DISCRETE APPROXIMATIONS OF BV SOLUTIONS TODOUBLY NONLINEAR DEGENERATE PARABOLIC EQUATIONSSteinar Evje, Kenneth Hvistendahl KarlsenDepartment of Mathematics, University of BergenJohs. Brunsgt. 12, N{5008 Bergen, NorwayE-mail: fsteinar.evje, kenneth.karlseng@mi.uib.noAbstract. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly non-linear degenerate, parabolic problems of the form(P) @tu + @xf(u) = @xA (b(u)@xu) ; u(x; 0) = u0(x); A(s) = Z s0 a(�)d�; a(s) � 0; b(s) � 0;under the very general structural condition A(�1) = �1. To mention only a few examples: the heat equation,the porous medium equation, the two-phase ow equation, hyperbolic conservation laws and equations arising fromthe theory of non-Newtonian uids are all special cases of (P). Since the di�usion terms a(s) and b(s) are allowed todegenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions arenot uniquely determined by their data. For these reasons we work within the framework of weak solutions that areof bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of theCauchy problem (P) in this class of so-calledBV entropy weak solutions follows from a work of Yin [18]. The discreteapproximations are shown to converge to the unique BV entropy weak solution of (P).Contents1. Introduction2. Mathematical Preliminaries3. The Discrete Approximations4. Regularity Estimates5. Convergence Resultsx1. Introduction.In this paper we present and analyse certain �nite di�erence schemes for a class of scalar, doubly nonlineardegenerate, parabolic equations in one spatial dimension. Nonlinear parabolic evolution equations arise in avariety of applications, ranging frommodels of turbulence, via tra�c ow, �nanical modelling and ow in porousmedia, to models for various sedimentation processes. The problem we study here is of the form(1) ( @tu+ @xf(u) = @xA(b(u)@xu); (x; t) 2 QT = R� h0; T i ;u(x; 0) = u0(x); x 2 R;where A(s) = Z s0 a(�) d�; a(s) � 0; b(s) � 0:1991 Mathematics Subject Classi�cation. 65M12, 35K65, 35L65.Key words and phrases. doubly nonlinear degenerate parabolic equation,BV solutions, entropy condition, implicit �nite di�er-ence schemes, convergence.The research of the second author has been supported by VISTA, a research cooperation between the Norwegian Academy ofScience and Letters and Den norske stats oljeselskap a.s. (Statoil). Typeset by AMS-TEX1

2 S. EVJE, K. H. KARLSENWe assume that a(s), b(s), f(s) and u0(x) are appropiately smooth functions. The functions a(s) and b(s) areallowed to have in�nite number of degenerate intervals in R. Di�culties arise because of this double degeneracyas well as the double nonlinearity represented by the nonlinear functions a and b. By de�ning B asB(u) = Z u0 b(�) d�;we may write (1) as(2) @tu+ @xf(u) = @xA (@xB(u)) :Examples of such equations include the heat equation, the porous medium equation and, more generally,convection-di�usion equations of the form(3) @tu+ @xf(u) = @2xB(u):Included are also hyperbolic conservation laws(4) @tu+ @xf(u) = 0;as well as certain equations arising from the theory of non-Newtonian uids,(5) @tu = @x�@xum j@xumjn�1�; n � 1; m � 1;which corresponds to the case A(v) = vjvjn�1 and B(u) = um.Kalashnikov [10] has established the existence of continuous solutions of the Cauchy problem for (2) whenf = 0 under some smoothness and boundedness conditions on the initial data u0 and some structural conditionson a(s) and b(s). In particular, these conditions imply that a(s) and b(s) may have degeneracy at and only atthe origin s = 0. We also refer to some recent work by Lu [14] for results concerning the regularity of solutionswhen the equations are degenerate at points at which u and @xu vanish.The more interesting cases are those in which a(s) and b(s) may have in�nite or uncountable points ofdegeneracy. A striking feature of such nonlinear strongly degenerate parabolic equations is that the solutionwill generally develop discontinuities in �nite time, even with smooth initial data. This feature can re ect thephysical phenomenon of breaking of waves and the development of shock waves. Consequently, due to the loss ofregularity, one needs to work with weak solutions. However, for the class of equations under consideration, weaksolutions are in general not uniquely determined by their data. Therefore an additional condition, the so-calledentropy condition (see (b) below), is needed to single out the physically relevant weak solution. Hence attentionfocuses on �nding a physically reasonable framework which incorporates discontinuous solutions and at thesame time guarantees uniqueness. The concept of a (weak) solution, which we adopt to the Cauchy problem(1) in this paper, is that of BV entropy weak solutions as formulated by Yin [18] for the initial-boundary valueproblem. We shall say that u(x; t) is a BV entropy weak solution (see x2 for precise statements) if(a) u(x; t) is in BV (QT ) and B(u) is uniformly H�older continuous on QT .(b) @tju� cj+ @x�sign(u� c)�f(u) � f(c) � A (@xB(u))�� � 0 (weakly).Letting k ! �1 in (b), we see that (1) holds in the usual weak sense. Yin [18] has shown well-posedness ofthe initial-boundary value problem assuming only the (very general) structural condition(6) A(+1) = +1 and A(�1) = �1:The well-posedness for the Cauchy problem (1) in the class of functions satisfying the conditions (a) and (b)follows by a similar analysis, see x2. Here we should also note, as pointed out by Yin, that the assumption (6)on A is needed only for the existence result. Under the additional assumption that B(s) is strictly increasing,which permits b(s) to become zero in some set of measure zero, BV solutions are continuous. Esteban andVazquez [7] studied the occurrence of �nite velocity of propagation for the solutions of the special case (5). Inparticular, they showed that the interface of the equation is nondecreasing and Lipschitz continuous. Wang andYin [16] have investigated the properties of the interface of the solution for the general problem (2) when f = 0.Since the di�usion term @xA(b(u)@xu) can degenerate both in a and b, di�erent kinds of interactions betweennonlinear convection and nonlinear di�usion will take place. The (lack of) smoothness of the solution is a result

DOUBLY NONLINEAR DEGENERATE EQUATIONS 3of the (lack of) balance between the convective and di�usive uxes. In the following we will brie y discuss somesimple numerical examples whose purpose is to demonstrate the e�ect of the degeneracy in a and b on intervals.As long as the di�usion term is nondegenerate (a; b > 0), there is a perfect balance between the convective anddi�usive uxes and the equation then has a classical smooth solution. The degeneracy which may occur in aor/and b, implies that there is a loss of regularity in the solution.First we discuss the e�ect of degeneracy in a. For this purpose, let us consider the equation (1) when b(u) = 1.We then have equations of the form(7) @tu+ @xf(u) = @xA(@xu):Let f be the Burgers ux f(s) = s2 and A the continuous function(8) A(s) =8>>>>><>>>>>: s + 4; for s 2 h�1;�5i;�1; for s 2 [�5;�1i;s; for s 2 [�1; 1];+1; for s 2 h+1;+5];s � 4; for s 2 h+5;+1i:Hence A satis�es (6) and degenerates on the two intervals [�5;�1] and [1; 5]. In Figure 1 (left) we have plottedthe solution at time T = 0:15. The degeneracy introduces only a 'mild' loss of regularity in the solution dueto the fact that the convective and di�usive uxes will be in balance for large gradients. Hence no jumps willarise in the solution.Next we consider the general problem (1). When b(s) is zero on an interval, jumps will in general occur inthe solution. Let f be the Burgers ux function as before, while A is the function given by (8) and b is thecontinuous function given by b(s) = 8><>: 0; for s 2 [0; 0:5i;2:5s� 1:25; for s 2 [0:5; 0:6i;0:25; for s 2 [0:6; 1]:In Figure 1 we have plotted the solution of this degenerate parabolic problem (right) at time T = 0:15. It isinstructive to compare this solution with the solution of the corresponding conservation law (4), see Figure 1(middle). In particular, we observe that the solution of the degenerate problem has a 'new' increasing jump,despite the fact that f is convex. In that sense the solution of the degenerate problem has a more complexstructure than the solution 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 is determined solely by f (Rankine-Hugoniotcondition), the speeds of the jumps in the solution of the degenerate problem are determined by both f andA(@xB(u)), see x2 for precise statements of the jump conditions.−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

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axisFigure 1. Left: The solution of Burgers' equation with a di�usion term A which degenerates on intervals. Middle: The solutionof the inviscid Burgers' equation. Right: The solution of Burgers' equation with di�usion terms A and B which degenerate onintervals.Convergence of explicit monotone �nite di�erence schemes has been established recently [8] for the special caseA(s) = s. To the best of our knowledge, for the general case no convergence results for discrete approximationsare available. The analysis presented here follows along the lines of [8]. Both works were inspired by the theorydeveloped by Crandall and Majda [4]. However, due to the double degeneracy as well as the double nonlinearity,the analysis in the present case is signi�cantly more involved than in [4,8].

