Nonlinear Evolution Governed by Accretive Operators in Banach Spaces: Error Control and Applications

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1 Nonlinear Evolution Governed by Accretive Operators in Banach Spaces: Error Control and Applications Ricardo H. Nochetto Department of Mathematics and Institute for Physical Science and Technology, University of Maryland Giuseppe Savaré Dipartimento di Matematica, Università di Pavia, and I.M.A.T.I., C.N.R. Abstract Nonlinear evolution equations governed by m-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order O τ. Applications to scalar conservation laws and degenerate parabolic equations with or without hysteresis in L 1, as well as to Hamilton-Jacobi equations in C are given. The error analysis relies on a comparison principle, for the novel notion of relaxed solutions, which combines and simplifies techniques of Benilan and Kružkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall-Liggett error estimate. 1 Introduction Let B be a Banach space with norm and let F be a possibly multivalued operator in B with domain DF B. Given an initial datum u in the closure of DF and a forcing function f L 1, T ; B, we analyze the approximation of the Cauchy problem { u t + Fut ft, CP u = u, by a variable step implicit or explicit Euler method. More precisely, given the partition P = {t := < t 1 <... < t N 1 < t N := T } 1.1 College Park, MD 2742, USA rhn@math.umd.edu. Partially supported by NSF Grants DMS and DMS Via Ferrata 1, 271 Pavia, Italy savare@imati.cnr.it, Web: savare. Partially supported by M.I.U.R. Grants Cofin1999,22 1

2 of the time interval [, T ] with variable step-size τ n := t n t n 1, τ := max n τ n, 1.2 and given U DF and the sequence {F n } N n=1 B, we consider the sequence {U n } N n= defined recursively by the Implicit Euler Scheme or by the Explicit Euler Scheme U n U n 1 τ n + FU n F n, 1 n N, IS 1 U n U n 1 τ n + FU n 1 F n, 1 n N. ES 1 Observe that IS 1 requires to solve at each step a problem of the type given w B find v DF : v + λfv w 1.3 where λ is one of the time steps, so that it is natural to suppose the map I + λf is surjective for < λ τ. IS 2 In order to guarantee solvability of an arbitrary number of iterations of ES 1 and to avoid ambiguities we have to assume that DF = B, F is single-valued. ES 2 In both cases, our crucial assumption relies on a contractivity assumption of the maps U n 1 U n defined by IS 1 or ES 1 when F n =. In fact, for the implicit scheme IS 1 we ask that v i + λfv i w i i = 1, 2 v 1 v 2 w 1 w 2, < λ τ, IS 3 whereas for the explicit scheme ES 1 we impose v i = w i λfw i i = 1, 2 v 1 v 2 w 1 w 2, < λ τ. ES 3 It is well known that assumptions IS 2,3 characterize the class of m-accretive operators in B and their validity extend to every λ > ; it turns out see 3 that ES 3 implies IS 3 and that F is Lipschitz continuous with constant L F 2τ 1. Consequently, in the explicit case, we are in fact imposing a constraint τ 2L 1 F on the maximum of the time steps, which may be viewed as an abstract CFL condition: we will call an operator F satisfying ES 3 explicitly τ-contractive. In any case, IS 2,3 and even more ES 2,3 assures the existence and uniqueness of a mild/integral solution of CP, which yields a well posed problem [15, 4, 2, 3]. Estimates for the error u U of order O τ for IS 1 were derived by Crandall and Liggett [15] for uniform time-steps, and then extended to nonuniform partitions in [2]. Crandall and Evans [12] showed that the core of the theory lies on some estimates concerning the degree of approximation by solutions of difference schemes to the exact solution of a boundary value problem involving the differential operator / s + / t. All these error estimates are a priori, i.e. the rate depends on the data f and u, but not explicitly on U n. 2

3 Since they are expressed in terms of τ and not τ n, they cannot be used to select the time-step τ n according to the local behavior of IS 1. In this paper we adopt the opposite point of view with respect to w.r.t. [12]: we regard the solutions of an approximation scheme related to CP as continuous solutions of a suitably relaxed formulation of CP, which we study in an abstract form, in order to derive the main estimates in a unified way. If the method of [12] resembles the Kru zkov s technique of doubling the variables [22], our key idea is to double the unknowns: for us, a relaxed solution of CP is a couple of functions satisfying a suitable evolution dissipation inequality related to CP up to an error which we call discrepancy. In this vein, a strong or integral solution u of CP is associated to a strong or weak, respectively relaxed solution with discrepancy. The other main feature of our construction goes back to the notion of integral solution due to Benilan [4]: discrete solutions are then characterized by a system of evolution inequalities involving the distance from an arbitrary test element V of B. In the Implicit case IS 1 we will consider the distance between the piecewise linear interpolant of the discrete values {U n } N n= from V as in [27]; in the Explicit case ES 1 we will invert this point of view, and we will consider the piecewise linear interpolation of the values of the discrete distances { U n V } N n=. This idea has then also been applied in the more general context of metric spaces [1, Chap. 5],[29], where vector linear interpolation is not allowed. This new viewpoint leads to a unified treatment of existence, uniqueness, and error analysis issues via a comparison principle between strong and weak relaxed solutions see 6. As a by-product we obtain novel error estimates, which address the following fundamental numerical issues: a A posteriori estimates for u U : computable quantities which depend solely on time-steps {τ n } N n=1, discrete solution {U n } N n=1, and data f, u, F; b Optimal rates of convergence: a posteriori bounds converging to zero as τ with an optimal rate w.r.t. the regularity of the data; c Minimal regularity: for IS 1 F is solely assumed to be an accretive operator in B, i.e. to satisfy IS 2 and IS 3 no other regularity assumptions, such as Lipschitz continuity, are made on F nor on the dimension of B; for ES 1 F is just supposed to satisfy ES 2 and ES 3 ; d Uniform stability and explicit constants: the stability and error constants are uniform with respect to possible space discretizations and they are explicitly determined without need of solving any auxiliary, or dual, problem; e Variable time-steps: no a priori constraints between consecutive time-steps, which could just be taylored to the a posteriori error estimators alone. We refer to a and b collectively as optimal a posteriori error estimates. We now state our primary results. Let P, τ n, τ be defined as in 1.1, 1.2, and let Ut be the piecewise linear interpolant of {U n } N n= on the grid P, 1.4 Ūt, F t be the piecewise constant functions which take the value U n, F n in the interval t n 1, t n ]

