KRUŽKOV S ESTIMATES FOR SCALAR CONSERVATION LAWS REVISITED

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICA SOCIETY Volume 350, Number 7, July 1998, Pages S (98) KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED F. BOUCHUT AND B. PERTHAME Abstract. We give a synthetic statement of Kružov-type estimates for multidimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in h 1/4 nown for finite volume methods on unstructured grids. es estimations de Kružov pour les lois de conservation scalaires revisitées Résumé Nous donnons un énoncé synthétique des estimations de type de Kružov pour les lois de conservation scalaires multidimensionnelles. Nous l appliquons pour obtenir d estimations nombreuses pour problèmes différents d approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en h 1/4 connue pour les méthodes de volumes finis sur des maillages non structurés. 1. Introduction and main results This paper deals with error estimates for a multidimensional scalar conservation law { t v +divf(v)=0 in ]0, [ R N, (1.1) v(0,x)=v 0 (x). Here the flux function f : R R N is assumed to be ipschitz continuous for simplicity. The equation has to be supplemented with the entropy inequalities, for any S : R R convex and ipschitz: (1.2) t S(v)+divη(v)0 in ]0, [ R N, with η = S f. Throughout the paper, all partial differential equations and inequations are understood in the usual sense of distributions. After the wors of P.D. ax [12], O.A. Oleini [15], A.I. Vol pert [20], S.N. Kružov [10], it is well-nown that for any v 0 1 loc (RN ), there exists a unique solution v C([0, [, 1 loc (RN )) to (1.1)-(1.2). We are given an approximate solution u and we wish to estimate the difference u v. The famous method of S.N. Kružov [10] has been used in many situations to obtain such estimates. It relies on a doubling of the variables and on a penalization Received by the editors July 8, Mathematics Subject Classification. Primary 65M15, 3565, 35B30. Key words and phrases. Multi-dimensional scalar conservation laws, error estimates, entropy inequalities, finite volumes. Mots-clés. ois de conservation scalaires multidimensionnelles, estimations d erreur, inégalités d entropie, volumes finis c 1998 American Mathematical Society

2 2848 F. BOUCHUT AND B. PERTHAME procedure. When applied to numerical approximations (such as finite volumes or finite elements), following the ideas of N.N. Kuznetsov [11], it becomes extremely intricate because the technicalities due to the large number of variables are added to notational difficulties for the numerical approximation itself. This is especially true for unstructured grids. In this context, the derivation of the rate of convergence for first-order finite volumes methods, in several dimensions and for unstructured grids, is still complicated. On the other hand the final result, a convergence rate in h 1/4 rather than h 1/2 in one space dimension (see R. Sanders [17]), is very easy to explain. This loss just comes from the BV estimate, which blows lie h 1/2 (while it is bounded in one dimension); see S. Champier, T. Gallouët, R. Herbin [3], B. Cocburn, F. Coquel, P. e Floch [4][5], A. Szepessy [18], J.-P. Vila [19], B. Cocburn, P.-A. Gremaud [7], R. Eymard, T. Gallouët, R. Herbin [9]. For fluctuation splitting schemes this also applies; see B. Perthame [16]. In order to simplify these proofs, we propose to formalize Kružov s method as follows. We assume that u solves the entropy inequalities with error terms which are partial derivatives. For R (1.3) t u +divsgn(u )[f(u) f()] t G +divh +K + 2 (ij) x i in ]0, [ R N, where G,H (j),k, (ij) measures, are local Radon measures, and satisfy, in the sense of (1.4) G (t, x) α G (t, x), H (j) K (t, x) α K (t, x), (ij) (t, x) α(j) (t, x), H (t, x) α (ij) (t, x), with α G, α (j) H, α K, α (ij) are non-negative -independent Radon measures. Using Kružov s method, we claim that one can estimate u v 1 in terms of α G Mt,x, α H Mt,x, etc. And it is the very striing and fundamental idea behind all the error estimates that the right estimates are always of the type u v 1 C sup G Mt,x + C sup H Mt,x +... (see the precise statement in Theorem 2.1). The derivatives in the error terms are paid, for the final estimate, only by a square root of the appropriate norm, which is taen in M t,x ( ). This is the main contribution of this paper, to indicate how to reduce the estimates to the form above. More precisely, we claim that in many situations, it is not necessary to perform the doubling of variables explicitly it is enough to use the abstract estimate of Theorem 2.1, and this greatly improves the understanding. This strategy was initiated in a paper by the authors and C. Bourdarias [2]. This approach is shown to be successful in the most classical situations, which we develop independently in 3, 4, 5. Moreover, we are able to improve the corresponding results. For multidimensional monotone finite volume methods, it is also possible to write an equation of the form (1.3) see the recent paper of R. Eymard, T. Gallouët, R. Herbin [9].

