In H. Freistüler and G. Warnecke, editors, "Hyperbolic problems: theory, numerics, applications", Int. Series of Numerical Mathematics, Vol. 140 (2001), 337-346, Birkhäuser Verlag A continuous dependence result for nonlinear degenerate parabolic equations with spatially dependent flux function S. Evje, K. H. Karlsen, and N. H. Risebro Abstract. We study entropy solutions of nonlinear degenerate parabolic equations of form u t + div k(x)f(u) = A(u), where k(x) is a vector-valued function and f(u);a(u) are scalar functions. We prove a result concerning the continuous dependence on the initial data, the flux function k(x)f(u), and the diffusion function A(u). This paper complements previous work [7] by two of the authors, which contained a continuous dependence result concerning the initial data and the flux function k(x)f(u). 1. Introduction In this paper we are concerned with entropy solutions of the initial value problem u t + div k(x)f(u) = A(u); u(x; 0) = u 0 (x); (1.1) for (x; t) 2 = R d (0;T) with T > 0 fixed. In (1.1), u(x; t) is the scalar unknown function that is sought, k(x)f(u) is the flux function, and A = A(u) is the diffusion function. We always assume that k : R d! R d, f : R! R, and A : R! R satisfy ( k 2 W 1;1 loc (Rd ); k; divk 2 L 1 (R d ); f 2 Lip loc (R); f(0) = 0; (1.2) A 2 Lip loc (R) and A( ) is nondecreasing with A(0) = 0: Since A 0 ( ) isallowed to be zero on an interval [ff; fi] (the scalar conservation law is a special case of (1.1)), solutions may become discontinuous in finite time even with a smooth initial function. Consequently, one needs to interpret (1.1) in the weak sense. However, weak solutions are in general not uniquely determined by their initial data and an entropy condition must be imposed to single out the physically correct solution. Definition 1.1. A measurable function u(x; t) is an entropy solution of (1.1) if D.1 u 2 L 1 ( ) L 1 ( ) C(0;T; L 1 (R d )) and A(u) 2 L 2 (0;T; H 1 (R d )).
2 Evje, Karlsen, and Risebro D.2 For all c 2 R and all non-negative test functions in C 1 0 ( ), ZZ ju cjffi t + sign (u c) k(x) f(u) f(c) rffi + ja(u) A(c)j ffi sign (u c)divk(x)f(c)ffi dt dx 0: (1.3) D.3 Essentially as t # 0, ku( ;t) u 0 (x)k L 1 (R d )! 0. Following Kru»zkov [9] and the recent work of Carrillo [3], two of the authors proved in [7] a uniqueness result for entropy solutions of the more general equation u t +divf(x; t; u) = A(u)+q(x; t; u); (1.4) where the flux function f = f(x; t; u) may have a non-smooth spatial dependence, see [7] for the precise assumptions on f and q in (1.4). Moreover, in the L 1 (0;T; BV (R d )) class of entropy solutions, the authors of [7] proved continuous dependence on the initial function u 0 and flux function in the case f(x; t; u) = k(x)f(u). However, in [7] the question of continuous dependence with respect to the diffusion function A was left open. Recently, Cockburn and Gripenberg [4] have obtained such a result when k(x) = 1. Their result does not, however, imply uniqueness of the entropy solution (from reasons that will become apparent later). Let us also mention that results regarding continuous dependence on the fluxfunction in scalar conservation laws (A 0 0) have been obtained in [11, 1, 8]. The purpose of the present paper is to combine the ideas in [7] with those in [4] and prove a version of Theorem 1.3 in [7] which also includes continuous dependence on the diffusion function A.To state our continuous dependence result, let us introduce v t +div l(x)g(v) = B(v); v(x; 0) = v 0 (x): (1.5) We assume that l : R d! R, g : R! R, and B : R! R satisfy the same conditions as k; f; A, see (1.2). Wenow state our main result: Theorem 1.1. Let v; u 2 L 1 (0;T; BV (R d )) be the unique entropy