4 If g BV, T ; B, then Var g stands for the total variation of g on [, T ] see the paragraph 4.3b below and [7, Appendix]. Theorem 1 Error Estimates Let U DF, let {U n } N n= be the solution of the implicit IS 1,2,3 or of the explicit ES 1,2,3 Euler scheme and let U, F be defined as in 1.4, 1.5. If u is the unique mild/integral solution of CP, then we have the following a posteriori error estimate max t T u Ut u U + f F L1,T ;B + 2 F 1 FU + Var F 1/2 N n=1 1/ τ n U n U n 1 Moreover, the a posteriori error estimator is bounded a priori as follows: N n=1 τ n U n U n 1 T F 1 FU + Var F τ 1.7 where, only for 1.8, we have chosen F n := ft n 1. T f FU + Varf τ, 1.8 As a particular case of Theorem 1, we find the a priori error estimate in the case u = U DF, f = F = max t T u Ut 2 Fu T τ. 1.9 For IS 1,2,3, this coincides exactly with the estimate of [15, eq. 1.1] in the case of a uniform mesh; this is also the asymptotic behavior proved by Kuznetsov [23] for scalar conservation laws. These results extend to accretive operators in Banach spaces the optimal a posteriori error estimates derived by Nochetto, Savaré and Verdi [26, 27] for subgradient and angle-bounded operators in Hilbert spaces; the rate of convergence proved in [26, 27] is Oτ provided u DF. Since such an order cannot be better than O τ for general monotone operators in Hilbert spaces [28], our present results are optimal. We refer to [25] for simpler results, and to [1, 29] for extensions to the Wasserstein metric. This paper is organized as follows. In 2 we present the basic ideas leading to Theorem 1 for ODE in R d ; this simplified derivation is later regarded as a reference for more technical developments. In 3 we exhibit applications of Theorem 1 to PDE such as scalar viscous conservation laws, degenerate parabolic PDEs with or without hysteresis, and Hamilton-Jacobi equations. We recall some instrumental results in functional analysis in 4; in particular we will discuss some properties related to ES 2,3. We motivate the notion of relaxed solution in 5 via IS 1, ES 1, and the Yosida regularization of CP, and we prove a comparison principle for relaxed solutions in 6. In 7 we apply this comparison principle and further stability estimates to derive a more general version of Theorem 1 and an estimate for the Yosida regularization of CP. They slightly extend and unify the classical approaches to existence and uniqueness of mild/integral solutions of CP for m-accretive operators developed by Crandall, Liggett, Evans, and Bénilan. 4

5 2 Basic Ideas: Proof of Theorem 1 for ODE We consider the simplified finite dimensional setting B := R d endowed with a differentiable but not necessarily euclidean norm. We let j : R d R d be the differential of the norm we formally define j :=, which satisfies v = v, jv for all v R d, and F : R d R d be a Lipschitz and accretive vector field, thereby satisfying Fu Fv, ju v u, v R d. 2.1 Supposing for simplicity f, F, we are thus approximating the system of nonlinear ODE s u t + Fut =, u = u 2.2 by the implicit scheme which can also be rewritten as U n U n 1 τ n + FU n =, 2.3 U t + FŪt = a.e. in, T, 2.4 where U, Ū is the couple of discrete solutions defined by 1.4,1.5. Failure of the Trivial Strategy. Let us first remark that the usual strategy to derive stability estimate, i.e. take the difference of the equations 2.2 and 2.4 and multiply it by jut Ut does not work here since we need to multiply by jut Ūt in order to take advantage of 2.1, but with this choice the term u U, ju Ū is no longer the derivative of the norm u U. If one tries to control this difference by standard perturbation arguments, the difficulty of dealing with two nonlinearities F and j comes up, and this does not allow for estimates independent of the dimension d and of the Lipschitz constant of F. Therefore, we are forced to deal with the continuous and the discrete equations separately. We describe now the crucial steps of our argument. I. Evolution Equation for the Continuous Solution. Since j is the differential of the norm, 2.2 is in fact equivalent of the system of evolution equations d dt ut v = Fut, jut v v R d, 2.5 which we will try to reproduce in the discrete setting, starting from 2.4. II. Evolution Inequalities for the Discrete Solutions. The map s Us V is convex in each interval t n 1, t n ] and therefore its derivative is not decreasing and it is bounded by its value at s = t n ; in particular d ds Us V = U s, jus V U s, jūs V V Rd, a.e. in, T. Combining 2.4 with 2.6 we end up with 2.6 d ds Us V FŪs, jūs V V Rd, a.e. in, T

6 III. Doubling Variables. Now we are ready to combine 2.5 and 2.7 with 2.1: as usual in such monotonicity argument, we would like to choose the elements v, V as v := U and V := u and to sum up the equations. The main point here is that it is impossible to write 2.5 and 2.7 at the same time t = s, since we derived the equations with respect to time-independent test elements v, V. Therefore we keep two distinct time variables t, s and we choose v := Ūs, V := ut obtaining, by 2.1 s Us ut + Ūs ut. 2.8 t IV. Penalization. Now we introduce a parameter ε, T, and we integrate this inequality in the two-dimensional strip Q ε,t := { s, t R 2 : t T, t ε s t }. 2.9 t t = T ε s = t ε Q ε,t t = s ε s = T s Figure 1: Strip Q ε,t corresponding to penalization about the diagonal {t = s}. In order to deal with negative values of s, we extend U, Ū in ε, by Ūs := U, Us := U sfu 2.1 so that 2.4 still holds. Applying the Gauss-Green formula in Q ε,t gives T ε Ūs ut ds ε u U + + Ūt ut Ut ut dt Ut ε ut Ūt ε ut dt V. Stability. We take the difference between two consecutive equations 2.3, and multiply by ju n U n 1, to arrive at 1 Un U n 1, ju n U n 1 + FU n FU n 1, ju n U n 1 τ n = 1 τ n 1 Un 1 U n 2, ju n U n 1. 6

7 Making use of 2.1, together with property v, jw v for all v R d with equality for w = v, we get 1 U n U n 1 1 U n 1 U n 2, 1 n N. τ n τ n 1 If we now choose U 1 = U τ FU with τ ε, which is consistent with 2.1, recursion yields the following stability estimate for the discrete derivative: sup U t = t,t sup 1 n N VI. A Posteriori Error Estimate. 1 τ n U n U n 1 FU Since Ūt ut Ut ut Ūt Ut, Ut ε ut Ūt ε ut Ūt ε Ut ε, and ε Ūt Ut dt = ε 2 FU, a simple manipulation of 2.11 implies 1 T Us ut ds 1 Ūs ut + ε T ε ε Ūs Us ds T ε Since 1.4 and 1.5 imply and 2.12 leads to we readily infer that u U + ε 2 FU + 2 ε Ut Ūt dt = 1 2 N τ n U n U n 1, n=1 Ūt Ut dt. 1 T Us UT ds ε ε T ε 2 FU, 2.13 ut UT u U + ε FU + 1 ε N τ n U n U n 1. n=1 Upon optimizing ε, we conclude the desired a posteriori error bound N ut UT u U + 2 FU 1/2 1/2. τ n U n U n VII. A Priori Error Estimate. To show that the above error bound exhibits the correct asymptotic order of convergence, we resort to 2.12 to deduce n=1 N τ n U n U n 1 τt FU. n=1 This yields the following celebrated a priori estimate of Crandall and Liggett [15], originally derived for constant time steps: ut UT u U + 2 FU τt