3 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2849 In 3, we study the following problem. We assume that u is an entropy solution of (1.5) t u +divg(u)=0 in ]0, [ R N, with g another ipschitz flux function. Then using Theorem 2.1, we are able to estimate u v 1 in terms of g f. We indeed recover the estimate in lip(g f) that was used by C.M. Dafermos [8] and by B.J. ucier [14] in the context of front tracing methods. But our formalism also enables to get a new estimate, in g f 1/2. We refer to 3 for a precise statement. In 4, we consider the nonlinear diffusion equation (1.6) t u +divf(u) φ(u)=0 in ]0, [ R N, with φ : R R a ipschitz and nondecreasing function. We refer to Ph. Benilan, R. Gariepy [1] for recent results on that equation. We here consider an entropy solution in the sense that for any S : R R convex and ipschitz, (1.7) t S(u)+divη(u) ν(u)0 in ]0, [ R N, with η = S f and ν = S φ. It is easy to see that it is equivalent to requiring that (1.7) holds for the entropies S (ξ) = ξ, R: (1.8) t u +divsgn(u )[f(u) f()] φ(u) φ() 0 in ]0, [ R N. Notice also that the equation (1.6) is recovered by letting ±. Our approach allows us to recover the usual estimate of u v 1 in ε 1/2, i.e., lip(φ) 1/2 for v, u t (BV (RN )), and also the recent result of B. Cocburn, P.-A. Gremaud [6] which states that v t (BV (RN )) and only u t (1 (R N )) is enough to get the same estimate. Moreover our result (see (ii) below) explains how the regularity of φ is involved in this matter, and we also give a new result for irregular diffusions φ. We have the following results. Theorem 1.1. Assume that u loc ([0, [,1 (R N )) is right-continuous with values in 1 loc (RN ) and is an entropy solution in the sense of (1.8). etvbe an entropy solution of (1.1) with initial data v 0 1 BV (R N ). Then, for any T 0, we have: (i) If TV(u(t,.)) V for any t 0, then (1.9) u(t,.) v(t,.) 1 u 0 v 0 1 +C TV(v 0 )V T lip(φ). (ii) If u(t,.) 1 U for any t 0, then, denoting Q =sup φ(ξ) φ(0) / ξ, ξ 0 (1.10) u(t,.) v(t,.) 1 u 0 v 0 1 +C N TV(v 0 ) 2/3 (TQU) 1/3. (iii) The following inequality holds: (1.11) u(t,.) v(t,.) 1 u 0 v 0 1 +CTV(v 0 ) Tlip(φ). Here and throughout this paper, the total variation TV(v 0 ) of a function v 0 BV (R N ) is defined by N TV(v 0 )= v 0 j=1 R x N j. The letter C denotes various absolute constants, while C N denotes a possible dependence in the dimension N.

4 2850 F. BOUCHUT AND B. PERTHAME Notice that if v is bounded, say a v b, then the ipschitz constant of φ in (1.11) can be replaced by lip (φ) by using Appendix A2 (ii) instead of (i) in the [a,b] proof (see 4). For the sae of completeness, let us also recall that another classical analysis leading to local estimates in lip(φ) 1/4 can be carried out when φ(ξ) =σξ, σ > 0, by using the entropy dissipation equation (for u 0 2 (R N )) u 2 ) t (σ 2 +divη(u) u2 = σ u 2, 2 with η (ξ) =ξf (ξ) (see [3], [18], [4], [5], [19], [7], [9] and 5). To deal with a general φ, the inequalities (1.7) are not sufficient since the entropy dissipation is neglected. In 5, we present a relaxation model towards piecewise constant functions. It does not involve notations as complicated as those of finite volumes, but leads to basically the same ind of theoretical difficulties. We show that error estimates can again be reduced to studying (1.3)-(1.4). We are given a general grid (C i ) i I of R N : I is a countable set; C i is a Borel set for any i I; C i C j =0fori j; R N =N i I C i with N =0;and (1.12) h=sup diam(c i ) <. i I For technical reasons we assume that the following 1 BV condition is satisfied: there exists a constant K h 0 such that (1.13) i I I Ci BV (R N ),TV(I Ci )K h C i. Notice that this is not a regularity assumption on the grid since K h is allowed to blow up very fast when h 0. It only means that there is no degeneracy at infinity. We define the piecewise constant projector P 0 : 1 loc (RN ) 1 loc (RN )by (P 0 u) Ci = 1 (1.14) u. C i C i We consider for a given ε>0 the unique solution u C([0, [, 1 (R N )) of t u +divf(u)= P0 u u (1.15) in ]0, [ R N, ε u(0,.)=u 0 1 (R N ), with the entropy inequalities t S(u)+divη(u)S (u) P0 u u (1.16) in ]0, [ R N ε for any S convex, ipschitz and C 1 (or equivalently for any entropy S (ξ) = ξ ). Again we may estimate the difference from the exact solution. Theorem 1.2. With the above notations and assumptions, if u (R N ) then P 0 u u T ω (1.17) ε 2ε u0 2 ]0,T [ ω for any T>0and any bounded subset ω of R N. Denoting by v an entropy solution of (1.1) with initial data v 0 BV (R N ), we have for any T > 0, x 0 R N and

5 R>0 (1.18) KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2851 x x 0 <R u(t,x) v(t,x) dx u 0 (x) v 0 (x) dx x x 0 <R+MT+ Rl + C N l [TV(v 0 ) R+ u 0 2 RN/4 1/2 ( R ] N/4 + l N/4 ) with M = lip(f), R = R + MT +h and l = h T/ε. In other words, the estimate is of order h 1/2 /ε 1/4.Whenε h, which is similar to the limitation imposed by a CF condition for finite volumes, we recover the usual rate in h 1/4. However, the estimate (1.18) is not really interesting because we have in mind to approximate the solution v by a piecewise constant function. Thus, it is more appropriate to estimate P 0 u v. This can be performed by combining (1.17) and (1.18), and yields an estimate in 1 (]0,T[ B(x 0,R)). Notice also that it is very natural to choose u 0 = P 0 v 0. Then, assuming that v BV (R N ), the 2 norm of u 0 in (1.18) can be majorized by v 0 2. Thus, if the grid is regular in the sense that C i is open and convex, and (1.19) diam(c i ) N+1 κ C i h, then the initial error can be estimated by (1.20) u 0 v 0 1 (R N ) C N κt V (v 0 )h. This is just a straightforward application of the Poincaré-Wirtinger inequality for a convex domain. The paper is organized as follows. In 2 we state and prove our abstract version of Kružov s estimates. In 3, 4 and 5 we treat in detail the above independent applications. The appendix is devoted to two independent lemmas. The first deals with a special inverse of the div operator which furnishes the lin between finite volumes and the formulation (1.3). It is needed for the proof of Theorem 1.2. The second deals with a general estimate on the gradient [g(u)], where g is ipschitz and u BV. It is used frequently in this paper. 2. An error estimate for partial derivatives in the right-hand side This section is devoted to the main estimate of this paper, the reformulation of Kružov type estimates. We give two versions of the result. Theorem 2.1. et u, v loc ([0, [,1 loc (RN )) be right-continuous with values in 1 loc (RN ). Assume that u solves the entropy inequalities with right-hand side (1.3)-(1.4), and that v is an exact solution: for R (2.1) t v +divsgn(v )[f(v) f()] 0 in ]0, [ R N. Moreover, assume that (2.2) α G loc ([0, [,1 loc (RN )). Then, for any T 0,x 0 R N,R>0, >0,δ >0,ν 0, denoting