4 S. EVJE, K. H. KARLSENIn what follows, we restrict our attention to implicit three-point di�erence schemes. That is, we considerdiscretizations of (2) of the following form (see x3 for more details )(9) Un+1j � Unj�t +D� �h(Un+1j ; Un+1j+1 )� A(D+B(Un+1j ))� = 0;where h denotes a monotone and consistent numerical ux function, �x;�t are the mesh sizes and D+; D�are the usual forward and backward di�erence operators respectively. Extension to general p-point monotoneschemes follows easily. Note here that we choose to discretize the di�usion term written on its conservativeform. In [8] we observed that this seems to be essential in order to ensure that the scheme is consistent withthe entropy condition. In this paper we show that (9) satis�es a cell entropy inequality consistent with theentropy inequality (b). In addition we establish several regularity estimates for the approximate solutions whichare su�cient to guarantee convergence (of a subsequence) to a limit. The main di�culty here is to show thatthe discrete di�usion term possesses the regularity properties which ensure that the approximate solutions arein BV (QT ). This is obtained by deriving and carefully analysing a linear di�erence equation satis�ed by thenumerical ux of the di�erence scheme (9). In addition it turns out that due to the double nonlinearity theinterpolants must be chosen carefully when constructing the approximate solutions. As a by-product of ouranalysis, we also establish the existence and regularity properties of solutions of the Cauchy problem (1), andin that respect complement the work of Yin [18] on the intial-boundary value problem.We should emphasise that this paper and the companion papers [8,9] (on strongly degenerate convection-di�usion equations) are intended as preliminary theoretical thrusts at the numerical approximation of non-classical solutions of degenerate parabolic equations, and they utilise discrete approximations which could besomewhat 'too crude' for practical applications. Having said this, we are currently looking into the issue ofdevising higher order di�erence schemes for degenerate parabolic equations. Another important issue that isunder investigation is the problem of deriving rigorous error estimates for our schemes. We also mention thatour interest in degenerate parabolic equations is partially motivated by the recent e�orts made in developingmathematical models for the settling and consolidation of a occulated suspensions in solid-liquid separationvessels (so-called thickeners). We refer to B�urger and Wendland [1] and Concha and B�urger [2] for an overviewof the activity centring around these sedimentation models, whose main ingredients are degenerate parabolicequations.−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

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axisFigure 2. The solution of the inviscid Burgers' equation with a bounded di�usion term A and b(s) = 1.Before ending this introduction, we should make some comments concerning the structural condition (6) on A.Let us for a moment return to equation (7). Such equations have been studied more recently by Kurganov, Levy,Rosenau [13,12] under the condition that A is bounded. In particular, they observe by numerical experimentsand analysis how the solutions develop in�nite spatial derivatives in �nite time from smooth initial conditions.For an example of this phenomenon, see Figure 2 where we have plotted the solution of the problem (7), butnow with A bounded. Intuitively it is obvious what happens. In this case the equation imposes an upperbound on the amount of the di�usive ux while the convective ux may be as large as desired. When the uxesare no longer in balance, smooth upstream-downstream transit becomes impossible and a subshock is formed.The importance of (6) used in this paper, is that under this condition it is possible to obtain an estimatej@xB(u(x; t))j � Const from the estimate jA(@xB(u(x; t))j � Const. This is obviously not true if A is bounded.The rest of this paper is organised as follows: In x2 we give a brief summary of the theory of doublynonlinear degenerate parabolic equations. We also recall some classical results needed from the Crandall andLiggett theory [3]. In x3 we present and discuss the discrete approximations. In section x4 we derive a numberof regularity estimates satis�ed by the discrete approximations. In x5 we exploit these estimates to prove theconvergence (compactness) of the approximate solutions to the unique solution of (1).

DOUBLY NONLINEAR DEGENERATE EQUATIONS 5x2. Mathematical Preliminaries.In this section we recall the known mathematical theory of double nonlinear degenerate parabolic equations.To this end, let be an open subset of Rd (d > 1). The space BV () of functions of bounded variation consistsof all L1loc() functions u(y) whose �rst order partial derivatives @u@y1 ; : : : ; @u@yd are represented by (locally) �niteBorel measures. The total variation jujBV () is by de�nition the sum of the total masses of these Borelmeasures. Moreover, BV () is a Banach space when equipped with the norm jjujjBV () = jjujjL1()+ jujBV ().It is well known that the inclusion BV () � Ld=(d�1)() holds for d > 1 and that BV () � L1() for d = 1.Furthermore, BV () is compactly imbedded into the space Lq() for 1 � q < d=(d� 1). Finally, we will alsoneed the H�older space C1;12 (QT ) consisting of bounded functions z(x; t) on R� [0; T ] which satis�esjz(y; � )� z(x; t)j � L�jy � xj+ j� � tj 12 �; 8 x; t; y; �;for some constant L > 0 (not depending on x; y; t; � ).In what follows, we shall always assume, if not otherwise stated, that the structural condition (6) holds. Dueto possibly strong degeneracy, we seek solutions of the Cauchy problem (1) in the following sense.De�nition 2.1. A bounded measurable function u(x; t) is said to be a BV entropy weak solution of (1) providedthe following two requirements hold:1. u 2 BV (QT ) and B(u) 2 C1;12 (QT ).2. For all test functions � � 0 with support in R� [0; T i and any c 2 R,(10) ZZQT �ju� cj@t�+ sign(u� c)�f(u) � f(c) � A(@xB(u))�@x��dt dx+ ZRju0 � cj dx � 0:De�nition 2.1 is similar to the one used by Yin [18] who studied the initial-boundary value problem. Theuniqueness proof for the Cauchy problem follows from the analysis of the corresponding initial boundary valueproblem. In fact, the Cauchy problem is simpler since the BV solutions of the boundary value problem mustsatisfy some extra conditions on the boundary. The following characterization of the set of discontinuity points(jumps) of u can be proved along the lines of Yin [18].Theorem 2.2 [Yin]. Let �u be the set of jumps of u; � = (�x; �t) the unit normal to �u; u�(x0; t0) andu+(x0; t0) the approximate limits of u at (x0; t0) 2 �u from the sides of the half-planes (t�t0)�t+(x�x0)�x < 0and (t� t0)�t + (x� x0)�x > 0 respectively; ul(x; t) and ur(x; t) denote the left and right approximate limits ofu(�; t) respectively. Let int(�; �) denote the closed interval bounded by � and �. Furthermore, de�nesign+(s) = � 1; if s > 0;0; if s � 0; sign�(s) = � 0; if s � 0;�1; if s < 0:Finally, let H1 be the one-dimensional Hausdor� measure. Then H1 - almost everywhere on �u(11) b(u) = 0; 8u 2 int(u�; u+) and �x 6= 0;(12) (u+ � u�)�t + (f(u+)� f(u�))�x � �A(@xB(u))r � A(@xB(u))l�j�xj = 0;(13) ju+ � cj�t + sign(u+ � c)�f(u+)� f(c) � �A(@xB(u))r sign+ �x � A(@xB(u))l sign� �x���x� ju� � cj�t + sign(u� � c)�f(u�) � f(c) � �A(@xB(u))l sign+ �x � A(@xB(u))r sign� �x���x:By explicitly making use of the above jump conditions, the following stability result, from which uniquenessfollows, can be obtained along the lines of Yin [18].Theorem 2.3 [Yin]. Let u1 and u2 be BV entropy weak solutions of (1) with initial functions u0;1 and u0;2respectively. Then for any t > 0,ZRju1(x; t)� u2(x; t)j dx � ZRju0;1(x)� u0;2(x)j dx:Finally, we note that the jump conditions in Theorem 2.2 can be more instructively stated as follows.

6 S. EVJE, K. H. KARLSENCorollary 2.4. Assume that b(u) = 0 for u 2 [u�; u�] for some u�; u� 2 R. Let u be a piecewise smooth solutionof (1) and let �u be a smooth discontinuity curve of u. A jump between two values ul and ur of the solutionu, which we refer to as a shock, can occur only for ul; ur 2 [u�; u�]. This shock must satisfy the following twoconditions:1. The shock speed s is given by(14) s = �f(ur) �A(@xB(u))r�� �f(ul) �A(@xB(u))l�ur � ul :2. For all c 2 int(ul; ur), the following entropy condition holds(15) �f(ur) �A(@xB(u))r�� f(c)ur � c � s � �f(ul)�A(@xB(u))l� � f(c)ul � c :Proof. For u 2 L1(QT ) \BV (QT ) it can be shown that the following relation between u+,u�,ur and ul holdsH1 almost everywhere on ��u = f(x; t) 2 �u : �x 6= 0g(16) u+(x; t) = ur(x; t) sign+ �x � ul(x; t) sign� �xu�(x; t) = ul(x; t) sign+ �x � ur(x; t) sign+ �x:These identities are non-trivial and we refer to [17] for a proof. Since j�xj = (sign+ �x + sign� �x)�x, (12) canbe written as(17)(u+ � u�)�t + �f(u+) � f(u�)� �x � �wru sign+ �x � wlu sign� �x� �x + �wlu sign+ �x �wru sign� �x� �x = 0;where wru = A(@xB(u))r and wlu = A(@xB(u))l . For c 2 int(u�; u+) = int(ul; ur) (by (16)) we have the relationsign(u+ � c) = � sign(u�� c). In light of this and (17), we now use (13) and perform the following calculation.sign(u+ � c)�(u+ � c)�t + �f(u+)� f(c)� �x � �wru sign+ �x � wlu sign� �x� �x�� � sign(u+ � c)�(u� � c)�t + �f(u�)� f(c)� �x � �wlu sign+ �x �wru sign� �x� �x�= � sign(u+ � c)�(u� � u+)�t + �f(u�)� f(u+)� �x + �wru sign+ �x �wlu sign� �x� �x� �wlu sign+ �x �wru sign� �x� �x�� sign(u+ � c)�(u+ � c)�t + �f(u+) � f(c)� �x � �wru sign+ �x �wlu sign� �x��x+ �wlu sign+ �x � wru sign� �x� �x � �wlu sign+ �x �wru sign� �x� �x�= � sign(u+ � c)�(u+ � c)�t + �f(u+)� f(c)� �x � �wru sign+ �x �wlu sign� �x� �x�:Hence sign(u+ � c)�(u+ � c)�t + �f(u+)� f(c)� �x � �wru sign+ �x � wlu sign� �x� �x� � 0:Dividing by ju+ � cj yields�t + (f(u+) � f(c)) � �wru sign+ �x � wlu sign� �x�u+ � c �x � 0or(18) �f(u+) � (wru sign+ �x � wlu sign� �x)� � f(c)u+ � c �x � ��t:Similarly, we can show that(19) ��t � �f(u�) � (wlu sign+ �x � wru sign� �x)� � f(c)u� � c �x:

DOUBLY NONLINEAR DEGENERATE EQUATIONS 7Combining (18) and (19) we have for c 2 int(ul; ur) that(20)�f(u+)� (wru sign+ �x �wlu sign� �x)�� f(c)u+ � c �x � ��t � �f(u�) � (wlu sign+ �x �wru sign� �x)�� f(c)u� � c �x:Invoking (16) it is not di�cult to see that (12) can be written on the form(ur � ul)�t + �f(ur)� f(ul)� �x � �wru �wlu� �x = 0:Let s = � �t�x , then (14) follows. Finally, in view of (16), we see that (20) is equivalent to (15). Hence the proofis completed. �The jump conditions (14) and (15) represent a generalization of the Rankine-Hugoniot condition and Oleinik'sentropy condition for conservation laws. The geometric interpretation of (14) and (15) is as follows:Corollary 2.5. Let (ul; ur) be a jump which satis�es the jump condition (14). Then the entropy condition (15)holds if and only if(i) in case ur < ul:The graph of y = f(u) over �ur; ul� lies below or equals the chord connecting the point(ur ; f(ur)) to (ul; f(ul)� A(@xB(u))l);(ii) in case ul < ur:The graph of y = f(u) over �ul; ur� lies above or equals the chord connecting the point(ul; f(ul)) to (ur; f(ur)� A(@xB(u))r).We close this section by brie y recalling a few key results from the Crandall and Liggett theory, since itwill be used later in the discussion of properties of the di�erence schemes. If X is a Banach space, a dualitymapping J : X ! X� has the properties that for all x 2 X, kJ(x)kX� = kxkX and J(x)(x) = kxk2X . A possiblymulti-valued operator A, de�ned on some subset D(A) of X, is said to be accretive if for every pair of elements(x;A(x)) and (y;A(y)) in the graph of A, and for every duality mapping J on X,J(x� y) (A(x)� A(y)) � 0:If, in addition, for all positive �, I + �A is a surjection, then A is m-accrective.Let (; d�) be a measure space. Then recall that, since the dual of L1() is L1(), any duality mapping Jin L1() is of the form J(u)(v) = R J(u)(x)v(x) d�, whereJ(u)(x) = kukL1()8><>: 1; if u(x) > 0;�1; if u(x) < 0;�(x); if u(x) = 0;where �(x) is any measurable function with j�(x)j � 1 for almost every x 2 . Later we shall rely heavily onthe following well-known results (see [3,5,15]) about m-accretive operators on X = L1():Theorem 2.6. Let (; d�) be a measure space. Suppose that the nonlinear and possibly multi-valued operatorA : L1()! L1() is m-accretive. Then for any � > 0 and any u 2 L1() the equationT (u) + �A(T (u)) = u;has a unique solution T (u). Furthermore, suppose that A satis�es RA(u) d� = 0 and commutes with transla-tions. Then T : L1()! L1() possesses the following properties:(1) R T (u) d� = R u d�,(2) kT (u) � T (v)kL1() � ku� vkL1(),(3) kT (u)kBV () � kukBV (),(4) u � v a.e. implies that T (u) � T (v) a.e.,(5) kT (u)kL1() � kukL1().

8 S. EVJE, K. H. KARLSENx3. The Discrete Approximations.Selecting mesh sizes �x > 0, �t > 0, the value of our di�erence approximation at (xj ; tn) = (j�x; n�t) willbe denoted by Unj . Capital letters U , V etc. will denote functions on the lattice � = fj�x : j 2 Zg. The valueof U at (xj; tn) will be written Unj . Thus Un is a function on � with values Unj . The following notations willbe used on occasions: � = �t�x; � = �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:a1 = supmin u0���maxu0 ja(�)j <1; b1 = supmin u0���maxu0 jb(�)j <1:To approximate (1) we consider three-point implicit di�erence schemes of the form(21) 8>>><>>>: Un+1j � Unj�t +D��h(Un+1j ; Un+1j+1 ) �A(D+B(Un+1j ))� = 0; (j; n) 2Z� f0; : : : ; N � 1g;U0j = 1�x Z (j+1)�xj�x u0(x) dx; j 2Z:We assume that the numerical ux h(u; v) satis�es the consistency condition(22) h(u; u) = f(u)and the monotonicity conditions(23) @uh(u; v) � 0; @vh(u; v) � 0:We will see later that (23) ensures that the solution operator of (21) is monotone. An example of a schemewhich satis�es these conditions is provided by a variant of the Engquist-Osher scheme where the numerical uxh(u; v) is given by h(u; v) = f+(u) + f�(v)where f+(u) = f(0) + Z u0 max(f 0(s); 0) ds; f�(u) = Z u0 min(f 0(s); 0) ds:For another example, assume that �; are strictly increasing and nondecreasing respectively, and considerthe numerical ux h given by h(u; v) = f(u) + f(v)2 � �x2� �� (v) � (u)�x �:This corresponds to a central (space) di�erencing of@tu+ @xf(u) = @xA(@xB(u)) + "@x�(@x (u));where " is chosen as �x22�t . Notice that this scheme is monotone provided that ��f 0(u) + �0(v) 0(u) � 0 for allu; v. When the problem is nondegenerate (a; b > 0) we can use the numerical ux given byh(u; v) = f(u) + f(v)2 ;which corresponds to central (space) di�erencing of (1). In this case the monotonicity assumptions are givenby the weaker assumptions (compared to (23))(24) 1�xa(r4)b(r3) + @uh(u; v)j(r1;r2) � 0; 1�xa(r4)b(r3)� @vh(u; v)j(r1;r2) � 0;where r1; r2; r3; r4 are arbitrary numbers in R. It follows that (24) is satis�ed provided �xjf 0j � 2a1b1.

DOUBLY NONLINEAR DEGENERATE EQUATIONS 9x4. Regularity Estimates.In this section we establish the regularity estimates which will be needed later for showing convergence of thediscrete approximations. In the following we treat the case where u0 has compact support and f;A;B are locallyC1. Then at the end of section x5 we brie y discuss the general case where u0 is not necessarily compactlysupported and f;A;B are locally Lipschitz continuous. If not otherwise stated, we will always assume, withoutloss of generality, that f(0) = 0. The function space that contains u0 will be taken as(25) B(f;A;B) = �z 2 L1(R)\BV (R) : jf(z) �A(@xB(z)jBV (R) <1:Convergence in L1loc of a subsequence of the family u� of approximate solutions generated from (21) is obtainedby establishing three estimates for fUnj g:(a) a uniform L1 bound,(b) a uniform total variation bound,(c) L1 Lipschitz continuity in the time variable,and two estimates for the discrete total ux term h(Un+1j ; Un+1j+1 ) �A �D+B(Un+1j )�:(d) a uniform L1 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) isused to obtain the H�older continuity in time and space of the discrete di�usion term B(Unj ).For later use, recall that the L1(Z) norm, the L1(Z) norm and the BV (Z) semi-norm of a lattice functionU is de�ned respectively as U L1(Z) = supj2Z��Uj��; U L1(Z) =Xj2Z��Uj��;��U ��BV (Z) =Xj2Z��Uj � Uj�1�� � �x D�U L1(Z):If not speci�ed, i; j will always denote integers fromZ;m;n; l integers from f0; : : : ; Ng; x; y; c real numbers fromR and t; � real numbers from [0; T ]. Throughout this paper C will denote a positive constant, not necessarilythe same at di�erent occurrences, which is independent of the discretization parameters involved.The following lemma deals with the question of existence, uniqueness and properties of the solution of the(nonlinear) system (21).Lemma 4.1. If (23) is satis�ed, then for any U there is a unique U� satisfying the following equation(26) U�j � Uj�t +D��h(U�j ; U�j+1) �A(D+B(U�j ))� = 0; j 2Z:Furthermore, the solution U� of (26) possesses the following properties:(a) Uj � Vj 8j 2Zimplies that U�j � V �j 8j 2Z,(b) U� L1(Z) � U L1(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 terms of an m-accretive operator andan associated contraction solution operator, i.e., we shall use the Crandall and Liggett theory [3]. A similartreatment of implicit di�erence schemes for conservation laws has been given earlier by Lucier [15] and forstrongly degenerate convection-di�usion equations in [9].For a �xed n, let us now rewrite the di�erence equation (21) as (supressing the �x dependence)(27) Un+1j +�tA(Un+1; j) = Unj ; j 2Z;where the operator A : L1(Z)! L1(Z) is de�ned byA(U ; j) = D��h(Uj ; Uj+1)� A (D+B(Uj))�:We �rst show that the operator A is accretive. To this end, it is su�cient to establish that for any U; V withU � V 2 L1(Z), Xj2Zsign(Uj � Vj)�A(U ; j)� A(V ; j)� � 0:

10 S. EVJE, K. H. KARLSENAs a �rst step to achieve this goal, we perform the following calculation(28) Xj2Zsign(Uj � Vj)�A(U ; j) �A(V ; j)�=Xj2Zsign(Uj � Vj) �A(U ; j)� A(V ; j) � c�Uj � Vj�� + cXj2Z��Uj � Vj��;� �Xj2Z��cWj � �A(U ; j)� A(V ; j)���+ cXj2Z��Wj��;where Wj denotes Uj � Vj and c = c(�x) > 0 is a number chosen so that(29) c � 1�x �max(u;v) @uh(u; v)� min(u;v)@vh(u; v)�+ 2�x2a1b1:Next, we observe that(30) A(U ; j)� A(V ; j)= 1�x�hu(�j; Uj+1)Wj + hv(Vj ; ~�j+1)Wj+1 � hu(�j�1; Uj)Wj�1 � hv(Vj�1; ~�j)Wj�� 1�x2�a( j)�b(�j+1)Wj+1 � b(�j)Wj�� a( j�1)�b(�j)Wj � b(�j�1)Wj�1��;where �j; ~�j; �j 2 int(Uj ; Vj) and j 2 int(D+B(Uj); D+B(Vj)). Inserting this into inequality (28) yields thedesired result:Xj2Zsign(Uj � Vj)�A(U ; j)� A(V ; j)�� cXj2Z��Wj���Xj2Z���h 1�xhu(�j�1; Uj) + 1�x2a( j�1)b(�j�1)iWj�1+ hc� 1�x (hu(�j; Uj+1)� hv(Vj�1; ~�j)) � 1�x2 b(�j) (a( j) + a( j�1)]�Wj+ h 1�x2a( j)b(�j+1)� 1�xhv(Vj ; ~�j+1)iWj+1���� cXj2Z��Wj���Xj2Zh 1�xhu(�j�1; Uj) + 1�x2a( j�1)b(�j�1)i��Wj�1���Xj2Zhc� 1�x (hu(�j; Uj+1)� hv(Vj�1; ~�j)) � 1�x2 b(�j) (a( j) + a( j�1))i��Wj���Xj2Zh 1�x2a( j)b(�j+1)� 1�xhv(Vj ; ~�j+1)i��Wj+1�� � 0;due to the monotonicity conditions (23) and the choice of c given by (29). From (30) we observe that theoperator A is Lipschitz continuous, A(U ) �A(V ) L1(Z) � � 2�xL(h) + 4�x2L(A;B)� U � V L1(Z);where L(h) = max jhuj + max jhvj and L(A;B) = a1b1. This implies that A is not only accretive but alsom-accretive, see [6]. We can now invoke Theorem 2.4 to conclude the existence of a unique monotone solutionoperator S associated with (21) such that U�j = S(U ; j);which proves the �rst part of the lemma. Since Pj2ZA(U ; j) = 0 and A commutes with translations, thesecond part of the lemma also follows from Theorem 2.4. �As a direct consequence of Lemma 4.1 the following lemma is established.

DOUBLY NONLINEAR DEGENERATE EQUATIONS 11Lemma 4.2. We have(31) Un+1 L1(Z) � U0 L1(Z) ; ��Un+1��BV (Z) � ��U0��BV (Z):Next we establish a regularity property for the total ux h(Un+1j ; Un+1j+1 ) �A �D+B(Un+1j )�. As mentioned,this regularity property is of fundamental importance when proving convergence of the scheme (21). Let us �rstindicate how this regularity estimate can be derived at the continuous level in the case of classical solutions. Tothis end, consider the uniformly parabolic equation(32) @tu+ @xf(u) = @xA (@xB(u)) ; A0; B0 > 0;and recall that this equation has a unique classical solution u. By di�erentiating (32) with respect to t andsubsequently integrating with respect to x, we �nd that the quantityv(x; t) = Z x�1 @tu(�; t)d�satis�es the linear, variable coe�cients, uniformly parabolic cequation(33) @tv + f 0(u)@xv = a(@xB(u))@x(b(u)@xv):From the maximum principle for this equation it follows thatkv(�; t)kL1(R) � kv0kL1(R) :From (32) and the de�nition of v we see that v = �f(u) +A(@xB(u)), which implies thatkA (@xB(u(�; t)))kL1(R) � C;where C = 2max jf j + kA (@xB(u(�; 0)))kL1(R). This is merely formalism since the solution of (1) in generalonly exists in a weak sense. However, these calculations clearly motivate the next lemma whose content is auniform L1 bound as well as a BV bound for the discrete total ux h(Un+1j ; Un+1j+1 )� A(D+B(Un+1j )).Lemma 4.3. We have(34) h(Un+1j ; Un+1j+1 )� A(D+B(Un+1j )) L1(Z) � h(U0j ; U0j+1)� A(D+B(U0j )) L1(Z);(35) ��h(Un+1j ; Un+1j+1 ) �A(D+B(Un+1j ))��BV (Z) � ��h(U0j ; U0j+1) �A(D+B(U0j ))��BV (Z):Proof. To prove these regularity properties for the approximate solutions, we introduce two auxiliary sequencesfWnj g and fV nj g given by Wn+1j = Un+1j � Unj�t ; V n+1j = �x jXk=�1Wn+1k :Using the �nite di�erence scheme (21) we observe(36) Wn+1k �x = ����h(Un+1k ; Un+1k+1 ) �A(D+B(Un+1k ))�:Summing over all k = �1; : : : ; j and having in mind that Unk = 0 for su�ciently large k and h(0; 0) = f(0) = 0,we get(37) V n+1j = ��h(Un+1j ; Un+1j+1 )� A(D+B(Un+1j ))�:From this relation it is clear that it is su�cient to establish L1 and BV estimates for V n. As a �rst step towardthat end, we derive an equation for the auxiliary sequence fV nj g. For this purpose, consider the di�erenceequation given by (21) and subtract the corresponding equation at time tn. Then we obtainWn+1k �x = Wnk�x����h(Un+1k ; Un+1k+1 ) � h(Unk ; Unk+1)�+���A(D+B(Un+1k )) �A(D+B(Unk ))�:

12 S. EVJE, K. H. KARLSENAgain we sum over all k = �1; : : : ; j, yielding(38) V n+1j = V nj � �h(Un+1j ; Un+1j+1 )� h(Unj ; Unj+1)� + �A(D+B(Un+1j ))� A(D+B(Unj ))�:We now rewrite the two last terms. To this end, we �rst we observe that(39) �xUn+1j � Unj�t = �xWnj = �x jXk=�1Wnk ��x j�1Xk=�1Wnk = V nj � V nj�1:Then we have(40) h(Un+1j ; Un+1j+1 )� h(Unj ; Unj+1)= �h(Un+1j ; Un+1j+1 )� h(Un+1j ; Unj+1)�+ �h(Un+1j ; Unj+1) � h(Unj ; Unj+1)�= @vh(Un+1j ; ~�n+12j+1 ) �Un+1j+1 � Unj+1�+ @uh(�n+12j ; Unj+1) �Un+1j � Unj �= ��@vh(Un+1j ; ~�n+ 12j+1 )[V n+1j+1 � V n+1j ] + @uh(�n+ 12j ; Unj+1)[V n+1j � V n+1j�1 ]�= �t �hn+1v;j D+V n+1j + hn+1u;j D+V n+1j�1 � ;where(41) hn+1v;j = @vh(Un+1j ; ~�n+ 12j+1 ); hn+1u;j = @uh(�n+ 12j ; Unj+1); �n+ 12j ; ~�n+12j 2 int(Unj ; Un+1j ):Similary, we rewrite the last term of (38).(42) A(D+B(Un+1j )) �A(D+B(Unj ))= a( n+ 12j )D+�B(Un+1j )� B(Unj )�= a( n+ 12j )D+�b(�n+ 12j )[Un+1j � Unj ]�= �t � a( n+ 12j )D+�b(�n+ 12j )D�V n+1j �= �t � an+1j D+�bn+1j D�V n+1j �;where(43) an+1j = a( n+ 12j ); n+ 12j 2 int(D+B(Unj ); D+B(Un+1j ));bn+1j = b(�n+ 12j ); �n+ 12j 2 int(Unj ; Un+1j ):From (38), (40) and (42) we obtain the following linear di�erence equation for fV nj g.(44) V n+1j � V nj�t + �hn+1u;j D+V n+1j�1 + hn+1v;j D+V n+1j � = an+1j D+ �bn+1j D�V n+1j � :This equation can be written as(45) cn+1j V n+1j�1 + dn+1j V n+1j + en+1j V n+1j+1 = V nj ;where cn+1j = � ��hn+1u;j + �an+1j bn+1j � ;dn+1j = �1 + � �hn+1u;j � hn+1v;j �+ �an+1j �bn+1j+1 + bn+1j �� ;en+1j = � ��an+1j bn+1j+1 � �hn+1v;j � :By the monotonicity assumption (23), we have(46) cn+1j + dn+1j + en+1j = 1; cn+1j ; en+1j � 0; dn+1j � 0:Thanks to (46), the linear system (45) is strictly diagonal dominant. Consequently, there exists a unique solutionV n+1. Furthermore, this solution satis�es a maximum principle:cn+1j jV n+1j�1 j+ dn+1j jV n+1j j+ en+1j jV n+1j+1 j � jV nj j =) V n+1 L1(Z) � kV nkL1(Z) :