8 3 Applications of Theorem 1 to PDE In this section we present several concrete examples. We provide some basic background and references where the assumptions IS 2 and IS 3 are shown. In some case it is easier to define F by a closure procedure, starting from a smaller selection F defined in a subset DF of a new Banach space B B dense in B. More precisely, if F : DF B 2 B is a multivalued operator satisfying w B, λ > v DF : v + λf v w, 3.1a v i DF, v i + λf v i w i B v 1 v 2 w 1 w 2, 3.1b then it is not difficult to see that the strong closure of the graph of F in B B produces an m-accretive operator F, which is therefore defined as w Fv w n F v n : v n, w n v, w strongly in B B Scalar Conservation Laws Consider the Cauchy problem for the following viscous conservation law in R d t u + div φu ɛ u = f, u, = u, 3.3 where ɛ >, u L 1 R d L R d, and the nonlinear function φ satisfies φ C 1 R, R d, φ =. 3.4 Consider the Banach spaces B := L 1 R d, B := L 1 R d L R d and the operator F v := div φv ɛ v with domain DF := {v B H 2 R d : F v B }. 3.5 Then F satisfies 3.1a and 3.1b [11, Prop. 2.2, Lemma 2.4, Cor. 2.2] and u = ux, t is a bounded variational solution of 3.3 if and only if t u, t solve the abstract CP for F given as in 3.2. The limit case ɛ = is more delicate and was first studied by Kru zkov [22], and later by Crandall [11] within the context of accretive operators in L 1. The operator F is now defined [11, Def. 1.1, Cor. 2.2] on v B = L 1 R d L R d as the set of w B such that as φvx sign vx k φk ϕx + wxϕx dx 3.6 R d for all ϕ C R d, ϕ, and k R; here sign =, sign x = x x if x. 3.7 As before, ux, t is a bounded entropy solution of 3.3 for ε = if t u, t is an integral solution of CP. In both cases, ɛ > and ɛ =, the operator F is m-accretive in B. Our error estimates of Theorem 1 for the scheme U n U n 1 τ n + div φu n ε U n = F n 3.8 are thus uniform in the viscosity parameter ɛ. 8

9 3.2 A Finite Volume Method for Conservation Laws Let T be decomposition of a bounded domain of R d into simplices K and let d be the cardinal of T. For each pair of adjacent volumes K, L T there is a discrete flux g K,L C R 2 satisfying the structural properties g K,L u, v = g L,K v, u, 3.9 g K,L is increasing w.r.t. u and decreasing w.r.t. v; 3.1 we set g K,L u, v if K, L are not adjacent [19], [21]. We consider the following semidiscrete piecewise constant finite volume discretization of 3.3 with ɛ =. Let ut := {u K t} K T be the semidiscrete solution of K d dt U Kt + L T K L g K,L U K t, U L t = K f K t, 3.11 where K L stands for the measure of the common side between K and L K L = if K, L are disjoint. The implicit version of 3.11 reads K U n,k U n 1,K τ n + L T K L g K,L U n,k, U n,l = K F n,k If u := {u K } K T, U n := {U n,k } K T, and FU := {F K U} K T : R d R d is defined by F K U := 1 K L g K,L U K, U L U := {U K } K T R d, K L T then 3.11 and 3.12 correspond to CP and IS 1 respectively for F. Observe that 3.9 yields the properties K F K U =, K T K U n,k = K U n 1,K + τ n K F n,k. K T K T K T 3.13 Let us show that 3.9 and 3.1 imply F is m-accretive w.r.t. the L 1 -type norm v 1 := K T K U K in R d. Since F is continuous, it is enough to check accretivity. We invoke an equivalent characterization of accretivity via the so called duality map J : R d 2 Rd see 4.1 and 4.11 in 3: setting see 3.7 J U := {sign U K } K T, which belongs to JU, 3.14 the accretivity of F follows provided we show that for all U, V FU FV, J U V = K T FK U F K V sign U K V K By 3.9 we have FU FV, J U V = = g K,L U K, U L g K,L V K, V L sign U K V K K T L T = g L,K U L, U K g L,K V L, V K sign U K V K. K T L T 9

10 Consequently, if A K,L := g K,L U K, U L g K,L V K, V L, B K,L := sign U K V K sign U L V L, we then infer 3.15 because FU FV, J U V = 1 2 A K,L B K,L. K,L T If W K := U K V K, the last inequality results from 3.1 and the properties: W K, W L > B K,L =, W K, W L < B K,L =, W K, W L B K,L, A K,L, W K, W L B K,L, A K,L Since F is m-accretive in R d w.r.t. the norm 1, our error estimates of Theorem 1 1 are valid for this fully discrete finite volume method. Let us now switch to the explicit scheme K U n,k U n 1,K τ n + L T K L g K,L U n 1,K, U n 1,L = In order to apply Theorem 1, we have to check if F is explicitly τ contractive for some τ >. Besides 3.9 and 3.1, we add now that the flux functions g K,L are Lipschitz continuous, i.e. g K,L u 1, v g KL u 2, v λ K,L u 1 u 2 u 1, u 2, v R If we set then it is possible to prove that λ K := 1 K L λ K,L, K L T λ := sup λ K, 3.19 K T if τ λ 1 then F is explicitly τ contractive Degenerate Parabolic Equations Let Ω be a smooth bounded domain of R d, f BV, T ; B with B := L 1 Ω. Consider the initial-boundary problem for the following degenerate parabolic equation in Ω t u βu = f, u, = u, 3.21 with β a maximal monotone graph in R 2 such that β. The operator Fv := βv, defined in DF := {v L 1 Ω : βv W 1,1 Ω, βv L 1 Ω}, was shown to be m-accretive by Brezis and Strauss [9]; see also [3]. This setting models a number of important physical processes. Within the class β W 1, R, R, we have the Stefan problem for which βs = s + 1 s +. If, instead, β 1 s = 1 s 1 we have an elliptic-parabolic equation which describes filtration with partial saturation. 1

11 Our error estimates of Theorem 1 apply to the scheme U n U n 1 τ n βu n = F n, 3.22 irrespective of the regularity of the maximal monotone graphs β or of the solutions. Equations 3.3 and 3.21 may be combined together in R d t u + div φu βu = f, u, = u. It is shown by Cockburn and Gripenberg [1] that under suitable assumptions on φ and β the ensuing operator Fv := div φu βu is m-accretive in B. Therefore our error estimates are also valid in this case for the related scheme. 3.4 Parabolic Equations with Hysteresis We describe briefly a model due to Visintin [31]. Let γ, γ + be two maximal monotone possibly multivalued graphs in R 2 such that inf γ v sup γ + v v R, 3.23 and let ϕv, w be the hysteresis loop in R 2 {+ } if w < inf γ + v, [, ] if w γ v\γ + v, {} if sup γ v < w < γ + v, ϕv, w := [, ] if w γ v\γ + v, { } if w > sup γ v, [, + ] if w γ v γ + v. Let Ω and f be as in 3.3. We consider the parabolic system in Ω, T : 3.24 t u + ξ u = f, t w ξ =, ξ ϕu, w, 3.25 with Dirichlet boundary condition for u and initial condition u, w. Equivalently, if V := u, w and F := f,, we can write 3.25 in vector form t V + AV + LV F, 3.26 where { DA := {V = u, w R 2 : inf γ u w sup γ + u} = Dϕ, AV := {ξ, ξ : ξ ϕv R} V DA, and LV := u,, DL := {u, w L 1 Ω; R 2 : u W 1,1 Ω, u L 1 Ω}. If DA, then A is m-accretive in R 2 [31]; this follows from an argument similar to that in Moreover, if γ, γ + do not grow faster than linearly at ±, namely z C 1 v + C 2 for all v R and z γ v, γ + v with constants C 1, C 2 >, then the nonlinear multivalued operator F := A+L is m-accretive in L 1 Ω; R 2. Our error estimates of Theorem 1 are valid for 3.26 and are the first ones for this problem. 11