6 2852 F. BOUCHUT AND B. PERTHAME (2.3) M = lip(f), B t = B(x 0,R+M(T t)+ +ν), we have (2.4) u(t,x) v(t,x) dx x x 0 <R B 0 u(0,x) v(0,x) dx + C(E t + E x + E G + E H + E K + E ), with C an absolute constant and (2.5) E t = E x = sup 0<s t<δ t=0 or T sup h < t=0 or T v(s, x) v(t, x) dx, B t B t v(t, x + h) v(t, x) dx, (2.6) E K = x B t 0<tT E = 1 2 α K (t, x), x B t 0<tT E H = 1 α (ij) (t, x), N j=1 x B t 0<tT α (j) H (t, x), (2.7) E G = ( 1+ T δ + MT ) +ν sup 0<t<2T B t α G (t, x) dx. Before we prove Theorem 2.1, some comments are in order. 1. If u is an exact solution, the right-hand side of (1.3) is zero. Hence, only the error terms E t and E x remain in (2.4). Therefore, we can choose ν =0and let δ, 0 so that by the assumed regularity of v we get E t,e x 0. We hence recover Kružov s estimate [10] (2.8) u(t,x) v(t,x) dx u(0,x) v(0,x) dx. x x 0 <R x x 0 <R+MT When the right-hand side of (1.3) is not zero, we can use the well-nown TVD property of the exact solution v. If v 0 BV (R N ), we obtain (2.9) E t MTV(v 0 )δ, E x TV(v 0 ). Then, we choose the time and space regularization parameters δ and so as to minimize the error terms in (2.4). The parameter ν either is chosen to be 0 for a local estimate or tends to + for a global one. 2. Our assumption that u and v are 1 loc right-continuous is motivated by numerical schemes; it is well suited for such problems. We refer to T.-P. iu, M. Pierre

7 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2853 [13] for uniqueness results in one dimension when the initial data is recovered only in wea For numerical schemes, the assumption (2.2) on α G is somehow related to an estimate on the time modulus of continuity of u. In fact, α G is comparable with t t u,with tthe time step (see [2]). However, when no such estimate is available, it is possible to get an estimate of u v 1 t,x with only the regularity of α G stated in (1.4) and an estimate of α G only for small t. In order to obtain this result, just average (2.4) with respect to T and then observe that when averaged, the last inequality of (2.28) in the proof below can be replaced by a suitable one since χ (t) Y ε(t)+y ε(t T). It is also possible to choose a different test function χ, seer.eymard,t.gallouët, R. Herbin [9]. Proof of Theorem 2.1. et us introduce two test functions, Φ Cc (]0, [ R N ), Φ 0, and ζ Cc (],0[ RN ), ζ 0, to be chosen later on, and set (2.10) ϕ(t, x, s, y) =Φ(t, x)ζ(t s, x y). This choice is slightly different from that of S.N. Kružov in [10], and is inspired by that of N.N. Kuznetsov [11] (who taes Φ = I 0<t<T ). Notice that ϕ Cc ((]0, [ R N ) 2 ). For each (s, y) ]0, [ R N and R, letustaeϕas a test function in (t, x) for (1.3). We get with (1.4) (2.11) [ ] u t ϕ +sgn(u )[f(u) f()] x ϕ (t, x) dtdx ]0, [ R N ]0, [ R N ]0, [ R N G t ϕ H x ϕ+k ϕ+ N α G t ϕ + j=1 α (j) H ϕ + α Kϕ + (ij) 2 ϕ (t, x) dtdx x i 2 ϕ x i (t, x) dtdx. α (ij) Similarly, for each (t, x) ]0, [ R N and l R, wetaeϕas a test function in (s, y) for (2.1), and obtain (2.12) [ ] v l s ϕ +sgn(v l)[f(v) f(l)] y ϕ (s, y) dsdy 0. ]0, [ R N Now we tae = v(s, y) in (2.11) and integrate with respect to (s, y); then we tae l = u(t, x) in (2.12) and integrate with respect to (t, x). By summing up the results we obtain

8 2854 F. BOUCHUT AND B. PERTHAME (2.13) [ u(t, x) v(s, y) t Φ(t, x) ] +sgn(u(t, x) v(s, y))[f(u(t, x)) f(v(s, y))] x Φ(t, x) ζ(t s, x y) dsdtdxdy [ α G (t, x) t Φ(t, x)ζ(t s, x y)+φ(t, x) t ζ(t s, x y) + N j=1 H (t, x) j Φ(t, x)ζ(t s, x y)+φ(t, x) j ζ(t s, x y) α (j) + α K (t, x)φ(t, x)ζ(t s, x y) + α (ij) (t, x) 2 ij Φ(t, x)ζ(t s, x y)+ i Φ(t, x) j ζ(t s, x y) R α. ] + j Φ(t, x) i ζ(t s, x y)+φ(t, x) ijζ(t 2 s, x y) dsdtdxdy et us now mae the choice of Φ precise. et θ>0andsety θ (t)= t Y θ (s) ds with Y θ (t) = 1 θ Y 1( t θ )andy 1 Cc (]0, 1[), Y 1 0and Y 1 =1. We introduce another parameter ε>0 and a function χ Cc (]0,T+ε[),χ 0, to be chosen later. We define Φ(t, x) =χ(t)ψ(t, x), (2.14) ( ψ(t, x) =1 Y θ x x0 R /2 M(T t) ) 0. Then Φ C (R R N )assoonasmε R+ /2. We have t Φ(t, x) =χ (t)ψ(t, x) Mχ(t)Y θ(), (2.15) x Φ(t, x) = χ(t)y θ () x x 0 x x 0. Therefore, by the ipschitz condition on f we get (2.16) u(t, x) v(s, y) t Φ(t, x) +sgn(u(t, x) v(s, y))[f(u(t, x)) [ f(v(s, y))] x Φ(t, x) = u(t, x) v(s, y) χ (t)ψ(t, x) χ(t)y θ () M u(t, x) v(s, y) x x ] 0 +sgn(u(t, x) v(s, y))[f(u(t, x)) f(v(s, y))] x x 0 u(t, x) v(s, y) χ (t)ψ(t, x). Together with (2.13), this yields (2.17) u(t, x) v(s, y) χ (t)ψ(t, x)ζ(t s, x y) dsdtdxdy R α. Now, by the triangle inequality we get (2.18) 0 I + R t + R x + R α