DOUBLY NONLINEAR DEGENERATE EQUATIONS 13In view of (37) we can now conclude that (34) is satis�ed. Next we prove that the solution of (44) has boundedvariation on Z. Introduce the quantity Znj = V nj � V nj�1 and observe thatZn+1j � Znj�t +D��hn+1u;j Zn+1j + hn+1v;j Zn+1j+1 � = D� �an+1j D+�bn+1j Zn+1j �� :Similarly to (45), we can write this equation as(47) �cn+1j Zn+1j�1 + �dn+1j Zn+1j + �en+1j Zn+1j+1 = Znj ;where �cn+1j = � ��hn+1u;j�1 + �an+1j�1bn+1j�1 � ;�dn+1j = �1 + � �hn+1u;j � hn+1v;j�1�+ �bn+1j �an+1j�1 + an+1j �� ;�en+1j = � ��an+1j bn+1j+1 � �hn+1v;j � :Again, due to the monotonicity assumption, we see that(48) �cn+1j+1 + �dn+1j + �en+1j�1 = 1; �cn+1j ; �en+1j � 0; �dn+1j � 0:Therefore, from (47), it follows that�cn+1j jZn+1j�1 j+ �dn+1j jZn+1j j+ �en+1j jZn+1j+1 j � jZnj j =)Xj2ZjZn+1j j �Xj2ZjZnj j;which immediately implies (35). This concludes the proof of the lemma. �An immediate consequence of (35), and (21) is that the discrete approximations (21) are L1 Lipschitz con-tinuous in time, and thus contained in BV (QT ).Lemma 4.4. We have(49) Um � Un L1(Z) � ��h(U0j ; U0j+1) �A �D+B(U0j )���BV (Z)�t�x jm� nj:Proof. Suppose that m > n. Using (21), we readily calculate thatXj2Z��Umj � Unj �� � �tm�1Xl=n Xj2Z��D��h(U l+1j ; U l+1j+1) �A �D+B(U l+1j )����� ��h(U0j ; U0j+1) �A �D+B(U0j )���BV (Z)�t�x (m� n);where the BV estimate (35) has been used. This concludes the proof of the lemma. �Lemma 4.5. If (23) is satis�ed, then the following cell entropy inequality holds(50) ��Un+1j � c��� ��Uj � c��+�tD��h �Un+1j _ c; Un+1j+1 _ c�� h �Un+1j ^ c; Un+1j+1 ^ c����tD��sign(Un+1j � c)A �D+B(Un+1j )�� � 0:Proof. The arguments are as follows. First, observe that(51) h(Un+1j _ c; Un+1j+1 _ c) � h(Un+1j ^ c; Un+1j+1 ^ c) � sign(Un+1j � c)�h(Un+1j ; Un+1j+1 ) � h(c; c)�;(52) ��h(Un+1j�1 _ c; Un+1j _ c) � h(Un+1j�1 ^ c; Un+1j ^ c)� � sign(Un+1j � c)�h(c; c)� h(Un+1j�1 ; Un+1j )�:These two inequalities follow from the monotonicity of h. Due to the similarity, we only show the �rst inequality(51). The proof is based upon examining several cases depending on whether Un+1j+1 is larger or smaller thanUn+1j . If c =2 int(Un+1j ; Un+1j+1 ), then the left hand side of (51) is equal to the right hand side.

14 S. EVJE, K. H. KARLSENNext, assume that c 2 int(Un+1j ; Un+1j+1 ) and Un+1j � Un+1j+1 . Thenh(Un+1j _ c; Un+1j+1 _ c)� h(Un+1j ^ c; Un+1j+1 ^ c)= h(c; Un+1j+1 )� h(Un+1j ; c)= h(c; c)� h(Un+1j ; Un+1j+1 ) + �h(c; Un+1j+1 )� h(c; c)�+ �h(Un+1j ; Un+1j+1 )� h(Un+1j ; c)�= sign(Un+1j � c)�h(Un+1j ; Un+1j+1 )� h(c; c)�+ Qn+1jwhere Qn+1j = �h(c; Un+1j+1 ) � h(c; c)�+ �h(Un+1j ; Un+1j+1 ) � h(Un+1j ; c)�= hn+1v;j (Un+1j+1 � c) + ~hn+1v;j (Un+1j+1 � c)and hn+1v;j = @vh(c; �n+1j+1 ); ~hn+1v;j = @vh(Un+1j ; ~�n+1j+1 ); �n+1j ; ~�n+1j 2 int(c; Un+1j ):Due to the monotonicity assumption (23) and the fact that c � Un+1j+1 , we conclude that Qn+1j � 0 and thedesired inequality is obtained. Similary, we can show that this inequality holds when Un+1j � Un+1j+1 .For the discrete di�usion term we have the following inequality.(53) sign(Un+1j�1 � c)A �D+B(Un+1j�1 )� � sign(Un+1j � c)A �D+B(Un+1j�1 )� :In order to see this, consider the relationsign(Un+1j�1 � c)A �D+B(Un+1j�1 )� = sign(Un+1j � c)A �D+B(Un+1j�1 )�+ Rn+1j ;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+1j�1 � �an+1j� 12 b(�n+1j� 12 )and �an+1j�12 = Z 10 a(�D+B(Un+1j�1 ))d� � 0; b(�n+1j�12 ) � 0; �n+1j� 12 2 int(Un+1j�1 ; Un+1j ):Now we observe that Rn+1j = 0 unless c is between Un+1j�1 and Un+1j . If c is in this interval, it is easy to checkthat Rn+1j is nonpositive. Invoking (51),(52), (53) and (21) we obtainjUn+1j � cj+�tD��h(Un+1j _ c; Un+1j+1 _ c)� h(Un+1j ^ c; Un+1j+1 ^ c)���tD��sign(Un+1j � c)A(D+B(Un+1j ))�� jUn+1j � cj+ � sign(Un+1j � c)�h(Un+1j ; Un+1j+1 )� h(c; c)�+ � sign(Un+1j � c) �h(c; c)� h(Un+1j�1 ; Un+1j )�� � sign(Un+1j � c)A �D+B(Un+1j )� + � sign(Un+1j � c)A �D+B(Un+1j�1 )�= sign(Un+1j � c)�Un+1j � c+�tD��h(Un+1j ; Un+1j+1 )� A(D+B(Un+1j ))��= sign(Un+1j � c)�Unj � c�� jUnj � cj:Hence the proof is complete. �Remark. The estimates of Lemmas 4.2, 4.3 and 4.4 have been obtained without making use of the structuralassumption (6) on A. From these estimates it is not di�cult to show that there is a subsequence of the approxi-mate solutions which converges to a limit function u. However, we do not have estimates on the di�usion termwhich ensures that A(D+B(Unj )) converges in some appropiate sense to the di�usion term A(@xB(u)).In the following we will discuss continuity properties of the discrete di�usion term fB(Unj )g. From (34) andthe assumption that u0 is contained in B(f;A;B) it follows that(54) A(D+B(Unj )) L1(Z) � ~C;where ~C is a constant independent of �. An immediate consequence of (54) and the assumption (6) is thefollowing lemma.

DOUBLY NONLINEAR DEGENERATE EQUATIONS 15Lemma 4.6. We have(55) D+B(Unj ) L1(Z) � C:Remark. The assumption (6) cannot be removed in establishing convergence to the BV entropy weak solutionin the sense of De�nition 2.1. In other words, the problem may not have BV entropy weak solutions if (6)is not assumed. Recall the example with A unbounded from section 1 (Figure 2). Here B(s) = s, but clearlyD+B(Unj ) = D+Unj is not uniformly bounded because of the appearance of a discontinuity. Hence this problemcannot have a solution in the class given by De�nition 2.1.Knowing that the discrete di�usion term fB(Unj )g is Lipschitz continuous in the space variable, the questionarises how to obtain information about the regularity in the time variable. One strategy would be to continueworking with the linear equation for v = f(u)�A (@xB(u)) and try to derive a result concerning the continuityof v with respect to the time variable from the known modulus of continuity in space. This technique, introducedby Kruzkov [11], was used for the simple degenerate case [8], i.e. when A(s) = s. To illustrate some of the addeddi�culties introduced by the double nonlinearity, let us see why this technique does not work in the generalcase. To this end, let �(x) be a test function on R and multiply (33) by � and integrate over R. Then we have(56) ZR�(x)@tv dx = � ZRf 0(x; t)@xv � �(x) dx+ ZRa(x; t)@x (b(x; t)@xv) � �(x) dx;where f 0(x; t); a(x; t); b(x; t) denote f 0(u(x; t)); a(@xB(u(x; t))); b(u(x; t)) respectively. The �rst term on the righthand side of (56) is bounded since v is of bounded variation. For the case when A(s) = s, that is a(x; t) = 1,the second term is bounded since one derivative can be moved over to the test function �. However, in thegeneral case a(x; t) = a(@xB(u(x; t))) is not constant and therefore it is not possible to bound this term. Hencewe have to choose another approach to this problem. We will employ a discrete version of a technique used byYin [18] which combines the scheme (21) and the estimate (34). For this purpose, de�ne u� as the interpolantof the discrete values fUnj g given by(57) u�(x; t) = 8<: Unj + Unj+1�Unj�x (x� xj) + Un+1j+1 �Unj+1�t (t� tn); (x; t) 2 TLj;n;Unj + Un+1j+1 �Un+1j�x (x � xj) + Un+1j �Unj�t (t� tn); (x; t) 2 TUj;n:Here TLj;n denotes the triangle with vertices (xj; tn),(xj+1; tn) and (xj+1; tn+1) while TUj;n denotes the trianglewith vertices (xj; tn),(xj; tn+1) and (xj+1; tn+1). LetRnj = [xj; xj+1]� �tn; tn+1�and note that Rnj = TLj;n [ TUj;n. Later we will use the notation Rx;t in order to denote a rectangle Rnj , notnecessarily unique, which contains the point (x; t). In particular, we note that u� is continuous everywhere anddi�erentiable almost everywhere in QT .Lemma 4.7. We have(58) ��B(Umi )�B(Unj )�� � C�jxi � xjj+pjtm � tnj+�x�:Proof. We have that��B(Umi ) �B(Unj )�� � ��B(Umi )� B(Uni )��+ ��B(Uni ) �B(Unj )�� =: I1 + I2:Clearly I2 = O (jxi � xjj) by using (55). Now we focus on how to estimate I1. Consider the interval [xi; xi + �],where � will be speci�ed later. Then for some x� 2 [xi; xi + �] (that also will be speci�ed later) we have(59)I1 = jB(u�(xi; tm))� B(u�(xi; tn))j� jB(u�(xi; tm))� B(u�(x�; tm))j+ jB(u�(x�; tm)) �B(u�(x�; tn))j+ jB(u�(x�; tn))� B(u�(xi; tn))j� 2C (jxi � x�j+�x) + jB(u�(x�; tm)) �B(u�(x�; tn))j� 2C (�+�x) + jB(u�(x�; tm)) �B(u�(x�; tn))j;