12 3.5 Hamilton-Jacobi Equations Let H CR, R be a Hamiltonian and let B =BUCR d be the space of bounded uniformly continuous functions over R d with the sup norm. We consider the Cauchy problem t u + H u = f, u, = u, 3.27 with u B. Viscosity solutions of 3.27 have been constructed by Crandall, Evans and Lions [13], [16]; see also [14]. If we define the domain DF of F as the set of all viscosity solutions u B of H u = f for some f B, and Fu B the set of all such f, then F is m-accretive in B [13]. Therefore, our error estimates of Theorem 1 apply to the scheme U n U n 1 τ n + H U n = F n A general way to approximate 3.27 [3] by a so called monotone scheme is to introduce a family of maps Sh : BR d BR d, h >, here BR d denotes the space of bounded real functions defined on R d which satisfies the properties u v Shu Shv u, v BR d, 3.29a Shu + k = Shu + k u BR d, k R, 3.29b φ Shφ h H φ as h φ C R d BR d. 3.29c In this approach, the approximation scheme is given by U := u and U n = 1 τ n Un 1 + τ n h h ShU n 1 + τ n F n, 3.3 which correspond to ES 1 for the operator F h v := v Shv h A result of Crandall and Tartar [17] see also paragraph 4.2b shows that F h satisfies ES 3 for τ = max n τ n h. Theorem 1 can thus be applied and provides a flexible choice of the mesh P, satisfying the CFL-like condition τ h. 4 Differential Calculus in Banach Spaces In this section we collect some basic functional analytic facts of Banach spaces that will be instrumental in the subsequent discussion. We refer to [2], [18], [24], [3] for more details and proofs. Notation 4.1 Multivalued Maps We will often deal with multivalued maps J : X 2 Y, X, Y being given sets: we will use the symbol Jx to indicate any selection y Jx, the same at every occurence of Jx in a given formula. 12

13 4.1 Accretive Operators. 4.1a Duality Map. We will denote by B the dual space of B, endowed with the dual norm ;, is the duality pairing between B and B. The duality map [3, p. 91] J : B 2 B is defined by Jv = { v B : v 1, v, v = v } ; 4.1 for every v B, Jv is a nonempty, weakly compact, and convex set of B, with Jv = 1 if v. We present now three relevant examples. 1. If B is a Hilbert space, then we can identify B with B and in this case Jv = for every v. v v 2. If B = L 1 Ω, Ω being an open subset of R d, then B = L Ω and Jv = signv. 3. If B = C Ω is the completion in the maximum norm of the space of continuous functions with compact support, then B = MΩ is the space of finite signed measures µ with Jordan decomposition µ = µ + µ and, for v, µ Jv µ + Ω + µ Ω = 1 supp µ + {x Ω : vx = v C Ω}, supp µ {x Ω : vx = v C Ω}. We thus see that Jv is composed of measures with unit mass and support in the set of extremal points of v. Examples 2 and 3 show that the duality map J is in general multivalued. 4.1b Directional Derivatives of the Norm, Pseudo-Scalar Product. The duality map is strictly related to the differentiability properties of the norm of B. First of all, we note that the map λ R w + λv is convex for all v, w B. Then, the directional derivatives of the norm satisfy: w + λv w [v, w] + := lim = inf λ λ λ> w w λv [v, w] := lim = sup λ λ λ> [, ] ± are called pseudo-scalar products. We observe that and w + λv w, λ 4.2a w w λv ; λ 4.2b [v, w] + = [ v, w] + = [ v, w] = [v, w], 4.3 [v, w] ± v, [v, v] = [v, v] + = v v, w B. 4.4 The duality map and the pseudo-scalar product are related by [v, w] = min v, w Jw w, [v, w] + = max v, w Jw w

14 In light of 4.2, 4.5 shows that J is the subdifferential of the norm of B [2, Chap. II, 2.2]. Moreover, we have the sub-super additivity properties [v, z] + [w, z] [v + w, z] 4.6a [v, z] + [w, z] + 4.6b [v + w, z] + [v, z] + + [w, z] + v, w, z B. 4.6c If t [, T ] ut B is an absolutely continuous and almost everywhere differentiable map, using 4.2 we see that d dt ut v = [u t, ut v] = [u t, ut v] + v B, a.e. on, T. 4.7 Since [, ] resp. [, ] + is the supremum resp. the infimum of a family of maps which are contractions with respect to the first argument and continuous w.r.t. the second one, it is also 1-Lipschitz continuous w.r.t. to the first argument and and l.s.c. resp. u.s.c. with respect to the second one, i.e. [v, z] [w, z] v w, lim w n = w n + [v, z] + [w, z] + v w, 4.8 lim inf [v, w n] [v, w], n + lim sup[v, w n ] + [v, w] +. n c Accretive Operators. Let F : B 2 B be a multi-valued operator with proper and nonempty domain DF := { v B : Fv }. With Notation 4.1 in mind, we see that F is accretive, i.e. it satisfies IS 3, if and only if [Fv 1 Fv 2, v 1 v 2 ] + v 1, v 2 DF. 4.1 This characterization cne be written in terms of the duality map J via 4.5 as follows w i Fv i, i = 1, 2, z Jv 1 v 2 : w 1 w 2, z Observe that the map λ v 1 v 2 + λ Fv 1 Fv 2 is not decreasing in [, +, 4.12 because it is convex and has a minimum at λ =. F is closed if its graph is a closed subset of B B. An accretive F is maximal if [f Fw, v w] + w DF f Fv Finally, as mentioned in the introduction, F is m-accretive if 4.1 holds and w B, ε > v DF : v + εfv w Every m-accretive operator is also maximal-accretive [3, Chap. IV, Prop. 7.2] the converse is false in general, but it is true in the framework of Hilbert spaces and every maximal-accretive operator is closed. 14