9 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2855 with (2.19) I = u(t, x) v(t, x) χ (t)ψ(t, x)ζ(t s, x y) dsdtdxdy, R t = v(t, y) v(s, y) χ (t) ψ(t, x)ζ(t s, x y) dsdtdxdy, R x = v(t, x) v(t, y) χ (t) ψ(t, x)ζ(t s, x y) dsdtdxdy. We now prescribe ζ to be a regularizing sequence (2.20) ζ(t, x) =ζ t (t)ζ x (x), ζ t,ζ x C c, 0, ζ t dt = ζ x dx =1, ζ t (t)= 1 δ ζt 1 (t δ ), supp ζt 1 ] 1, 0[, ζ x (x) = 1 ( x Nζx 1, supp ζ1 ) x B (0, 1/4). We also choose ζ x to be a product ζ x 1 (x) = N j=1 ζxj 1 (x j). For χ we tae (2.21) χ(t) =Y ε (t) Y ε (t T) so that 0 χ 1. We are going to tae the lim sup ε 0 in (2.18), θ fixed. We notice that (2.22) and obtain R t I x x0 <R+M(T t)+ /2 ψ(t, x) I x x0 <R+M(T t)+ /2+θ, y x 0 <R+M(T t)+3 /4+θ Then by right continuity of v at 0 and T, (2.23) lim sup R t ε 0 t=0,t y x 0 <R+M(T t)+3 /4+θ v(t, y) v(s, y) (Y ε (t)+y ε(t T))ζ t (t s) dsdtdy. v(t, y) v(s, y) ζ t (t s) dsdy 2E t just by choosing θ = /4+ν. For the term R x we have R x v(t, x) v(t, y) (Y ε (t)+y ε (t T))ζx (x y) dtdxdy, x x 0 <R+M(T t)+ /2+θ and by the same continuity property of v (2.24) lim sup R x ε 0 t=0,t x x 0 <R+M(T t)+ /2+θ 2E x. v(t, x) v(t, y) ζ x (x y) dxdy

10 2856 F. BOUCHUT AND B. PERTHAME For I we have, using (2.22), I = u(t, x) v(t, x) [Y ε (t) Y ε (t T )]ψ(t, x) dtdx u(t, x) v(t, x) Y ε (t) dtdx x x 0 <R+M(T t)+ /2+θ x x 0 <R+M(T t)+ /2 u(t, x) v(t, x) Y ε (t T ) dtdx, and by right continuity of u and v at 0 and T, lim sup I u(0,x) v(0,x) dx ε 0 x x (2.25) 0 <R+MT+ /2+θ u(t,x) v(t,x) dx. x x 0 <R+ /2 It now remains to estimate R α defined in (2.13). Using the bounds (2.26) x Φ(t, x) C θ, tφ(t, x) χ (t) +C M θ, and denoting by Ω the set (2.27) Ω= we get the estimates (2.28) { } t, x; 0 <t<t+ε, x x 0 <R+M(T t)+ /2+θ, α K (t, x)φ(t, x)ζ(t s, x y) α K (t, x), Ω α (j) H (t, x)φ(t, x) jζ(t s, x y) C α (j) H (t, x), Ω α G (t, x)φ(t, x) t ζ(t s, x y) C α G (t, x), δ Ω α (j) H (t, x) jφ(t, x) ζ(t s, x y) C α (j) H (t, x), θ α G (t, x) t Φ(t, x) ζ(t s, x y) C (1 + M(T + ε)/θ) sup 0<t<T +ε x x 0 <R+M(T t)+ /2+θ Ω α G (t, x) dx. For the second-order terms of R α, we notice that in the support of x Φ(t, x) we have x x 0 R+ /2 Mε+C 0 θ C 0 θ for some C 0 > 0. This yields 2 (2.29) Φ x i C θ 2.

11 We hence get (2.30) KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2857 α (ij) α (ij) (t, x)φ(t, x) 2 ij ζ(t s, x y) C 2 (t, x) iφ(t, x) j ζ(t s, x y) C θ α (ij) (t, x) 2 ij Φ(t, x) C ζ(t s, x y) θ 2 Ω Ω Ω α (ij) (t, x), α (ij) (t, x), α (ij) (t, x). Taing into account that θ = /4+ν, we obtain, by combining (2.28) and (2.30), (2.31) lim sup R α C(E G + E H + E K + E ), ε 0 and the lim sup ε 0 in (2.18) together with the estimates (2.23), (2.14), (2.25) gives the result. Notice that in Theorem 2.1 and its proof, we never use any specific property of the exact solution v. We only use the entropy inequalities (2.1) in the distributional sense for any R. Therefore, our result actually contains Kružov s uniqueness theorem with the stated regularity. Moreover, it is possible to modify the proof in order to also consider error terms for v. The result is the following. Proposition 2.2. We mae the same assumptions as in Theorem 2.1, except that we replace the inequalities (2.1) on v by (2.32) t v +divsgn(v )[f(v) f()] t G +divh +K + 2 (ij) x i in ]0, [ R N, where G, H (j), K, (ij) are local measures, and (2.33) G β G, H (j) β (j) H, K β K, (ij) β (ij), with β G,β (j) H,β K,β (ij) being -independent non-negative measures. Then we have the same estimate (2.4), with the additional terms (2.34) where (2.35) (2.36) E K = x B 0 0<tT +δ E = 1 2 C(E G + E H + E K + E ), β K (t, x), x B 0 0<tT +δ E G = 1 δ E H = 1 β (ij) (t, x), x B 0 0<tT +δ N j=1 β G (t, x). x B 0 0<tT +δ β (j) H (t, x),