solutions of (1.5), (1.1) with initial data v 0 ;u 0 2 L 1 (R d ) L 1 (R d ) BV (R d ), respectively. Suppose that v; u take values in in the closed interval I ρ R and define V v = sup t2(0;t ) jv( ;t)j BV (R d ). Suppose k 2 Lip(Rd ) and divk 2 BV (R d ). Then for almost all t 2 (0;T), kv( ;t) u( ;t)k L 1 (R d )»kv 0 u 0 k L 1 (R d ) + C Conv t kl kk L 1 (R d) + jl kj BV (R d) + kg fk L 1 (I) + kg fk Lip(I) + C Diff p t fl flfl p B 0 p A 0 fl flfl L1 (I) ; (1.6) for some constants C Conv ;C Diff. Here C Conv depends on V v ; kkk L 1 (R d ); jkj BV (R d ); kgk L 1 (I) ; kgk Lip(I) and C Diff depends on V v ; kkk Lip(R d ); jdivkj BV (R d ); kgk L 1 (I) ;
Degenerate parabolic equations 3 kgk Lip(I). The explicit form of the constants C Conv ;C Diff proof of Theorem 1.1. can be traced from the We remark that existence of BV ( )entropy solutions of (1.1) (or (1.4)) can be proved by the vanishing viscosity method provided f; A; q; u 0 are sufficiently smooth, see Vo'lpert and Hudjaev [13]. Existence of L 1 (0;T; BV (R d )) entropy solutions of (1.1) is guaranteed if divk 2 BV (R d ). This follows from the results obtained by Karlsen and Risebro [6], who proved convergence of finite difference schemes for degenerate parabolic equations with rough coefficients. For an overview of the literature dealing with numerical methods for approximating entropy solutions of degenerate parabolic equations, we refer to [5]. In this connection, we should mention that the arguments used to prove Theorem 1.1 can be used to prove error estimates for numerical methods. This will be discussed elsewhere. For later use, we mention that the results in [6] can be used to prove the existence of L 1 (0;T; BV (R d )) entropy solutions of (1.1) bythevanishing viscosity method. To this end, consider the uniformly parabolic problem u μ t +div k(x)f(u μ ) = A(u μ )+μ u μ ; u μ (x; 0) = u 0 (x); (1.7) for μ>0. Provided k; f; A; u 0 are sufficiently smooth, it is well known that there exists a unique classical (and hence entropy) solution of (1.7) which possesses all the continuous derivatives occurring in the partial differential equation in (1.7). Using the space and time translation estimates derived in [6], it is not difficult to show that u μ converges in L 1 loc ( )asμ # 0toanentropy solution u of (1.1) (see also Vo'lpert and Hudjaev [13]). Convergence of the viscosity method and smoothness of the solution u μ of (1.7) will be used in the proof of Theorem 1.1. Finally, to relaxthe smoothness assumptions on k; f; A; u 0 needed by the vanishing viscosity method to those actually required by Theorem 1.1, one can approximate k; f; A; u 0 by smoother functions and then use Theorem 1.1 to pass to the limit as the smoothing parameter tends to zero. We will not go into further details about this limiting operation but instead leave this as an exercise for the interested reader. Also, in this paper we have exclusively treated the initial value problem but it is possible to treat various initial-boundary value problems. For work in this direction, we refer to Bürger, Evje, and Karlsen [2] and Rouvre and Gagneux[12]. Before ending this section, we present an immediate corollary of Theorem 1.1 concerning the convergence rate of the viscosity method. Corollary 1.1. Let u 2 L 1 (0;T; BV (R d )) be theuniqueentropy solution of (1.1) with initial data u 0 2 L 1 (R d ) L 1 (R d ) BV (R d ) and let u μ be the corresponding viscous approximation of u, i.e., u μ is the unique classical solution of (1.7). Suppose k 2 Lip(R d ) and divk 2 BV (R d ). Then for almost all t 2 (0;T), for some non-negative constant C. ku( ;t) u μ ( ;t)k L 1 (R d )» C p μ; (1.8) The remaining part of this paper is devoted to proving Theorem 1.1.