15 4.1d Yosida Regularization. For τ > we introduce the resolvent operator J τ := I + τf 1, v τ = J τ v v τ + τfv τ v, 4.15 which is a contraction of B, and the Yosida regularization of F defined as F τ : v B v J τ v, with the property F τ v FJ τ v τ It is well known that F τ is a 2τ 1 -Lipschitz accretive operator on B [3, Proposition 7.1]. Moreover, writing v τ + τfv τ v, subtracting the trivial identity v + τfv v + τfv, and using 4.1 in conjunction with 4.16, we obtain F τ v Fv v DF Explicitly Contractive Operators. 4.2a General Properties. First of all, let us recall that an operator H : B B is non expansive iff Hv Hw v w v, w B; 4.18 it is easy to see that every convex combination of non-expansive operators is non expansive again; moreover if H is non-expansive then I H is m-accretive use 4.6b and 4.4. For a fixed τ >, we say that F is a τ-explicitly contractive operator if DF = B and I τ F is non-expansive Since I λf = λ τ I τf + 1 λ τ I, λ τ, 4.2 is a convex combination of non-expansive operators, 4.19 is equivalent to ES 2 and ES 3. We say that F is strongly τ-explicitly contractive if the map λ v1 v 2 λ Fv 1 Fv 2 is non increasing in [, τ] Since the map defined by 4.21 is convex, in fact it is non increasing in, τ]. Of course, a strongly τ-explicitly contractive operator is also τ-explicitly contractive. As we already announced in the introduction, we have: Lemma 4.2 Every τ-explicitly contractive operator F is 2τ 1 -Lipschitz and satisfies [Fv Fw, v w] v, w B; 4.22 in particular, F is m-accretive. Proof. From ES 3 we get for every v 1, v 2 B, τ Fv 1 Fv 2 τfv 1 v 1 τfv 2 v 2 + v 1 v 2 2 v 1 v 2, which shows the Lipschitz character of F. In order to prove 4.22 we simply observe that just by definition v w v w λfv Fw [Fv Fw, v w] = lim, λ λ whence F is accretive because of 4.5 and 4.1. Finally, it is well known that every Lipschitz and accretive operator is also m-accretive [18, Cor. II.9.2]. 15

16 It is interesting to notice that every Yosida regularization of an m-accretive operator is explicitly contractive; in fact, it satisfies the stronger property: Lemma 4.3 If F is m-accretive, then F τ is strongly τ-explicitly contractive. Proof. We already know that F τ is an everywhere defined single-valued operator. Choose λ τ and observe that setting w i λ := v i λf τ v i we have w i λ = J τ v i + τf τ v i λf τ v i = J τ v i + τ λf τ v i. Since F τ v i FJ τ v i, by 4.12 we conclude the assertion. We can revert the previous lemma as follows. Lemma 4.4 Every strongly τ-explicitly contractive operator F can be written in a unique way as the τ-yosida regularization of an m-accretive operator G. Proof. We define G as Gv := F I τf 1 v = { Fw : v = w τfw }, 4.23 DG := { v B : v = w τfw for some w B } Now we check that G is accretive: so we fix v 1, v 2 DG, w 1, w 2 B with w i τfw i = v i, and note that v1 v 2 + λ Gv 1 Gv 2 = w1 τfw 1 w 2 τfw 2 + λ Fw 1 Fw 2 = w1 w 2 + λ τ Fw 1 Fw is a not decreasing function w.r.t. λ, by definition In order to check that G is also m-accretive, we simply observe that by definition v := w τfw is a solution of the equation v + τgv w, for every choice of w B Finally, if v is the unique solution of 4.26, as asserted. G τ w = w v τ = w w τfw τ = Fw, As a last result for these class of operators, we present a useful characterization of strongly τ-explicitly contractive operators in Hilbert spaces, which is intimately related to a coercivity type property. Proposition 4.1 Let B be a Hilbert space with scalar product,. Then F is a strongly τ-explicitly contractive operator iff Fv 1 Fv 2, v 1 v 2 τ Fv 1 Fv 2 2, v 1, v 2 B

17 Proof. We simply take the derivative of the square of the map 4.21 d v1 v 2 λ Fv 1 Fv 2 2 = dλ = 2 Fv 1 Fv 2, v 1 v 2 λ Fv 1 Fv 2, λ [, τ], and we deduce 4.27 upon choosing λ = τ. The converse is trivial from the previous equality. Corollary 4.5 Let B be a Hilbert space with scalar product,. Then F is a τ-explicitly contractive operator iff it is strongly τ/2-explicitly contractive. Proof. F is a τ-explicitly contractive operator iff v 1 v 2 τfv 1 Fv 2 2 v 1 v 2 2 v 1, v 2 B. Since B is a Hilbert space, the previous inequality is equivalent to 2τ v 1 v 2, Fv 1 Fv 2 + τ 2 Fv 1 Fv 2 2, i.e. v 1 v 2, Fv 1 Fv 2 τ 2 Fv 1 Fv 2 2. Applying Proposition 4.1, we deduce the assertion. Remark 4.6 In view of Lemmas 4.3, 4.4 and Corollary 4.5, the class of explicitly contractive operators in a Hilbert space coincides with the class of Yosida regularizations up to a factor 2 in the parameters. 4.2b Explicitly Contractive Operators w.r.t. L 1 and L Norms. Proposition 4.1 gives a direct characterization of explicitly contractive operators in the case of an Hilbert norm. Now we want to show other useful characterizations in the case of norms of L 1, L -type. First of all we recall a crucial result of Crandall and Tartar [17]. Lemma 4.7 Let Ω, S, µ be a measure space. 1. If H : L 1 Ω, S, µ L 1 Ω, S, µ is an operator satisfying Hu dµ = u dµ u L 1 Ω, S, µ Ω Ω then it is non expansive 4.18 if and only if it is order preserving, i.e. u v µ-a.e. in Ω Hu Hv µ-a.e. in Ω If H : L Ω, S, µ L Ω, S, µ is an operator satisfying Hu + c = Hu + c u L Ω, S, µ, c R, 4.3 then it is non expansive 4.18 if and only if it is order preserving Remark 4.8 In the second statement of the previous lemma it is always possible to replace L Ω, S, µ by BΩ the space of the bounded real functions defined on Ω with the sup norm or by any closed subspace containing the constant functions. In this case no measures are involved and the order preserving property 4.29 should be intended pointwise everywhere. 17

18 Corollary 4.9 Let F : R d R d be a given locally Lipschitz map with components F 1,..., F d : R d R, and let α 1,..., α d be given positive numbers; we denote by 1, respectively the norms in R d defined by d v 1 := α i v i, i=1 v := sup α i v i v = v 1,..., v d R d i d 1. Suppose that d α i F i v = v R d i=1 Then F is explicitly τ-contractive w.r.t. the 1 -norm if and only if F i x j a.e. in R d, for 1 i j d; 4.33 F i x i 1 τ a.e. in R d, for 1 i d Suppose that d j=1 F i x j = a.e. in R d, for i = 1,..., d, 4.35 or, equivalently, that for every choice of x = x 1, x 2,..., x d R d and i = 1,..., d F i x 1 + c, x 2 + c,..., x d + c = F i x 1, x 2,..., x d c R Then F is explicitly τ-contractive w.r.t. the -norm if and only if 4.33 holds. 4.3 Properties of Integral Solutions. 4.3a Inequalities in the Sense of Distributions. We recall here a definition which we will extensively use in the following. If ζ C, T, ζ L 1, T, then we say that d dt ζ + ζ in D, T if and only if β 4.37 ζβ ζα + ζt dt α β T. α Observe that if ζ is absolutely continuous, i.e. ζ W 1,1, T, then 4.37 is also equivalent to the more familiar d dt ζt + ζt for a.e. t, T