12 2858 F. BOUCHUT AND B. PERTHAME Proof. The inequality (2.12) becomes (2.37) ]0, [ R N ]0, [ R N ]0, [ R N [ ] v l s ϕ +sgn(v l)[f(v) f(l)] y ϕ (s, y) dsdy G l s ϕ H l y ϕ+k l ϕ+ N β G s ϕ + j=1 β (j) H and gives a new term in (2.13). (2.38) ϕ y j + β Kϕ + (ij) l R β = [ β G (s, y)φ(t, x) t ζ(t s, x y) + N j=1 β (j) H (s, y)φ(t, x) jζ(t s, x y) 2 ϕ (s, y) dsdy y i y j β (ij) 2 ϕ y i y j (s, y) dsdy, + β K (s, y)φ(t, x)ζ(t s, x y) + β (ij) (s, y)φ(t, x) 2 ij ζ(t s, x y) ] dsdtdxdy. Since where Φ(t, x)ζ(t s, x y) 0wehave(s, y) Ω, with (2.39) Ω = { } s, y; 0 <s<t+ε+δ, y x 0 <R+MT +3 /4+θ, we get (2.40) R β C δ β G (s, y) Ω + C N j=1 Ω β (j) H (s, y)+ Ω β K (s, y)+ C 2 Ω β (ij) (s, y). Therefore, lim sup ε 0 R β C(E G + E H + E K + E ), and we get the result. 3. Estimates for two different flux functions In this section we apply Theorem 2.1 to prove various estimates for the following problem. Given two globally ipschitz flux functions f,g : R R N and u 0, v 0 1 loc (RN ), we consider the entropy solutions u and v of (3.1) t u +divg(u)=0, t v +divf(v)=0 in ]0, [ R N with initial data u 0 and v 0 respectively. Then, what ind of estimate can we expect for u v in terms of g f? We have the following result.

13 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2859 Theorem 3.1. et v 0 BV (R N ). Then, with the above notations, for any T 0: (i) The following estimate holds: (3.2) (3.3) with u(t,.) v(t,.) 1 (R N ) u 0 v 0 1 (R N ) +CTV(v 0 )T lip(g f). (ii) Moreover, u(t,.) v(t,.) 1 (R N ) u 0 v 0 1 (R N ) +C N ( u 0 1 (R N ) TV(v 0 )TQ(f,g) ) 1/2, (3.4) Q(f,g)= sup ξ R\{0} g(ξ) f(ξ) g(0) + f(0) / ξ. (iii) If g f (R), then for x 0 R N and R>0 (3.5) u(t,x) v(t,x) u 0 (x) v 0 (x) x x 0 <R x x 0 <R+MT ) 1/2 +C N ((R +MT) N TV x x (v0 )T g f (g f)(0), 0 <R+MT with M = max(lip(f), lip(g)). The estimate (i) is nown (see C.M. Dafermos [8], B.J. ucier [14]), but the much stronger estimates (ii) and (iii) seem fairly new. Proof of Theorem 3.1. For R we have t v +div sgn(v )[f(v) f()] 0, t u +div sgn(u )[g(u) g()] 0. Therefore, we can write (3.6) t u +div sgn(u )[f(u) f()] div γ (u), with γ (ξ) =sgn(ξ )[(f g)(ξ) (f g)()] sgn( )[(f g)(0) (f g)()] (3.7) =sgn(ξ )[(f g)(ξ) (f g)(0)] +[sgn(ξ ) sgn( )][(f g)(0) (f g)()]. Notice that γ lip(r, R N ). (i) Assume that u 0 BV (R N ). Then γ (u) C([0, [, 1 loc(r N )) B([0, [,BV(R N )) (where B(X, Y ) denotes the set of bounded functions X Y ), and by Appendix A2 γ (j) x (u) lip(γ (j) ) u (3.8) i x i lip(g f) u x i.

14 2860 F. BOUCHUT AND B. PERTHAME Therefore, K div γ (u) (]0, [, M(R N )) with N K lip(g f) u x i α K. We estimate α K by α K (t, dx) = lip(g f)tv(u(t,.)) lip(g f)tv(u 0 ). R N Now, the inequalities (1.3)-(1.4) are satisfied with a single term K of order zero. Hence, we can apply Theorem 2.1. Then, we just let δ 0, then 0 and finally R in (2.4), and we obtain u(t,.) v(t,.) u 0 v 0 +CTlip(g f)tv(u 0 ). R N R N The estimate (i) follows by exchanging u and v. (ii) The right-hand side of (3.6) can also be viewed as div H, with (3.9) H = γ (u). We have (3.10) γ (ξ) (f g)(ξ) (f g)(0) +2I ξ (f g)(0) (f g)() Q(f,g) ξ +2I ξ Q(f,g) 3 Q(f,g) ξ. Therefore, we get H 3Q(f,g) u α H.Sinceu 0 1 (R N ), we find that α H (t, x) dx =3Q(f,g) u(t, x) dx 3 Q(f,g) u 0 1. R N R N Now we can apply Theorem 2.1 with a single divergence term in the right-hand side. Since v 0 BV (R N ), we can use (2.9). We let δ 0, and, for any > 0, we get x x 0 <R u(t,.) v(t,.) R N i=1 u 0 v 0 +C N ( TV(v 0 ) + 1 ) Q(f,g) u0 1 T. Finally, we let R tend to, we choose the optimal, and we obtain (ii). (iii) This estimate is a localized version of (ii), but we have to be careful in order to get the right domain of dependence. Assume that v 0 BV loc (R N ), and let us denote R = R + MT and B = B(x 0, R). There exists an extension v 0 of v 0 B such that v 0 1 BV (R N ), supp v 0 B(x 0, 2 R), and (3.11) v 0 1 (R ) C N N v 0 (TV B(v 1 ( B), TV R N( v 0 ) C 0 N )+ 1 R ) v 0 1 ( B). et us define û 0 (x) = { u 0 (x) if x B, v 0 (x) if x/ B,