4 Evje, Karlsen, and Risebro 2. Some preliminaries Let u be an entropy solution of (1.1). It easy to see from Definition 1.1 that the equality ZZ uffi t + k(x)f(u) ra(u) Λ rffi dt dx =0 (2.1) holds for all ffi 2 L 2 (0;T; H0(R 1 d )) W 1;1 (0;T; L 1 (R d )). Let h ; i denote the usual pairing between H 1 (R d ) and H0 1 (R d ). From (2.1), we conclude that @ t u 2 L 2 (0;T; H 1 (R d )), so that the equality (2.1) can restated as Z T Ω @t u; ffi ff ZZ k(x)f(u) Λ dt + ra(u) rffi dt dx =0 (2.2) 0 for all ffi 2 L 2 (0;T; H0(R 1 d )) W 1;1 (0;T; L 1 (R d )). The fact that an entropy solution u satisfies (2.2) is important for the uniqueness proof [3, 7]. We recall that after the work of Carrillo [3], the uniquness proof for entropy solutions of degenerate parabolic equations has become very similar to the doubling of variables" proof introduced by Kru»zkov [9] many years ago for first order hyperbolic equations. However, to apply the doubling of variables" device to second order equations, one needs a version of an important lemma stated and proved first in [3]. This lemma indentifies a certain entropy dissipation term that must be taken into account if the doubling device" is going to work. Before stating this lemma, we introduce some notation. For ">0, set 8 >< 1; fi <"; sign " (fi) = fi="; "» fi» "; >: 1 fi >": Moreover, we leta 1 : R! R denote the unique left-continuous function satisfying A 1 (A(u)) = u for all u 2 R. Define E = Φ r : A 1 ( ) discontinuous at r Ψ. Note that E is associated with the set of points Φ u : A 0 (u) = 0 Ψ at which the operator u 7! A(u) is degenerate elliptic. We can now state following lemma: Lemma 2.1 ([7]). Let u be an entropy solution of (1.1). Then, for any non-negative ffi 2 C 1 0 ( ) and any c 2 R such that A(c) =2 E, wehave ZZ ju cjffi t +sign(u c) k(x) f(u) f(c) ra(u) Λ rffi ZZ fi fi sign (u c) divk(x)f(c)ffi dt dx =lim fi ra(u) fi 2 sign 0 " (A(u) A(c)) ffidtdx: (2.3) Note that if (1.1) is uniformly parabolic (A 0 > 0), then the set E is empty and a weak solution is automatically an entropy solution. The idea of the proof of Lemma 2.1 is to use [sign " (A(u) A(c)) ffi] 2 L 2 (0;T; H 1 0(R)) as a test function
Degenerate parabolic equations 5 in (2.2) together with a weak chain rule" to deal with the time derivative and subsequently sending " # 0. We refer to [7] for details on the proof of Lemma 2.1 (see Carrillo [3] when k(x) 1). Although the identification of the entropy dissipation term (i.e., the righthand side of (2.3)) is the cornerstone of the uniqueness proof as well as the proof of continuous dependence on the fluxfunction k(x)f(u), it seems difficult to obtain continuous dependence on the diffusion function A(u) with this form of the dissipation term. However, it is possible to derive a version of (2.3) in which a different (form of the) entropy dissipation term appears. But this seems possible only if u is smooth or at least belongs to L 2 (0;T; H 1 (R d )). Consequently,Theorem 1.1 does not yield uniqueness of the entropy