19 4.3b Functions of Bounded Variation. We denote by BV, T ; B the Banach space of functions f : [, T ] B such that with the norm Var f := sup r <r 1<...<r J T J fr j fr j 1 < +, 4.39 j=1 f BV,T ;B := f + Var f. 4.4 By extending ft := f to,, it is not difficult to see that [7, Appendix] h ft f dt + = h ft ft h dt ft ft h dt h Var f h [, T ] c Evolution Equations: Strong and Integral Solutions. Let F be an accretive operator in B. Given an initial datum u DF and a function f L 1, T ; B, 4.42 we say that u C [, T ]; B is a strong solution of the Cauchy problem { u + Fu ft u = u CP if u is also absolutely continuous in [, T ] and it satisfies CP at almost every point; in particular ut DF for a.e. t, T. It is possible to write an integral formulation of CP which will turn out to be useful: since u t = ft ξt, ξt Fut, by 4.7 we get for all v B d dt ut v = [u t, ut v] = [ft ξt, ut v] a.e. in, T Therefore, in view of 4.3, if u is a strong solution then d dt ut v + [Fut ft, ut v] +, a.e. in, T, v B, 4.44 where Fut denotes the L 1, T ; B selection ξt introduced before. If v DF, by 4.3, 4.6c and 4.1 this last formula yields d dt ut v + [Fv ft, ut v], v DF We say that u C [, T ]; B is a weak integral solution in the sense of Bénilan [4] if it satisfies 4.45 in the sense of distributions It is clear that strong solutions are also integral solutions. Conversely, if F is maximal accretive and u is a differentiable integral solution, then u is also a strong solution. 19

20 4.3d Existence, Uniqueness and Regularity Results [15, 4, 12]. If F is m-accretive, then for every u DF and f L 1, T ; B there exists a unique weak integral solution u of CP. The map u, f u is non-expansive, i.e. if v is the integral solution w.r.t. v, g then sup ut vt u v + f g L1,T ;B t [,T ] Moreover if u DF and f BV, T ; B then we introduce the following measure of regularity and compatibility of data ρu, f := f ẑ +Var f u DF, f BV, T ; B, 4.47 inf ẑ Fu and we realize that u is also Lipschitz continuous with values in B see also [3, Chap. IV, 8], [24, Chap. 5, 4], [2, Chap. III, 2] and ut us t s ρu, f s, t [, T ] Consequently 4.46 and 4.48 suggest a simple way to estimate the modulus of continuity of a general integral solution u of CP in terms of the data. We introduce the following definition which is intimately related to the Peetre s K-functional [5] see also [6] for a similar setting. Definition 4.1 Modulus of Regularity For all v B, f L 1, T ; B, the modulus of regularity ω ; v, f is the non-decreasing, concave function defined in the interval [, + by { ωε; v, f := inf v z + f ψ L1,T ;B + ερz, ψ : } 4.49 z DF, ψ BV, T ; B. A standard density argument shows that u DF, f L 1, T ; B lim ε ωε; u, f =, 4.5 whereas, taking z = u, ψ = f one gets immediately u DF, f BV, T ; B ωε; u, f ερu, f Combining 4.46 and 4.48 it is easy to see that the modulus of continuity of the integral solution u of CP w.r.t. u, f can be estimated by t s ε ut us 2ωε/2; u, f e Piecewise Constant Functions. Let us denote by P P the space of piecewise constant functions on the grid P of 1.1 { } P P := Ψ : [, T ] B : Ψ Ψn if t t n 1, t n ], n = 1,... N There is a natural projection operator Π P : L1, T ; B P P defined as Ψ = Π Pψ Ψ n = tn t n 1 ψt dt,

21 which is non-expansive w.r.t. the L 1, T ; B-norm and variation diminishing, when we choose Ψ = Ψ = ψ Ψ = Π Pψ Var Ψ Var ψ Therefore, in case f = F P P it is easy to check that the computation of ω ; v, F just invokes a minimization on a finite number of variables in B: ωε; v, F { := inf v z + F Ψ L 1,T ;B + ερz, Ψ : z DF, Ψ } 4.56 P P. 5 Relaxed Solutions: Motivation In this section we exploit IS 1 and ES 1 to construct relaxed solutions to CP. We recall cf. 1.4, 1.5 that Ut is the piecewise linear interpolant of the values {U n } N n= on the grid P = {t n } N n= of 1.1, and that Ūt, F t are the piecewise constant functions which respectively take the values U n, F n in the interval t n 1, t n ]. 5.1 Implicit Euler Scheme. We first deal with the implicit method IS 1. Theorem 2 Let {U n } N n= be the discrete solution of the Euler implicit scheme IS 1,2,3 and let U, Ū, F be defined in 1.4 and 1.5. Then d dt Ut w +[FŪt F t, Ūt w] + in D, T, w B, 5.1 where FŪt denotes the L1, T ; B selection ξt := F t U t. Proof. Since Ut and Ut w are piecewise C 1 functions, we will show that 5.1 holds a.e. in, T. Let us suppose that t t n 1, t n ] for a fixed n between 1 and N. Since we deduce that U, Ū satisfy U t U n U n 1 τ n t t n 1, t n, U t + FŪt F t t t n 1, t n. 5.2 Moreover, since Ut is linear, for every w B the map t Ut w is convex in [t n 1, t n ]; in particular its time derivative is nondecreasing and is bounded above by the left derivative at t := t n. Therefore, in view of 4.2b, we have for a.e. t t n 1, t n ] d Ut n w Ut n λ w dt Ut w lim λ λ Ut n w Ut n w λu t = lim λ λ = [U t, Ut n w] = [U t, Ūt w]. 21

22 Taking 5.2 into account and setting ξt := F t U t FŪt, we get d dt Ut w +[ξt F t, Ūt w] + [U t, Ūt w] + [ U t, Ūt w] + =, 5.3 where we have used 4.3. We thus conclude that the pair U, Ū satisfies the dissipation inequality Explicit Euler Scheme. We now consider the explicit method ES 1 and we introduce the auxiliary piecewise linear function Ũt, w interpolating the discrete values { U n w } N n= for every w B; thus for every w B where Ũt, w := lt U n w + 1 lt U n 1 w, t [t n 1, t n ], 5.4 lt := t t n 1 τ n, t t n 1, t n ]. 5.5 The reader could compare this approach with the analogous one adopted in [1, Chap. 5]. Theorem 3 Let {U n } N n= be the discrete solution of the explicit Euler scheme ES 1,2,3 and let Ũ, Ū, F be defined in 5.4 and 1.5. For every choice of w DF d dtũt, w + [Fw F t, Ūt w] + in D, T. 5.6 Proof. From ES 1 we deduce U n = U n 1 τ n FU n 1 + τ n F n. 5.7 Since ES 3 yields U n 1 τ n FU n 1 w τ n Fw U n 1 w w B, with the aid of 4.2a and 5.7, we can write U n w U n 1 w U n w U n w + τ n Fw F n τ n [Fw F n, U n w] +. We have thus arrived at U n w U n 1 w + τ n [Fw F n, U n w] +, which can be equivalently written as the dissipation inequality