16 S. EVJE, K. H. KARLSENwhere the estimate of the �rst and third term of the second line follow from the monotonicity of B(s). Next wedescribe how jB(u�(x�; tn)) �B(u�(x�; tm))j can be estimated. For this purpose, we introduce the quantityQ(x) = xZ�1 �u�(�; tm) � u�(�; tn)� d�:Since u� is continuous, Q(x) is di�erentiable everywhere. Hence, there is a number x� in [xi; xi + �] such thatQ0(x�)� = Q(xi + �)�Q(xi) = Z xi+�xi �u�(�; tm) � u�(�; tn)� d�:We then have the following relation(60) jB(u�(x�; tn))� B(u�(x�; tm))j � b1ju�(x�; tn)� u�(x�; tm)j= b1jQ0(x�)j = b1� � ��� Z xi+�xi (u�(�; tm)� u�(�; tn)) d����:Since u� is di�erentiable in time almost everywhere on QT we have(61) Z xi+�xi �u�(�; tm)� u�(�; tn)� d�= Z xi+�xi Z tmtn @tu� dt dx = Z xjxi Z tmtn @tu� dt dx+ Z xi+�xj Z tmtn @tu� dt dx =: J1 + J2;where j is the integer such that 0 < (xi + �) � xj < �x. Now, in view of (59), (60) and (61) we want to showthat jJ1j; jJ2j � �2 and then choose � equal to pjm� nj�t. We haveJ1 = Z xjxi Z tmtn @tu� dt dx= j�1Xk=i m�1Xl=n �ZZTUk;l @tu� dt dx+ ZZTLk;l @tu� dt dx�= 12�x�t j�1Xk=i m�1Xl=n �U l+1k � U lk�t + U l+1k+1 � U lk+1�t �:Using the �nite di�erence scheme (21) and estimate (34) of Lemma 4.3, we obtain the following estimatejJ1j = 12�x�t���j�1Xk=i m�1Xl=n �U l+1k � U lk�t + U l+1k+1 � U lk+1�t ����� 4 � 12�tjm� nj h(U0j ; U0j+1)�A �D+B(U0j )� L1(Z)= 2C0jm� nj�t = 2C0�2;where(62) C0 = h(U0j ; U0j+1)� A �D+B(U0j )� L1(Z) ;and we have set � equal to pjm� nj�t. Repeating the arguments for J2 we also deduce that jJ2j � 2C0�2.From (60) and (61) we now conclude thatjB(u�(x�; tn)) � B(u�(x�; tm))j � 4C0�;and hence, from (59), we obtainI1 = jB(u�(xi; tm))� B(u�(xi; tn))j = O��+�x� = O�pjm� nj�t+�x�:Now the proof of (58) is completed. �

DOUBLY NONLINEAR DEGENERATE EQUATIONS 17x5. Convergence Results.Now we will employ the regularity properties established for fUnj g and fB(Unj )g in x5 to prove that theapproximate solutions generated by (21) in fact converges to the solution of (1) in the sense of De�nition 2.1.We start by showing that a subsequence of the family of approximate solutions converges to a function u andthat this limit inherits the properties of the approximate solutions (see Lemma 5.2). Finally, using the cellentropy inequality of Lemma 4.5 and the properties of the interpolant we show that this limit satis�es theentropy inequality of De�nition 2.1. The arguments needed to prove this turn out to be rather involved due tothe double nonlinearity of the problem. In particular, we will see that it is important how the linear interpolantis de�ned.Recall that u� denotes the interpolant of the discrete values fUnj g given by (57). Similary we de�ne w� asthe interpolant of the discrete values fB(Unj )g given by(63) w�(x; t) = 8<: B(Unj ) + B(Unj+1)�B(Unj )�x (x� xj) + B(Un+1j+1 )�B(Unj+1)�t (t� tn); (x; t) 2 TLj;n;B(Unj ) + B(Un+1j+1 )�B(Un+1j )�x (x� xj) + B(Un+1j )�B(Unj )�t (t � tn); (x; t) 2 TUj;n:For later use, observe that the following important relations hold(64) @xu� = D+Unj ; @xw� = D+B(Unj )on the parallelogram Pnj with vertices (xj ; tn�1), (xj; tn), (xj+1; tn) and (xj+1; tn+1), i.e., Pnj = TUj;n�1 [ TLj;n.Similary,(65) @tu� = Un+1j � Unj�ton the parallelogram Qnj with vertices (xj�1; tn),(xj; tn), (xj; tn+1) and (xj+1; tn+1), i.e., Qnj = TLj�1;n [ TUj;n.Note also that for (x; t) 2 Rnj neither w� nor B(u�) will introduce new minima or maxima, that is(66) min�B(Unj ); B(Unj+1); B(Un+1j ); B(Un+1j+1 )� � w�; B(u�) � max�B(Unj ); B(Unj+1); B(Un+1j ); B(Un+1j+1 )�:This follows from the de�nition of u�; w� and the fact that B(s) is monotone. The next technical lemma dealswith the interpolation error associated with the linear interpolant (63) of H�older continuous functions.Lemma 5.1. Assume that G(x; t) 2 C1;12 (QT ) and let ��G(x; t) denote the interpolant given by��G(x; t) = ( G(xj; tn) + G(xj+1 ;tn)�G(xj;tn)�x (x � xj) + G(xj+1 ;tn+1)�G(xj+1;tn)�t (t � tn); (x; t) 2 TLj;n;G(xj; tn) + G(xj+1 ;tn+1)�G(xj;tn+1)�x (x� xj) + G(xj;tn+1)�G(xj;tn)�t (t � tn); (x; t) 2 TUj;n:Then the following error estimate holdsk��G�GkL1(QT ) � C��x+p�t�:Proof. To see this, let (x; t) be an arbitrary point in QT . Then (x; t) is contained in some rectangle Rnj and wehave(67) j��G(x; t)� G(x; t)j � j��G(x; t)�G(xj; tn)j+ jG(xj; tn) �G(x; t)jFor the �rst term on the right hand side of (67) we havej��G(x; t)�G(xj; tn)j � ( jG(xj+1; tn)� G(xj; tn)j+ jG(xj+1; tn+1)� G(xj+1; tn)j; (x; t) 2 TLj;n;jG(xj+1; tn+1)� G(xj; tn+1)j+ jG(xj; tn+1)� G(xj; tn)j; (x; t) 2 TUj;n:Therefore, since G(x; t) 2 C1;12 (QT ), it follows that the �rst and the second term on the right hand side of (67)is of order �x+p�t. �Now we show that the following compactness and convergence results hold.

