefore, by usingan approximate procedure the result holds for allc.

This completes our discussionwhenu0 has compact support andf,A,Bare locallyC1. Foru0 ∈ B(f,A,B) not necessarily compactly supportedandf,A,B merely locally Lipschitz continuous, we approximateu0 by acompactly supported functionup0 andf, a, b by a smoother functionf

p, ap,bp, compute the difference approximation of the resulting problem and thenlet p→∞ and∆t,∆x→ 0.

We are now ready to state our main result:

Theorem 5.6 Supposeu0 ∈ B(f,A,B) and the fluxesf,A,B are locallyLipschitz continuous. Assume also thatA satisfies the structural condition

A(−∞) = −∞, A(+∞) = +∞.Then the entire sequenceu∆ defined by (21) and (57) converges inL1

loc(QT ) and pointwise a.e. inQT to the uniqueBV entropy weak solution ofthe problem

∂tu+ ∂xf(u) = ∂xA(b(u)∂xu), (x, t) ∈ QT = R× 〈0, T 〉 ,u(x, 0) = u0(x), x ∈ R,

where

A(s) =∫ s

0a(ξ) dξ, a(s) ≥ 0, b(s) ≥ 0.

Remark. From the proof of Lemma 5.2 it is clear that the following re-sults hold without assuming (6): There is a functionu(x, t) ∈ L∞(QT ) ∩BV (QT ) and a functionA(x, t) ∈ L1(QT ) such that

u∆(x, t)→ u(x, t), in L1loc(QT ) and pointwise a.e. inQT ,

A (∂xw∆) ∗A(x, t), in L∞

loc(QT ).


Furthermore, in view of Lemma 5.5 it follows that the following integralinequality is satisfied

∫∫QT

(|u− c|∂tφ+ sign(u− c)(f(u)− f(c)−A)

∂xφ)dt dx

+∫R

|u0 − c| dx ≥ 0.(80)

We letC(0, T ;L1(R)) denote the usual Bochner space consisting of allcontinuous functionsu : [0, T ]→ L1(R) forwhich thenorm‖u‖C(0,T ;L1(R))= supt∈[0,T ] ‖u(t)‖L1(R) is finite. A closer inspection of the arguments lead-ing to Theorem 5.6 will reveal thatU∆(t) converges inC(0, T ;L1(R))to the uniqueBV entropy weak solutionu(t), with u(0) = u0, of the ini-tial value problem (1). A reexamination of the proofs leading to Theorem5.6 also shows that we have proved the following result on existence andproperties of solutions of (1):

Corollary 5.7 Letf andA,B be locally Lipschitz continuous. Then for anyinitial function u0 ∈ B(f,A,B) there exists aBV entropy weak solutionu ∈ C(0, T ;L1(R)) of the initial value problem (1). Denoting this solutionbyStu0, we have the following properties:

(1) t → Stu0 is Lipschitz continuous intoL1(R) and ‖Stu0‖BV (R) ≤∥∥u0‖BV (R),

(2)∥∥Stu0 − Stv0

∥∥L1(R) ≤ ‖u0 − v0‖L1R),

(3) u0 ≤ v0 impliesStu0 ≤ Stv0,

(4) m ≤ u0 ≤M impliesm ≤ Stu0 ≤M .

Acknowledgement.We thank Nils Henrik Risebro for his reading and criticism of themanuscript.

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Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

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