23 5.3 Yosida Regularization. For τ > let J τ, F τ be the resolvent and the Yosida regularization of F introduced in 4.15 and For every u B, f L 1, T ; B Cauchy- Lipschitz-Picard theorem [8, Theorem VII.3] entails the existence of the strong W 1,1 [, T ]; B solution u τ of { u τ t + F τ u τ t = ft, t [, T ], 5.8 u τ = u. To prove a dissipation inequality for u τ, we introduce the auxiliary functions f τ t := 1 τ t e s t/τ fs ds, solutions of { τf τ t + f τ t = ft, f τ =. 5.9 Theorem 4 The functions ũ τ t, w := u τ t τf τ t w, ū τ t := J τ u τ t 5.1 satisfy the dissipation inequality d t, w + [Fū τ t f τ t, ū τ t w] + dtũτ in D, T, 5.11 where Fū τ t L 1, T ; B denotes the selection ξt := F τ u τ t Fū τ t. Proof. In view of 5.8 and 5.9 we have d dt u τ t τf τ t + F τ u τ t = f τ t t ], T [, 5.12 whence, invoking 4.7 and 4.16, d dtũτ t, w = [ d dt u τ t τf τ t, u τ t τf τ t w] = [f τ F τ u τ, u τ τf τ w] = [f τ u τ ū τ, u τ τf τ w] τ = 1 τ [ū τ w u τ τf τ w, u τ τf τ w] 1 τ [ū τ w u τ τf τ w, ū τ w] = [f τ F τ u τ, ū τ w], which implies In the last inequality, we used the monotonicity property [v w, w] [v w, v] v, w B, which follows directly from 4.6a,b,c. This completes the proof. Remark 5.1 Observe that in the homogeneous case f, equations 5.9 and 5.1 reduce to the considerably simpler form f τ t, ũ τ t, w := u τ t w, ū τ t = J τ u τ t

24 6 Relaxed Solutions: Definition and Comparison Principle The novel concept of relaxed solution of the Cauchy problem CP { u + Fu ft u = u CP for an accretive operator F is now introduced and fully discussed. This concept is inspired in 5.1, 5.6, and Definition 6.1 Relaxed solutions We say that a couple of functions is a u := ũ, ū C [, T ] B; R L 1, T ; B 6.1 strong relaxed solution of CP w.r.t. u B, f L 1, T ; B 6.2 if there exists a suitable selection ξt L 1, T ; B of Fūt s.t. d + in D, T, w B, 6.3 ũ, w u w w B. 6.4 We say that u = ũ, ū as in 6.1 is a weak relaxed solution of CP w.r.t. u B, f L 1, T ; B if d in D, T, w DF, 6.5 ũ, w u w w DF, 6.6 for all ζ Fw. The discrepancy of a relaxed solution is defined by u := δ u t dt, 6.7 where, for a.e. t, T, δ u t := sup w B ũt, w ūt w. The deviation of two relaxed solutions u, v at the time t [, T ] is defined by Dt; u, v := inf ũt, w + ṽt, w. 6.8 w B Example 6.2 Implicit Euler Scheme In view of Theorem 2, the pair u := Ũ, Ū with Ũ = U w is a strong relaxed solution w.r.t. U, F of CP, and δ u t = t n t U n U n 1 t n 1 < t t n, τ n u = 1 N τ n U n U n 1. 2 n=

25 Example 6.3 Explicit Euler Scheme In view of Theorem 3, the pair u := Ũ, Ū is a weak relaxed solution w.r.t. U, F of CP. Compared with Example 6.2, where Ũt, w is the norm of the piecewise linear function Ut w, now Ũt, w is the piecewise linear interpolant of the norms { U n w } N n=: Ũt, w = lt U n w + 1 lt U n 1 w. Using this expression, it is easy to see that δ u t and u also satisfy 6.9. Example 6.4 Yosida Regularization In view of Theorem 4, the pair u := ũ τ, ū τ defined in 5.1 is a strong relaxed solution w.r.t. u, f τ of CP with δ u t = u τ τf τ J τ u τ = τ F τ u τ f τ, u = τ Varu τ τf τ, 6.1 because F τ u τ f τ = F τ u τ f + f f τ = u τ + τf τ. Remark 6.5 A strong relaxed solution is also a weak relaxed solution. Notice that u is a strong or integral solution of CP, as defined in 4.3c, if and only if the couple ũt, w := ut w, ūt := ut, 6.11 is also a relaxed strong respectively, weak solution of the same equation with discrepancy u =. Remark 6.6 Observe that Dt; u, v, together with δ u t, δ v t, provide control of ū v. In fact, for every z B ūt vt ūt z ũt, z + ũt, z + ṽt, z + z vt ṽt, z, whence ūt vt δ u t + Dt; u, v + δ v t. In particular, we have the L 1 -error bound ū v L 1,T ;B u + v + Dt; u, v dt. Theorem 5 Let us assume that u := ũ, ū is a strong relaxed solution w.r.t. u, f in the interval, T, v := ṽ, v is a weak relaxed solution w.r.t. v, g in the interval, T. Then, the deviation DT ; u, v of u and v at time T satisfies DT ; u, v u v + f g L1,T ;B + 4Ω u v; u, f, 6.12 where the regularity function Ω associated with u, f is defined as follows in terms of the modulus of regularity ω of Definition 4.1: Ω δ; u, f := inf ωε/2; u, f + δ ε> 2ε δ

26 Remark 6.7 Recalling 4.47 and 4.51, it is easy to see that u DF, f BV, T ; B Ω δ; u, f δ ρu, f A standard density argument yields lim Ω δ; w, f = w DF, f L 1, T ; B δ The proof of Theorem 5 is based on the next three lemmas concerning extension, comparison, and stability of relaxed solution. Lemma 6.8 Extension Let u := ũ, ū be a strong weak relaxed solution w.r.t. u, f in, T and let z DF, ẑ Fz be given. If we extend u, f, ξ for t < as ũt, w := ũ, w, ūt := z, ξt ft := ẑ, 6.16 then u is a strong weak relaxed solution w.r.t. u, f in ε, T for all ε. Proof. It is a simple verification of 6.3 or 6.5 in D ε, T. Lemma 6.9 Comparison Let us fix ε > and assume that u := ũ, ū is a relaxed solution w.r.t. u, f in the interval ε, T, 6.17 v := ṽ, v is a relaxed solution w.r.t. v, g in the interval, T, 6.18 and at least one of them is strong. Then we have T ε ṽt, ūs + δ u s ds ṽ, ūs + δ u s ds ε + fs gt ds dt + 2 u + v, Q ε,t 6.19 where Q ε,t denotes the strip 2.9 of Figure 1. Remark 6.1 To see that this is indeed a comparison result, we apply it to Example 6.2 with u = U w, Ū and v = u w, u, u being an integral solution of CP see Remark 6.5. We thus get an estimate for the error T ε ut Us ds = T ε T ε ut Ūs + Ūs Us ds ṽt, ūs + δ u s ds. 6.2 Proof of Lemma 6.9. It is not restrictive to assume that u is a strong relaxed solution. Let e = e 1, e 2, h = h 1, h 2 be the auxiliary vector fields given by es, t := ũs, vt, ṽt, ūs, 6.21 hs, t := [ξs fs, ūs vt] +, [ξs gt, vt ūs]