18 S. EVJE, K. H. KARLSENLemma 5.2. There exists a function u 2 L1(QT ) \BV (QT ), with B(u) 2 C1;12 (QT ), such that(68) 8>>><>>>: (a) u�(x; t)! u(x; t); in L1loc(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 L1loc(QT ).(d) A (@xw�) �* A (@xB(u)) ; in L1loc(QT ).Proof. The functions u�(x; t) and w�(x; t) satisfy the following estimates:(69) jju�jjL1(QT ) � C; ju�jBV (QT ) � C;and(70) jw�(y; s) � w�(x; t)j � C�jx� yj +pjt� sj+�x+p�t�; 8x; y; s; t:The �rst estimate of (69) follows immediately from the de�nition of the linear interpolant u� and Lemma 4.2.The second estimate of (69) is a consequence of the following two estimates:ZZQT j@xu�j dt dx =Xj;n ZZPnj j@xu�j dt dx= �x�tXj;n jD+Unj j � T ju0jBV :Here we have used (64) and Lemma 4.2. Similary, by using (65) and Lemma 4.4 we obtain the estimateZZQT j@tu�j dt dx=Xj;n ZZQnj j@tu�j dt dx= �x�tXj;n jUn+1j � Unj j�t � C0 � T;where C0 = jh(U0j ; U0j+1) � A(D+B(U0j ))jBV (Z). The estimate (70) requires argument. Let (x; t) and (y; s) besome arbitrary given points and choose two rectangles Rx;t and Ry;s such that (x; t) 2 Rx;t and (y; s) 2 Ry;s(they may coinside). Moreover, let (xi; tm) and (xj; tn) denote vertices of Rx;t and Ry;s respectively, such thatjxj � xij+pjtn � tmj � jx� yj +pjt� sj:Then, we havejw�(y; s) � w�(x; t)j � jw�(y; s) �w�(xi; tm)j+ jw�(xi; tm)� w�(xj; tn)j+ jw�(xj ; tn)�w�(x; t)j=: E1 + E2 + E3Clearly, by (58) E2 = jB(Umi )� B(Unj )j � C�jxi � xjj+pjm� nj�t+�x�� C�jx� yj+pjt� sj+�x�:Now estimate (70) follows since we have, in view of (66), thatE1; E3 � C��x+p�t�:By virtue of estimates (69), fu�g is bounded inW 1;1(K) � BV (K) for each compact set K. Using thatBV (K)is compactly imbedded in L1(K) it is not di�cult to show that fu�g, passing if necessary to a subsequence,converges in L1loc(QT ) and pointwise almost everywhere in QT to a function u,u 2 L1(QT ) \BV (QT ):Next we discuss convergence properties of the sequence fw�g. By estimate (70) we can repeat the proof of theAscoli-Arzela theorem to conclude that there is a subsequence of fw�g and a limit w,w 2 C1; 12 (QT )such that w� ! w; uniformly on compact sets and pointwise in QT :

DOUBLY NONLINEAR DEGENERATE EQUATIONS 19By the continuity of w and the pointwise convergence, we conclude that w = B(u). To see this, let (x; t) be anarbitrary point such that u�(x; t)! u(x; t), i.e. B(u�(x; t))! B(u(x; t)). We havejB(u(x; t))� w(x; t)j � jB(u(x; t))�B(u�(x; t))j+ jB(u�(x; t))� w�(x; t)j+ jw�(x; t)�w(x; t)j:Since w�(x; t) ! w(x; t), we only have to check that jB(u�(x; t)) � w�(x; t)j must tend to zero. For thispurpose, assume that (x; t) is contained in a rectangle Rx;t. Then, in view of (66) we havejB(u�(x; t))� w�(x; t)j � jB(u�(xj ; tn))� w�(xi; tm)j = jB(Unj ) �B(Umi )j � C��x+p�t�;where (xj ; tn) and (xi; tm) are appropiate chosen vertices of the rectangle Rx;t. Hence w = B(u) almosteverywhere in QT . By the continuity of w, this must hold for all points in QT .Now we continue showing the convergence result (c) of (68). From (55) and (64) it follows thatk@xw�kL1(QT ) � C:Hence there is a limit function W such that kWkL1(QT ) � C and, passing if necessary to a subsequence@xw� �*W; in L1(QT ):Since ZZQT w�@x�dt dx! ZZQT B(u)@x�dt dx; � 2 C10 (QT );it is obvious that W = @xB(u) and (c) follows. Finally we show why (d) is satis�ed. Due to the fact thatkA(@xw�)kL1(QT ) � ~C;(see (54)) we know there is a function A(x; t) in L1(QT ) such that, again passing if necessary to a subsequence,A(@xw�) �* A; in L1(QT ):We now show that A = A(@xB(u)) by using a discrete version of the arguments used by Yin [18]. Let ��B(u)be the interpolant of the discrete values B(u(xj ; tn)) de�ned as in Lemma 5.1. For the moment, assume that Kis a compact subset of QT of the form K = [j;nPnj where (j; n) 2 fJ1; : : : ; J2g � fN1; : : : ; N2g. We then have(71) ZZK A(@xw�) (@xw� � @xB(u)) dt dx= ZZK A(@xw�) (@xw� � @x��B(u)) dt dx+ ZZK A(@xw�) (@x��B(u) � @xB(u)) dt dx=: E1 + E2:First, we estimate E1 as follows (recall (64)).E1 = ZZK A(@xw�)�@xw� � @x��B(u)� dt dx=Xj;n ZZPnj A(@xw�)�@xw� � @x��B(u)� dt dx= �x�tXj;n A(D+B(Unj ))�D+B(Unj ) �D+B(u(xj ; tn))�= ��x�tXj;n D�A(D+B(Unj )) �B(Unj )� B(u(xj ; tn))�+�x�tXn �A(D+B(UnJ2 )) �B(UnJ2 )� B(u(xJ2 ; tn))�� A(D+B(UnJ1 )) �B(UnJ1 )� B(u(xJ1 ; tn))��= ��x�tXj;n�Unj � Un�1j�t +D�h(Unj ; Unj+1)��w�(xj; tn)� B(u(xj ; tn))�+�x�tXn �A(D+B(UnJ2 )) �B(UnJ2 )� B(u(xJ2 ; tn))�� A(D+B(UnJ1 )) �B(UnJ1 )� B(u(xJ1 ; tn))�� ;

20 S. EVJE, K. H. KARLSENwhere we have used the �nite di�erence scheme (21) for the last equality. Hence, by Lemmas 4.2, 4.4 and (54)(72) jE1j � (C0T + T (max j@uhj+max j@vhj)ju0jBV) � kw� � B(u)kL1(K) +C�x:In order to estimate E2 let !�(x) denote a standard molli�er in the x variable with support in [��; �]. LetA�(�; t) = !�(�) �A(@xw�(�; t)). For E2 we then haveE2 = ZZK A(@xw�) (@x��B(u) � @xB(u)) dt dx= ZZK A�(x; t) (@x��B(u) � @xB(u)) dt dx+ ZZK �A(@xw�(x; t))� A�(x; t)� (@x��B(u) � @xB(u)) dt dx= � ZZK @xA�(x; t) (��B(u) �B(u)) dt dx+ ZZK �A(@xw�(x; t))�A�(x; t)� (@x��B(u)� @xB(u)) dt dx=: E2;1 +E2;2:Clearly, in view of Lemma 5.1,(73) jE2;1j � Z T0 jA�(�; t)jBV (R)dt � k��B(u) �B(u)kL1(K) = O(�x+p�t);due to the H�older continuity of B(u) and the fact thatjA�(�; t)jBV (R) � jA(@xw�(�; t))jBV (R) = jA(D+B(Unj ))jBV (Z) � C (for some appropiate n);which is true because of (35). Moreover, we have(74) jE2;2j � k@x��B(u) � @xB(u)kL1(K) � A�(x; t)� A(@xw(x; t)) L1(K) = O(�);since k@xB(u)kL1(QT ) � C. From (71),(72), (68)b, (73) and (74) it follows that(75) lim�!0 ZZK A(@xw�)�@xw� � @xB(u)� dt dx = 0:Note that for a general compact set K we can split K into two sets KP and �K such thatK = KP [�K; KP = [j;nPnj ; meas(�K) = O(�x+�t):Hence ZZK A(@xw�)�@xw� � @xB(u)� dt dx= ZZKP A(@xw�)�@xw� � @xB(u)� dt dx+ ZZ�K A(@xw�)�@xw� � @xB(u)� dt dx:In light of the analysis above, the �rst term tends to zero. Because the integrand of the last integral is uniformlybounded, it follows that this term is of order �x+�t and thus tends to zero. Hence (75) holds for all compactK � QT . On the other hand, since A(@xB(u)) is in L1(QT ) we have by (68)c(76) lim�!0ZZK A (@xB(u)) (@xw� � @xB(u)) dt dx = 0:From (75) and (76) it follows that(77) lim�!0 ZZK a��@xw� � @xB(u)�2 dt dx = lim�!0 ZZK �A(@xw�)� A(@xB(u))��@xw� � @xB(u)� dt dx = 0;

DOUBLY NONLINEAR DEGENERATE EQUATIONS 21where(78) a� = a�(x; t) = Z 10 a (�@xw� + (1� �)@xB(u)) d� = A(@xw�) �A(@xB(u))@xw� � @xB(u) :Using (78) and H�older's inequality we now deduce���ZZQT �A (@xw�)� A(@xB(u))�� dt dx��� � k�kL1(QT ) ZZsupp�pa� � ���pa��@xw� � @xB(u)���� dt dx� k�kL1(QT ) �ZZsupp� a� dt dx�12 ��ZZsupp� a��@xw� � @xB(u)�2 dt dx�12 :Since a� � a1 <1 it follows thatlim�!0ZZQT �A(@xw�)� A(@xB(u)))� dt dx = 0; � 2 C10 (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. Let � R2 and gj(x) ! g(x) a.e. in . Then there exists a set F , which is at most countable,such that for any c 2 RnF , sign(gj(x)� c)! sign(g(x) � c); a.e. in .The proof is elementary and is omitted.Lemma 5.4. Let ~u� be a piecewise constant interpolant of the discrete data points fUnj g de�ned such that~u���Pnj = Unj . Then, passing if necessary to a subsequence, ~u� ! u pointwise a.e. in QT , where u denotes thelimit function obtained in Lemma 5.2.Proof. Clearly we haveZZQT j~u� � u�j dt dx=Xj;n ZZPnj j~u� � u�j dt dx =Xj;n ZZTUj;n�1 j~u� � u�j dt dx+Xj;n ZZTLj;n j~u� � u�j dt dx=: S1 + S2:For S1 we have S1 =Xj;n ZZTUj;n�1 j~u� � u�j dt dx=Xj;n ZZTUj;n�1 ���(Unj+1 � Unj )�x� xj�x �+ (Unj � Un�1j )� t � tn�1�t � 1���� dt dx� 12�x�tXj;n �jUnj+1 � Unj j+ jUnj � Un�1j j� � T2 �ju0jBV�x+C0�t�;where C0 is given by (62). Similary, we have S2 � 12T �ju0jBV�x+ C0�t�. HenceZZQT j~u� � u�j dt dx � T �ju0jBV�x+ C0�t�;from which the lemma follows. �We continue by showing that the limit u satis�es the integral inequality (10).