solution. Provided (1.1) is uniformly parabolic and hence admits a unique classical solution, the following version of Lemma 2.1 holds: Lemma 2.2. Suppose (1.1) is uniformly parabolic (i.e., A 0 > 0). Let u be a classical solution of (1.1). Then, for any non-negative ffi 2 C 1 0 ( ) and any c 2 R, wehave ZZ ju cjffi t + sign (u c) k(x) f(u) f(c) ra(u) Λ rffi ZZ sign (u c) divk(x)f(c)ffi dt dx =lim A 0 fi fi (u) fi rufi 2 sign 0 " (u c) ffidtdx: (2.4) This lemma can proved by using [sign " (u c) ffi] as a test function (this is indeed a test function since u is smooth) in (2.2) and then sending " # 0. The proof of Lemma 2.2 is similar to the proof of Lemma 2.1 and it is therefore omitted. Notice the difference between the entropy dissipation terms in (2.3) and (2.4). 3. Proof of Theorem 1.1 We are nowinterested in estimating the L 1 difference between the entropy solution v of (1.5) and the entropy solution u of (1.1). In view of the discussion in Section 1, we will prove Theorem 1.1 under the assumption that B 0 ;A 0 > 0 so that (1.5) and (1.1) become uniformly parabolic problems and therefore admit unique classical solutions. To treat the degenerate parabolic case (B 0 ;A 0 0), we proceed via the vanishing viscosity method, i.e., we replace A(u) and B(v) in (1.1) and (1.5) by A(u) + u and B(v) + v, respectively, and then send # 0. The argument given below uses Lemma 2.2 and Kru»zkov's idea of doubling the number of dependent variables together with a penalization procedure. It is inspired by Carrillo [3] and Cockburn and Gripenberg [4]. Strictly speaking, we could have carried out the argument below under the assumptions that the (entropy) solutions v; u belong to L 2 (0;T; H 1 (R d )), i.e., v; u need not be (entirely!) classical solutions. Although we do not prove this here, all functions below that are integrated can be proven to be integrable!
6 Evje, Karlsen, and Risebro Following [9, 10], we now specify a non-negative test function ffi 2 C 1 0 ( ). To this end, introduce a nonnegative function ffi 2 C 1 0 (R) which satisfies ffi(ff) =ffi( ff), ffi(ff) 0 for jffj 1, and R R ffi(ff) dff =1.For ρ 0 > 0, let ffi ρ 0(ff) = 1 ffi ff. ρ0 ρ0 Pick two (arbitrary but fixed) points ν; fi 2 (0;T). For any ff0 > 0, we define W ff 0(t) = H ff 0(t ν) H ff 0(t fi), where H ff 0(t) = R t 1 ffi ff0(s) ds. With 0 <ρ 0 < min(ν; T fi) andff 0 2 (0; min(ν ρ 0 ;T fi ρ 0 )), we define the test function ffi(x; t; y; s) =W ff 0(t)ffi ρ (x y)ffi ρ 0(t s), ρ>0. One should observe that ffi t + ffi s = ffi ff 0(t ν) ffi ff 0(t fi) Λ ffi ρ (x y)ffi ρ 0(t s) andr x ffi + r y ffi 0. Applying Lemma 2.2 with v = v(x; t) and c = u(y; s) and then integrating the resulting equation with respect to (y; s) 2,weget jv ujffi t +sign(v u) l(x) g(v) g(u) r x B(v) Λ r x ffi = lim sign (v u)div x l(x)g(u)ffi dο B 0 fi (v) fi rx v fi fi 2 sign 0 " (v u) ffidο; (3.1) where dο = dt dx ds dy. Similarly, applying Lemma 2.2 with u = u(y; s) and c = v(x; t) and then integrating the resulting equation with