27 Then 6.17 and 6.18 yield whence with s e 1s, t + h 1 s, t, in D ε, T for a.e. t, T, 6.23 t e 2s, t + h 2 s, t, in D, T for a.e. s ε, T, 6.24 div es, t + hs, t in D ε, T, T, 6.25 hs, t := h 1 s, t + h 2 s, t [fs gt, ūs vt] + fs gt In order to prove 6.19, we simply have to apply a slightly modified version of the Divergence Theorem in the domain Q ε,t of Figure 1; see Lemma 6.15 at the end of this section for a rigorous proof. This gives, at least formally, T ε e 2 s, T ds ε e 2 s, ds + e 2 t, t e 1 t, t + e 1 t ε, t e 2 t ε, t dt + fs gt dsdt. Q ε,t 6.27 Using the definitions of δ u t and δ v t, we obtain e 2 t, t e 1 t, t = ṽt, ūt ūt vt + ūt vt ũt, vt δ v t + δ u t, and e 1 t ε, t e 2 t ε, t = ũt ε, vt ūt ε vt + ūt ε vt ṽt, ūt ε δ u t ε + δ v t Upon adding T ε δ usds to the left-hand side of 6.27, and extracting the quantity ε δ usds from 6.28, we get the asserted estimate Corollary 1 Let us fix ε > and assume that u := ũ, ū is a relaxed solution w.r.t. u, f in the interval, T, 6.29 v := ṽ, v is a relaxed solution w.r.t. v, g in the interval, T, 6.3 and at least one of them is strong. Then we have T ε ṽt, ūs + δ u s ds u v + f g L 1,T ;B + 2 ωε/4; u, f + u + v. ε

28 Proof. We fix z DF, ẑ Fz, extend ũ, ū, f for t < as in Lemma 6.8, and apply Lemma 6.9. In view of 6.4 and 6.6, the first two terms on the right-hand side of 6.19 become ε ṽ, ūs ds ε v z ε v u + ε u z, ε δ u s ds ε u z, whereas the integral term in Q ε,t can be bounded by fs gt ds dt ft gt + fs ft ds dt Q ε,t = ε Q ε,t ft gt dt + fs ft ds dt. Q ε,t The last integral can be estimated as follows for every ψ BV, T ; B: Q ε,t fs ft ds dt 2ε f ψ L1,T ;B + ε2 ψ ẑ +Var ψ In fact, setting ψt ẑ for t <, we have fs ft ds dt Q ε,t Q ε,t fs ψs + ψs ψt + ψt ft ds dt 2ε f ψ L 1,T ;B + ε ψt ψt h dh dt, 6.33 and, invoking 4.41, ε ε ε ψt ψt h dh dt h T ψt ẑ dt + ψt ψt h dt dh h T h T ẑ ψ + Var ψ dh ε2 2 Hereafter, we use the notation ẑ ψ + Var ψ a b := mina, b, a b := maxa, b. Substituting these estimates into 6.19 and taking the infimum w.r.t. z, ẑ the definition 4.49 of ω yields Now we establish a sort of stability estimate for strong relaxed solutions. 28

29 Lemma 6.11 Stability Let ε > be fixed and let u := ũ, ū be a strong relaxed solution w.r.t. u, f in, T. We have T ε ũt, ūs + δ u s ds 2 ωε/4; u, f + u ε Proof. We apply the same extension argument of the previous corollary and the same reasoning of Lemma 6.9 to the couples u = v = ũ, ū. We observe though that in this case the vector field e defined by 6.21 satisfies e 1 t, t e 2 t, t. Therefore, 6.27 and 6.28 become ũt, ūs + δ u s ds ũ, ūs ds T ε + + ε ε Q ε,t ũs, ūs + ε ūs ūs + ε ds fs ft ds dt + 2 u. In view of 6.16, we readily have ũ, ūs u z and ũs, w ūs w u z for ε < s <. Finally, arguing as in Corollary 1, we obtain Remark 6.12 To verify that Lemma 6.11 gives indeed an estimate of the time regularity of u, we apply it to Example 6.2 with u = U w, Ū. We obtain T ε UT Us ds T ε ũt, ūs + δ u s ds 2 ωε/4; U, F + u = ε F 1 FU + Var ε 2 F + 2 u ε. We then see that this extends 2.13 to nonzero forcing F, and realize the presence of the additional term 2 u ε. This result is not sharp for Example 6.2. Remark 6.13 If u is a weak integral solution of CP and u = u w, u, as in Remark 6.5, then 4.52 gives 6.35 directly. Proof of Theorem 5. We add 6.31 and 6.35, and observe that to end up with T ε ṽt, ūs + ũt, ūs ds εdt ; u, v, DT ; u, v u v + f g L1,T ;B + 4ωε/4; u, f + 4 u + 2 v. ε Taking the infimum w.r.t. ε >, we get the asserted estimate Remark 6.14 Since the comparison Lemma 6.9 and its Corollary 1 make no distinction between strong and weak solution, so that the modulus of regularity ω of either u or v could be used, we may wonder about the assumption that u is a strong relaxed solution. This enters in the main estimate 6.12 of Theorem 5 via Lemma 6.11, and is consistent with step V of 2. The following result reverts this situation provided u is a weak integral solution. 29

30 Corollary 2 Let u be an integral solution of CP and ũt, w := ut w, ūt := ut be as in 6.11 so that u := ũ, ū is a weak relaxed solution w.r.t. u, f in the interval, T with discrepancy u =. Let v := ṽ, v be a strong relaxed solution w.r.t. v, g in the interval, T. Then, the deviation DT ; u, v of u and v at time T satisfies DT ; u, v u v + f g L1,T ;B + 4Ω 1 2 v; u, f Proof. Argue as in Theorem 5 with Remark 6.13 instead of Lemma The following lemma establishes 6.27 in a weak context where 6.23 and 6.24 are only known to hold in the sense of distributions. This is not a difficult task because of the simple geometry of Q ε,t defined in 2.9. Lemma 6.15 A Weak Divergence Theorem Let e = e 1, e 2 and h = h 1, h 2 be integrable vector fields in ε, T, T with s e 1 s, t C [ ε, T ] for a.e. t, T, t e 2 s, t C [, T ] for a.e. s ε, T. If they satisfy 6.23 and 6.24, then 6.27 holds Proof. Let us start from 6.23 in the equivalent integral form 4.37 with α := t ε, β := t for a.e. t [, T ]: e 1 t, t e 1 t ε, t + t t ε h 1 s, t ds If we integrate this inequality from t = to t = T, we get e 1 t, t e 1 t ε, t dt + h 1 s, t ds dt We now write the integral form of 6.24 between α = s and β = s + ε T, and thereby obtain as before for a.e. s ε, T e 2 s, s + ε T e 2 s, s + Q ε,t s+ε T s h 2 s, t dt. Finally we perform another integration w.r.t. s from ε to T e 2 s, s + ε T e 2 s, s ds + h 2 s, t ds dt. 6.4 ε Q ε,t We observe that the first integral of this inequality can be rewritten as e 2 s, s + ε T e 2 s, s ds = ε e 2 t ε, t dt + e 2 s, T ds e 2 s, ds e 2 t, t dt. T ε ε Substituting 6.41 into 6.4, and adding 6.39, we get