22 S. EVJE, K. H. KARLSENLemma 5.4. Let � be a nonnegative test function with compact support on R� [0; T i and c 2 R. Then thelimit function u(x; t) of Lemma 5.2 satis�es the integral inequality (10).Proof. Let � be a suitable test function and put �nj = �(j�x; n�t). Multiplying the cell entropy inequality(50) by �nj�x, summing over all j and n and applying summation by parts, we get(79) �x�tN�1Xn=0 Xj2Z�jUn+1j � cjh�n+1j � �nj�t i+ �h(Un+1j _ c; Un+1j+1 _ c)� h(Un+1j ^ c; Un+1j+1 ^ c)�D+�nj� sign(Un+1j � c)A(D+B(Un+1j ))D+�nj �+�xXj jU0j � cj�0j � 0:For the �rst term we have�x�tN�1Xn=0Xj2ZjUn+1j � cjh�n+1j � �nj�t i = ZZQT j~u� � cj@t�dt dx+ O(�x+�t):Using Lemma 4.2 and the fact that h is consistent with f , we can obviously write�xXj2Z�h(Un+1j _ c; Un+1j+1 _ c)� h(Un+1j ^ c; Un+1j+1 ^ c)�D+�nj= �xXj2Zsign(Un+1j � c)�f(Un+1j )� f(c)�D+�nj + O(�x):Hence we have for the second term of (79)�x�tN�1Xn=0Xj2Z�h(Un+1j _ c; Un+1j+1 _ c)� h(Un+1j ^ c; Un+1j+1 ^ c)�D+�nj= ZZQT sign(~u� � c) (f(~u�) � f(c)) @x�dt dx+ O(�x+�t);For the discrete di�usion term of (79) we now have�x�tN�1Xn=0Xj2Zsign(Un+1j � c)A(D+B(Un+1j ))D+�nj= N�1Xn=0Xj2ZZZPn+1j sign(~u� � c)A(@xw�)@x�dt dx+ O(�x+�t)= ZZQT sign(~u� � c)A(@xw�)@x�dt dx+ O(�x+�t):Hence, we can replace (79) byZZQT j~u� � cj@t�+ sign(~u� � c) (f(~u�)� f(c)) @x�� sign(~u� � c)A (@xw�) @x�dt dx+ ZRju0 � cj�(x; 0) dx � �C��x+�t�:Using (68) and Lemma 5.3 and 5.4 we conclude that u satis�es (10) for almost all c 2 R. To complete the proof,note that A(@xB(u)) = 0 a.e. in Ec = f(x; t) 2 QT : u(x; t) = cg for any constant c. Therefore, by using anapproximate procedure the result holds for all c. �This completes our discussion when u0 has compact support and f;A;B are locally C1. For u0 2 B(f;A;B)not necessarily compactly supported and f;A;B merely locally Lipschitz continuous, we approximate u0 by acompactly supported function up0 and f; a; b by a smoother function fp; ap; bp, compute the di�erence approxi-mation of the resulting problem and then let p!1 and �t;�x! 0.We are now ready to state our main result:

DOUBLY NONLINEAR DEGENERATE EQUATIONS 23Theorem 5.5. Suppose u0 2 B(f;A;B) and the uxes f;A;B are locally Lipschitz continuous. Assume alsothat A satis�es the structural conditionA(�1) = �1; A(+1) = +1:Then the entire sequence fu�g de�ned by (21) and (57) converges in L1loc(QT ) and pointwise a.e. in QT to aBV entropy weak solution u of the initial value problem( @tu+ @xf(u) = @xA(b(u)@xu); (x; t) 2 QT = R� h0; T i ;u(x; 0) = u0(x); x 2 R;where A(s) = Z s0 a(�) d�; a(s) � 0; b(s) � 0:Remark. From Lemma 5.2 it is clear that the following results hold without assuming (6): There is a functionu(x; t) 2 L1(QT ) \BV (QT ) and a function A(x; t) 2 L1(QT ) such thatu�(x; t)! u(x; t); in L1loc(QT ) and pointwise a.e. in QT ,A (@xw�) �* A(x; t); in L1(QT ).Furthermore, in view of Lemma 5.4 it follows that the following integral inequality is satis�ed(80) ZZQT �ju� cj@t�+ sign(u� c)�f(u) � f(c) � A�@x��dt dx+ ZRju0 � cj dx � 0:We let C(0; T ;L1(R)) denote the usual Bochner space consisting of all continuous functions u : [0; T ]! L1(R)for which the norm kukC(0;T ;L1(R)) = supt2[0;T ] ku(t)kL1(R) is �nite. A closer inspection of the arguments leadingto Theorem 5.5 will reveal that fU�(t)g converges in C(0; T ;L1(R)) to the unique BV entropy weak solutionu(t), with u(0) = u0, of the initial value problem (1). A reexamination of the proofs leading to Theorem 5.5also shows that we have proved the following result on existence and properties of solutions of (1):Corollary 5.6. Let f and A;B be locally Lipschitz continuous. Then for any initial function u0 2 B(f;A;B)there exists a BV entropy weak solution u 2 C(0; T ;L1(R)) of the initial value problem (1). Denoting thissolution by Stu0, we have the following properties:(1) t! Stu0 is Lipschitz continuous into L1(R) and kStu0kBV (R) � u0kBV (R),(2) Stu0 � Stv0 L1(R) � ku0 � v0kL1R),(3) u0 � v0 implies Stu0 � Stv0,(4) m � u0 �M implies m � Stu0 �M .Acknowledgment. We thank Nils Henrik Risebro for his reading and criticism of the manuscript.References1. R. B�urger, W. L. Wendland,Mathematical problems in sedimentation, Proc. of the Third Summer Conf. `Numerical Modellingin Continuum Mechanics', Prague (Czech Republic), (1997).2. F. Concha, R. B�urger, Mathematical model and numerical simulation of the settling of occulated suspensions, Preprint,University of Stuttgart (1997).3. M. G. Crandall, T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J.Math. 93 (1971), 265{298.4. M. G. Crandall, A. Majda, Monotone di�erence approximations for scalar conservation laws, Math. Comp. 34 (1980), 1{21.5. M. G. Crandall, L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78(1980), 385{390.6. K. Deimling, Ordinary Di�erential Equations in Banach Spaces, Springer-Verlag, New York (1977).7. J. R. Esteban, J. L. Vazquez, On the equation of turbulent �ltration in one-dimensional porous media, Nonlinear Analysis,TMA 10(11) (1986), 1303-1325.8. S. Evje, K. Hvistendahl Karlsen, Monotone di�erence approximations of BV solutions to degenerate convection-di�usionequations, Submitted to Siam J. Num. Anal., Preprint, University of Bergen (1998).9. S. Evje, K. Hvistendahl Karlsen, Degenerate convection-di�usion equations and implicit monotone di�erence schemes, Sub-mitted to the Proc. of the Seventh Int. Conf. on Hyp. Prob., Preprint, University of Bergen (1998).10. A. S. Kalashnikov, Cauchy problem for second order degenerate parabolic equations with nonpower nonlinearities, J. SovietMath. 33(3) (1986), 1014{1025.11. S. N. Kruzkov,Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications,Mat. Zametki 6 (1969), 97{108.

24 S. EVJE, K. H. KARLSEN12. A. Kurganov, D. Levy, P. Rosenau,On Burgers-type equations with non-monotonic dissipative uxes, Comm. Pure Appl. Math.51 (1998), 443{473.13. A. Kurganov, P. Rosenau, E�ects of a saturating dissipation in Burgers-type equations, Comm. Pure Appl. Math. 50(8) (1997),753{771.14. Y.-G. Lu, H�older estimates of solutions of some doubly nonlinear degenerate parabolic equations, Submitted to Arch. RationalMech. Anal. (1998).15. B. J. Lucier, On non-local monotone di�erence schemes for scalar conservation laws, Math. Comp. 47 (1986), 19{36.16. Y. Wang, J. Yin, Properties on interface of solutions for a doubly degenerate parabolic equation, J. Partial Di�erentialEquations9(2) (1996), 186{192.17. Z. Wu, J. Yin, Some properties of functions in BVx and their applications to the uniqueness of solutions for degeneratequasilinear parabolic equations, Northeastern Math. J. 5 (1989), 395{422.18. J. Yin, On a class of quasilinear parabolic equations of second order with double-degeneracy, J. Partial Di�erential Equations3(4) (1990), 49{64.