15 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2861 and consider the entropy solutions û and v associated with the flux functions g and f, and with initial data û 0 and v 0. Since supp û 0 B(x 0, 2 R), for any t 0we have supp û(t,.) B(x 0, 2 R + t lip(g)). Therefore, for any t 0 γ (û(t, x)) 3 g f (g f)(0) I B(x0,2 R+tip(g)) (x) α H(t, x). Then we apply Theorem 2.1 to û and v with H = γ (û), and by letting δ 0we see that for any > 0 û(t,x) v(t,x) û 0 v 0 x x 0 <R R [ N +C N TV( v 0 ) + T B(x 0, 2 R ] + T lip(g)) g f (g f)(0). Choosing the optimal and taing into account the definition of û 0 and v 0 and the finite speed of propagation for û and v, weget u(t,x) v(t,x) x x 0 <R u 0 v 0 + C N [ RN TV( v 0 )T g f (g f)(0) ] 1/2. x x 0 < R It remains to estimate TV R N( v 0 ). By the Poincaré-Wirtinger inequality we have B v0 1 B v 0 C RTV B(v 0 N ). B Therefore, by (3.11), the result holds as soon as v B 0 =0.If not, let c = v B 0 / B. The functions u c and v c are entropy solutions associated to the flux functions g(ξ + c) g(0) and f(ξ + c) f(0), and have initial data u 0 c and v 0 c. Since B (v0 c) = 0, the result holds for these functions, and so it holds for u and v. 4. The nonlinear diffusion model This section is devoted to the proof of Theorem 1.1. et us prove (i). Since u B([0, [,BV(R N )) (B(X, Y ) denotes the set of bounded functions X Y ), we have g(u) B([0, [,BV(R N )) for any ipschitz continuous function g by Appendix A2. Moreover, g(u) (]0, [, M(R N )), and (4.1) g(u) lip(g) u. By choosing (4.2) g (ξ) = φ(ξ) φ() φ(0) φ(), we get (4.3) g (u) lip(φ) u α(j) H.

16 2862 F. BOUCHUT AND B. PERTHAME Now the error term in (1.8) can be written as (4.4) φ(u) φ() =divh, H = [g (u)], and we have H (j) α(j) H.Therefore we can apply Theorem 2.1 with this single error term. We choose ν =0andweletδ 0, so that E t 0. Then E x is estimated by (2.9), and E H T V lip(φ). We hence obtain that for any > 0 u(t,x) v(t,x) dx x x 0 <R u(0,x) v(0,x) dx + C (TV(v 0 ) + T ) lip(φ)v. x x 0 <R+MT+ We then let R, and by choosing the optimal value of we get (1.9). Now let us prove (ii). We consider the error term in (1.8) as (4.5) φ(u) φ() = 2 (ij) x i, with g defined in (4.2). Since we have (4.6) (ij) δ ij Q u α (ij), (ij) = δ ij g (u), we can apply Theorem 2.1. As above, we choose ν =0andletδ 0, so that E t 0. We have E T 2 NQU, and E x is estimated by (2.9). Therefore we get that for any > 0 u(t,x) v(t,x) dx x x 0 <R u(0,x) v(0,x) dx + C (TV(v 0 ) + T ) 2 NQU. x x 0 <R+MT+ We then let R, and by choosing the optimal value of we get (1.10). In order to prove (iii) we need a preliminary result, which is another version of Theorem 2.1. It formalizes the method of B. Cocburn and P.-A. Gremaud [6] to replace the BV regularity of u by the BV regularity of v. Proposition 4.1. Under the same hypothesis as in Theorem 2.1, assume moreover that (4.7) H = H(t, x, ), are Borel functions satisfying (ij) = (ij) (t, x, ) H(t, x, 1 ) H(t, x, 2 ) M H 1 2, (4.8) (ij) (t, x, 1 ) (ij) (t, x, 2 ) M (ij) 1 2. Assume also that v B([0, [ loc,bv loc (R N )).

17 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2863 Then, we can replace E H and E in (2.4) by E H and E,where (4.9) E H = 1 +ν N j=1 x B t 0<tT α (j) H (t, x), E = 1 ( + ν) provided that we add the term C(E H + E ), with x B t 0<tT α (ij) (t, x), (4.10) E H = M H N j=1 x B 0 0<tT +δ v (t, x), E = 1 M (ij) x B 0 0<tT +δ v (t, x). Moreover, if a v b, it is sufficient that (4.8) holds for a 1, 2 b. Notice that the only difference between E H and E H (respectively E and E ) is that we replace a factor 1/ by1/( + ν), which is bounded when 0. Then for a local estimate we choose for ν a finite positive value, and for a global estimate we just let ν. The result also holds when considering error terms for v as in Proposition 2.2. Proof of Proposition 4.1. In the proof of Theorem 2.1, instead of using (1.4) in (2.11), we eep the terms H(t, x, ) and (ij) (t, x, ). Hence, the terms we have to estimate are R H = H(t, x, v(s, y)) x ϕdsdtdxdy, (4.11) R = (ij) 2 ϕ (t, x, v(s, y)) dsdtdxdy. x i ij They can be written R H = R H + R H, R = R + R, with R H = H(t, x, v(s, y)) y ϕdsdtdxdy, (4.12) R = (ij) (t, x, v(s, y)) 2 ϕ dsdtdxdy, y i y j ij and R H,R tae into account x ϕ + y ϕ and 2 ϕ x i 2 ϕ y i y j. Then we estimate R H and R as in the proof of Theorem 2.1. The only difference is that we no longer have the terms corresponding to the maximal order of derivation for ζ. Therefore, in (2.28) and (2.30) we retain only terms in 1/θ and 1/ θ respectively, instead of 1/ and1/ 2 as before. We hence get lim sup ε 0 R H CE H, lim sup R CE. ε 0 Now, in order to estimate R H and R, we notice that for fixed s, t, x, the function v(s,.) belongs to BV loc. By the ipschitz conditions (4.8) and by Appendix A2, we obtain that H(t, x, v(s,.)) BV loc and (ij) (t, x, v(s,.)) BV loc. Therefore, we