respect to (x; t) 2, we get ju vjffi t +sign(u v) k(y) f(u) f(v) r y A(u) Λ r y ffi = lim sign (u v) div y k(y)f(v)ffi dο A 0 fi (u) fi ry u fi fi 2 sign 0 " (u v) ffidο: Following [7] when adding (3.1) and (3.2), weget jv uj ffi t + ffi s + IConv I 1 Diff dο = lim = lim B 0 (v) fi fi rx v fi fi 2 + A 0 (u) fi fi ry u fi fi 2 sign 0 " (v u) ffidο; p B 0 (v)r x v p 2 A 0 (u)r y u +2 p B 0 (v) p A 0 (u)r x v r y u sign 0 " (v u) ffidο;» lim I 2 dο; Diff (3.2) (3.3)
Degenerate parabolic equations 7 where I Conv = sign (v u) div x k(y)f(u) l(x)g(u) ffiλ div y l(x)g(v) k(y)f(v) ffiλ ; I 1 Diff =sign(v u) r x B(v) r x ffi +sign(u v) r y A(u) r y ffi; I 2 Diff = 2 p B 0 (v) p A 0 (u)r x v r y u sign 0 " (v u) ffi: By the triangle inequality, we get jv(x; t) u(y; s)j ffit + ffi s dο» I + Rt + R x ; where I = jv(y; t) u(y; t)j ffiff 0(t ν) ffi ff 0(t fi) Λ ffi ρ (x y)ffi ρ 0(t s) dο; R t = R x = ju(y; t) u(y; s)j ffiff 0(t ν) ffi ff 0(t fi) Λ ffi ρ (x y)ffi ρ 0(t s) dο; jv(x; t) v(y; t)j ffiff 0(t ν) ffi ff 0(t fi) Λ ffi ρ (x y)ffi ρ 0(t s) dο: It is fairly easy to see that lim ρ 0#0 R t =0and Z Z lim R x = jv(x; fi) v(y; fi)j jv(x; ν) v(y; ν)j ffi ρ (x y) dx dy; R d R d» ρ sup jv( ;t)j BV (R ); d lim I = kv( ;fi) u( ;fi)k L 1 (R d ) kv( ;ν) u( ;ν)k L 1 (R ): d We therefore get the following approximation inequality kv( ;fi) u( ;fi)k L 1 (R d ) where»kv( ;ν) u( ;ν)k L 1 (R d ) + ρ sup jv( ;t)j BV (R d ) + E Conv = I Conv dο; E Diff = I 1 Diff dο lim I 2 lim ff0;ρ0#0 E Conv + E Diff ; Diff dο =: E 1 Diff E 2 Diff: (3.4)
8 Evje, Karlsen, and Risebro Observe that E 2 Diff = lim = lim = lim 2 2 Z u 2r y sign 0 " (v ο) p A 0 (ο) dοp B 0 (v)r x vffi dο v Z u sign 0 " (v ο) p A 0 (ο) dοp B 0 (v)r x vr y ffidο v Z u sign 0 " (v ο) p A 0 (ο) dοp B 0 (v)r x vr x ffidο v p = 2sign (v u) B 0 (v) p A 0 (v)r x vr x ffidο: (3.5) Writing sign (u v) r y A(u) =r y fi fia(u) A(v) fi fi and using integration parts twice as well as the relation xffi = yffi, one can easily show that sign (u v) r y A(u) r y ffidο = sign (v u) r x A(v) r x ffidο: (3.6) From (3.5), (3.6), and fl flr x ffi ρ (x y) fl fll 1 (R d ) = 2d ρ,weget lim E Diff» lim» lim sign (v u) B 0 (v)+a 0 (v) 2 p B 0 (v) p A 0 (v) r x vr x ffidο p B 0 (v) p 2 A 0 fi (v) firx v fi fi fi fi rx ffi ρ (x y) fi fi Wff0(t)ffi ρ0(t s) dο 2d fl fl» (fi ν) sup jv( ;t)j BV (R d ) fl p B ρ 0 (v) p fl A 0 fl (v) fl 2 L1 (R ): d (3.7)
Degenerate parabolic equations 9 Arguing exactly as in [7], one can prove that E Conv = sign (v u) div y k(y) f(v) g(v) div y k(y) div x l(x) g(v) Λ + k(y) l(x) r x G(v; u)+k(y) r x F (v; u) G(v; u) ffidο = sign (v u) div y k(y) f(v) g(v) div x k(x) div x l(x) g(v) Λ + k(x) l(x) r x G(v; u)+k(y) r x F (v; u) G(v; u) ffidο + sign (v u) divx k(x) div y k(y) g(v)ffidο + sign (v u) k(y) k(x) rx G(v; u)ffidο =: E 1 Conv + E 2 Conv + E 3 : Conv (3.8) Following [7], we derive the estimate lim E 1 Conv» (fi ν) kgk Lip sup jv( ;t)j BV (R d ) kk lk L 1 (R d ) + kgk L 1 (I) jk lj BV (R d ) + jkj BV (R d ) kf gk L 1 (I) + kkk L 1 (R d ) sup jv( ;t)j BV (R )kf gk d Lip(I) : (3.9) Taking into account divk 2 BV (R d )andk 2 Lip(R d ), it is easy to show that lim E 2»jdivkj Conv BV (R )kgk d L 1 (I) (fi ν)ρ; (3.10) lim E 3 Conv»kkk Lip(R d )kgk Lip(I) sup jv( ;t)j BV (R d )(fi ν)ρ: (3.11) Inserting (3.7), (3.9), (3.10), and (3.11) into (3.4), minimizing the result with respect to ρ>0, and subsequently sending ν # 0, we get (1.6). This concludes the proof of Theorem 1.1 when B 0 ;A 0 > 0. Note that (1.6) does not depend on the smoothness of v; u. Hence the proof in the general case B 0 ;A 0 0 can proceed via the L 1 convergence of the viscosity method (see Section 1). References [1] F. Bouchut and B. Perthame. Kru»zkov's estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc., 350(7):2847 2870, 1998.
10 Evje, Karlsen, and Risebro [2] R. Bürger, S. Evje, and K. H. Karlsen. On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl., 247(2):517 556, 2000. [3] J. Carrillo. Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal., 147(4):269 361, 1999. [4] B. Cockburn and G. Gripenberg. Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations, 151(2):231 251, 1999. [5] M. S. Espedal and K. H. Karlsen. Numerical solution of reservoir flow modelsbased on large time step operator splitting algorithms. In Filtration in Porous Media and Industrial Applications (Cetraro,Italy,1998), volume 1734 of Lecture Notes in Mathematics, pages 9 77. Springer, Berlin, 2000. [6] K. H. Karlsen and N. H. Risebro. Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. Mathematical Modelling and Numerical Analysis, 35(2):239 270, 2001. [7] K. H. Karlsen and N. H. Risebro. On the uniqueness and stability ofentropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint, Department of Mathematics, University of Bergen, 2000. [8] R. A. Klausen and N. H. Risebro. Stability of conservation laws with discontinuous coefficients. J. Differential Equations, 157(1):41 60, 1999. [9] S. N. Kru»zkov. First order quasi-linear equations in several independent variables. Math. USSR Sbornik, 10(2):217 243, 1970. [10] N. N. Kuznetsov. Accuracy of some approximative methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. and Math. Phys. Dokl., 16(6):105 119, 1976. [11] B. J. Lucier. A moving mesh numerical method for hyperbolic conservation laws. Math. Comp., 46(173):59 69, 1986. [12] É. Rouvre and G. Gagneux. Solution forte entropique de lois scalaires hyperboliquesparaboliques dégénérées. C. R. Acad. Sci. Paris Sér. I Math., 329(7):599 602, 1999. [13] A. I. Vol'pert and S. I. Hudjaev. Cauchy's problem for degenerate second order quasilinear parabolic equations. Math. USSR Sbornik, 7(3):365 387, 1969. RF-Rogaland Research, Thormfihlensgt. 55, N 5008 Bergen, Norway E-mail address: Steinar.Evje@rf.no Department of Mathematics, University of Bergen Johs. Brunsgt. 12, N 5008 Bergen, Norway E-mail address: kennethk@mi.uib.no Department of Mathematics, University of Oslo P.O. Box1053, Blindern, N 0316 Oslo, Norway E-mail address: nilshr@math.uio.no