18 2864 F. BOUCHUT AND B. PERTHAME integrate by parts to get R H = (4.13) R = ij ] ϕdiv y [H(t, x, v(s, y)), ϕ [ ] (ij) (t, x, v(s, y)). y i y j Then, by Appendix A2 (i), or (ii) if a v b, N R H v ϕ M H (s, y) x j=1 j, (4.14) R ϕ y i M(ij) v (s, y). ij These terms are finally estimated as in the proof of Proposition 2.2, and we obtain lim sup ε 0 which ends the proof of Proposition 4.1. R H E H, lim sup R CE, ε 0 Proof of Theorem 1.1 (iii). We begin as in the proof of (ii). We notice that (ij) δ ij g (u) can be written (ij) = (ij) (t, x, ), with ] (ij) (t, x, ) =δ ij [ φ(u(t, x)) φ() φ(0) φ(). = Therefore, the ipschitz condition (4.8) is fulfilled with M (ij) =2δ ij lip(φ), and we can apply Proposition 4.1. We obtain u(t,x) v(t,x) x x 0 <R u 0 (x) v 0 (x) + C(E t + E x + E + E ). We have E x x 0 <R+MT+ +ν NQ u(t, x) dtdx, E 2(T + δ) ( + ν) ]0,T [ R N lip(φ)tv(v 0 ), and E x is estimated by (2.9). By letting δ 0andν (so that E 0) we get for any > 0 ( u(t,x) v(t,x) u 0 (x) v 0 (x) +C TV(v 0 ) + T ) lip(φ)tv(v0 ). x x 0 <R R N By choosing the optimal value of and letting R we obtain (iii).

19 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED A relaxation model for finite volume methods In this section we use Theorem 2.1 to deduce an estimate in h 1/4 for the finite volume type model presented in the introduction. We first state an existence result. Proposition 5.1. With the notations of the introduction, for any ε>0and any u 0 1 (R N ) there exists a unique solution u C([0, [, 1 (R N )) of (1.15) satisfying the entropy inequalities (1.16) for all convex, ipschitz and C 1 functions S, or equivalently for all Kružov entropies. Since this result uses very standard techniques, we only give a short setch of the proof. First let us notice that for an arbitrary convex and ipschitz S, S (u) is not well defined because of possible jumps of S. That is why we require S to be C 1. Then, if (1.16) holds for these C 1 test functions, it is easy to see that it also holds for the entropies S, with the convention that S (ξ) =sgn(ξ )and sgn(0) = 0. The uniqueness is obtained by Kružov s method. We actually get that for two solutions u, v of (1.16), t u v +div sgn(u v)[f(u) f(v)] (5.1) P 0 (u v) u v in ]0, [ R N, ε which yields the contraction property (5.2) u(t,.) v(t,.) 1 u 0 v 0 1. For existence, we use the small diffusion approximation method. Here the 1 BV hypothesis (1.13) on the grid is involved. It is actually equivalent to asserting that P 0 maps 1 (R N )intobv (R N ) with (5.3) TV(P 0 w) K h w 1, w 1 (R N ). This yields the following a priori estimate for a solution u of (1.15)-(1.16) of initial data u 0 1 BV (R N ): (5.4) u (t,.) M e t/ε u0 M +(1 e t/ε )K h u 0 1. This allows us to prove existence for u 0 1 BV. Then by the contraction property (5.2) we obtain the existence of a solution for any u 0 1 (R N ). Now we can prove the convergence rate to the continuous solution. Proof of Theorem 1.2. If u 0 1 p (R N )forsomep,1p<,we can actually choose S(ξ) = ξ p in (1.16). Integrating this inequality with respect to x, weget d u p dx S (u) P 0 u u dx in ]0, [. dt R N R ε N Then, since S (P 0 u) is constant in each cell, and since the integral of P 0 u u on each cell vanishes, we have S (u) P 0 u u [ = S (u) S (P 0 u) ] P 0 u u (5.5) 0. R ε N R ε N Therefore, we obtain d u p [ + S (P 0 u) S (u) ] P 0 u u 0 in ]0, [, dt R N R ε N

20 2866 F. BOUCHUT AND B. PERTHAME and for any T 0 (5.6) u(t,.) p + R N ]0,T [ R N [ S (P 0 u) S (u) ] P 0 u u ε In the case where u (R N )(p=2),we find that (P 0 u u) 2 RN u 0 2 (5.7) ε 2. ]0, [ R N u 0 p. R N The announced inequality (1.17) easily follows by Hölder s inequality. Now we write the entropy inequalities, for R (5.8) t u +div sgn(u )[f(u) f()] sgn(u ) P 0 u u ε = [ sgn(u ) sgn(p 0 u ) ] P 0 u u +sgn(p 0 u ) P0 u u ε ε sgn(p 0 u ) P 0 u u. ε Since the integral in each cell of the right-hand side vanishes (for a fixed t), we can invert the div operator as in Appendix A1, and therefore sgn(p 0 u ) P 0 u u (5.9) =divh ε for some H (]0, [, 1 (R N )), which can be bounded above by (A.7) and (1.17): T B(x0,R+2h) H (t, x) dtdx h u ε ]0,T [ B(x 0,R) By examining the proof in Appendix A1, it is easy to see that since the sign in (5.9) is bounded by 1, we have indeed (5.10) H (t, x) α H (t, x) for some α H independent of, and which also satisfies T B(x0,R+2h) (5.11) α H (t, x) dtdx h u ε ]0,T [ B(x 0,R) Therefore, combining (5.8) and (5.9), we may apply Theorem 2.1 with the sole error term (5.9). By choosing ν = 0 and letting δ 0(sothatE t 0) we obtain, using (2.9) for E x and (5.11) for E H, that for any > 0 (5.12) u(t,x) v(t,x) dx u 0 (x) v 0 (x) dx x x 0 <R + C N ( x x 0 <R+MT+ T (R + MT + +h) N TV(v 0 ) + h ε ) u 0 2.

21 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2867 Finally, we choose (5.13) and we get the inequality (1.18). = Rl Remar. It is an open problem to obtain an estimate in h 1/2 (except in one dimension see R. Sanders [17]). Notice that Proposition 4.1 cannot be applied, since the term H is not ipschitz continuous in. When all the cells are star-shaped, denoting by H(t, x) the function obtained Appendix A1 such that div H = P 0 u u (5.14), ε we actually have, by the support property (A.4), (5.15) H =sgn(p 0 u )H. So in this case the bound (5.11) is obvious. Appendix. Inversion of the div operator; composition of ipschitz and BV functions A1. Inversion of the div operator. We consider a question arising from the finite volume type of approximations in 5. In this context, we need the result below in order to use the error estimates stated in 2. emma A1.1. et (C i ) i I be a general grid of R N as described in the introduction. Assume that m 1 (R N ) satisfies (A.1) m(x) dx =0, i I. C i Then there exists a vector field H(x) such that (A.2) div H = m in R N, (A.3) H 1 (R N ) h m 1 (R N ). Remar A1.2. (1) If all the cells C i are star-shaped (there is a point x i C i such that for any x C i the segment [x i,x] is included in C i ), then the construction of H is purely local. More precisely, (A.4) H = i I H i, H i =0 in R N \C i, with (A.5) div H i = mi Ci, (A.6) H i 1 (R N ) h mi Ci 1 (R N ). (2) The estimate (A.3) can be localized: for any bounded Borel set ω R N, we have (A.7) H 1 (ω) h m 1 (ω ), h (A.8) ω h = dist(c i,ω)h C i.

22 2868 F. BOUCHUT AND B. PERTHAME Actually, H i =0inωas soon as dist(c i,ω) >h, with H i defined in the proof below. Proof of emma A1.1. To solve (A.2) in the distributional sense means that we loo for a function H such that, for all test functions ϕ Cc (R N ), H(x) ϕ(x)dx = m(x)ϕ(x) dx R N R N = m(x)(ϕ(x) ϕ(x i )) dx i I C i (A.9) = 1 m(x) ϕ (x i + θ(x x i )) (x x i ) dθdx i I C i 0 = 1 ( (mi Ci ) x i + y x ) i ϕ(y) y x i dθ dy i I R N 0 θ θ θ N. Here x i denotes an arbitrary point of C i, but in the star-shaped case it should possess the property in Remar A1.2 (1). Therefore we choose (A.10) H = i I H i, 1 ( H i (y) = (mi Ci ) x i + y x ) (A.11) i y xi 0 θ θ N+1 dθ. We may majorize the 1 norm of H i, 1 ( H i (y) dy ( m I Ci ) x i + y x ) i y xi R N R N 0 θ θ N+1 dθdy 1 (A.12) = ( m I Ci )(x) x x i dxdθ 0 R N diam(c i ) m(x) dx. C i Adding these inequalities, we obtain the statement of emma A1.1. In the starshaped case, we notice from formula (A.11) that if H i (y) 0,then z θ = x i + y xi θ C i for some θ ]0, 1[. This means that y = x i + θ(z θ x i ) also belongs to C i.this proves (A.4). The properties (A.5) and (A.6) are contained in the above proof. A2. Composition of ipschitz and BV functions. et Ω be an open subset of R N and consider a function u BV loc (Ω), and g : R R ipschitz continuous. The well-nown theory of A.I. Vol pert [20] allows us to compute the gradient of g(u) provided that g C 1. However, for a general ipschitz function g we have the following simple result. emma A2.1. With the above notations, g(u) belongs to BV loc (Ω) and: (i) in the sense of measures, (A.13) [g(u)] lip(g) u. (ii) if a u b, then (A.14) [g(u)] lip (g) u. [a,b]

23 KRUŽKOV S ESTIMATES FOR SCAAR CONSERVATION AWS REVISITED 2869 Proof. et us define g n = ρ n g, u n = ρ n u, whereρ n 0 is a standard smoothing sequence (in 1 or N dimensions). Since these functions are smooth, we have g n (u n ) = g n(u n ) u n g n u n lip(g) u n. From the formula un = ρ n u we get un ρn u, and hence (A.15) g n (u n ) lip(g) ρ n u. Therefore, for any test function ϕ Cc (Ω) we have [g n (u n )],ϕ lip(g) ρ n u, ϕ, and by letting n u (A.16) [g(u)],ϕ lip(g), ϕ. Since the right-hand side is bounded by C ϕ,wegetthat [g(u)] is locally a measure, and hence g(u) BV loc (Ω). Then by density (A.16) still holds for any ϕ C c (Ω). By approximation in 1 ( j [g(u)] + j u ), (A.16) is also true for any ϕ measurable and bounded with compact support in Ω. Therefore, [g(u)] E lip(g) u (A.17) E for any Borel set E with compact closure in Ω, and (A.13) follows from the definition of the absolute value of a measure. In order to prove (ii) we define g(ξ) if a ξ b, g(ξ) = g(a) if ξ a, g(b) if ξ b. Then lip( g) = lip (g), and g(u) =g(u). We get the result by applying (i) to g and [a,b] u. References 1. Ph. Benilan, R. Gariepy, Strong solutions in 1 of degenerate parabolic equations, J.Diff.Eq. 119 (1995), MR 96c: F. Bouchut, Ch. Bourdarias, B. Perthame, A MUSC method satisfying all the numerical entropy inequalities, Math. Comp. 65 (1996), MR 97a: S. Champier, T. Gallouët, R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh, Numer. Math. 66 (1993), MR 95b: B. Cocburn, F. Coquel, P. e Floch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), MR 95d: B. Cocburn, F. Coquel, P. e Floch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), MR 97f: B. Cocburn, P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach, Math. Comp. 65 (1996), MR 96g: B. Cocburn, P.-A. Gremaud, Error estimates for finite element methods for scalar conservation laws, SIAMJ. Numer. Anal. 33 (1996), MR 97e:65096

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