A result of existence for an original convection-diffusion equation. G. Gagneux and G. Vallet

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1 RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 99 (), 2005, pp Matemática Aplicada / Applied Matematics Comunicación Preliminar / Preliminary Communication A result of existence for an original convection-diffusion equation G. Gagneux and G. Vallet Abstract. In tis paper, we are interested in te matematical analysis of a non-classical conservation law. Te model describes stratigrapic processes of te geology, taking into account a limited weatering condition. Firstly, we present te pysical topic and te matematical formulation, wic lead to an original conservation law. Ten, te definition of a solution and some matematical tools in order to prove te existence of a solution. Un resultado de existencia para una ecuación original de difusión-convección Resumen. En este artículo se estudia el análisis matemático de una ley de conservación que no es clásica. El modelo describe procesos estatigráficos en Geología y tiene en cuenta una condición de tasa de erosión limitada. En primer lugar se presentan el modelo físico y la formulación matemática (posiblemente nueva). Tras enunciar la definición de solución, se presentan las erramientas que permiten probar la existencia de soluciones. Introduction In tis paper, we are interested in te matematical analysis of a stratigrapic model developed by te Institut Français du Pétrole (IFP). Te model concerns geologic basin formation by te way of erosion and sedimentation. By taking into account large scale in time and space and by knowing a priori, te tectonics, te eustatism and te sediments flux at te basin boundary, te model as to state about te transport of sediments. One may find, on te one and, in D. Greanjeon and al. [0] and R. Eymard and al. [7] te pysical and te numerical modelling of te multi-litological case, and, on te oter and, in S.N. Antontsev and al. [2], G. Gagneux and al. [8], [9] a matematical analysis of te mono-litological case. Let us consider in te sequel a sedimentary basin, denoted by Ξ wit base a smoot, bounded domain in R N (N =, 2), assumed orizontal. In order to describe te teoretical topograpy u(t, x) of te basin for a.e. (t, x) in Q =]0, T [, te gravitational model as to take into account: i) a sediment flux assumed to be proportional to q t = [u + τ t u], were τ is a nonnegative parameter, ii) an erosion speed, t u in its nonpositive part, tat is underestimated by E, were E is a given nonnegative bounded measurable function in Q (a weatering limited process): i.e. t u + E 0 in Q. Te original aspect of tis model is its weater limited condition on te erosion rate. Presentado por J. I. Díaz Díaz el 2 de marzo de Recibido: 5 de febrero de Aceptado: de junio de Palabras clave / Keywords: Stratigrapic models, Weater limited, Degenerated conservation laws. Matematics Subject Classifications: 35K65, 35L80, 35Q35. c 2005 Real Academia de Ciencias, España. 25

2 G. Gagneux and G. Vallet In order to join togeter te constraint and a conservative formulation, D. Greanjeon [0] proposes to correct te diffusive flux q t by introducing a dimensionless multiplier λ : te new flux is ten q cor = λ [u + τ t u], were λ is an unknown function wit values a priori in [0, ]. Ten, we will ave to add a relevant law of state in te form λ = λ( t u) for posing satisfactory closure conditions. Terefore, te matematical modelling as to express respectively: te mass balance of te sediment: t u div(λ [u + τ t u]) = 0 in Q. te boundary conditions on = Γ e Γ s specifying te input and output fluxes: q cor = f on ]0, T [ Γ e, t u + E 0, f q cor and (f q cor )( t u + E) = 0 on ]0, T [ Γ s. te weater limited condition (moving obstacle depending on climate, batymetry,... ): t u E in Q. () te initial condition u t=0 = u 0 and te closure of te model by defining te role of te flux limitor λ. In order to simplify, one considers in te sequel omogeneous Diriclet conditions on te boundary. G. Gagneux, D. Etienne and G. Vallet [6] ave considered boundary conditions of unilateral type wit τ = 0. Te matematical analysis is inspired by te capter 2 of G. Duvaut and J. L. Lions [5] and by te new problems of J. L Lions [, p. 420], dealing of termic. Ten, te problem of te regularity of t u arises in view of defining te trace of t u on Γ s. If τ = 0, t u belongs a priori to L 2 (Q) and not to L 2 (0, T, H ()). Wat justifies te interest for te study of te case τ > 0. In order to give a matematical modelling of λ, T. Gallouët and R. Masson [7] propose to consider te following global constraint, leading to a non-standard free boundary problem: t u + E 0, λ 0 and ( t u + E)( λ) = 0 in Q. (2) It means tat te flux as to be corrected because of te constraint (i.e. if λ <, ten t u + E = 0). If te erosion rate constraint is inactive, te flux is equal to te diffusive one. Ten S. N. Antontsev, G. Gagneux and G. Vallet [2] propose (if τ = 0) te following conservative formulation tat contains implicitly (). See G. Vallet [4] too. If H denotes te maximal monotone grap of te Heaviside function, ten (λ, ) formally solves: 0 = t u div(λ [u + τ t u]) were λ H( t u + E) in Q. Tis general problem remains an open problem and te aim of tis paper is to give some matematical results wen H is replaced by a continuous function a, an approximation Yosida of H for example. Anoter pysical motivation can be found in [2] and [3] for describing te ysteresis penomena of pore water. 2 Presentation of te model Tus, te problem of te stratigrapic motion may be written: find u a priori in L 2 (0, T, V ) wit t u in L 2 (0, T, V ) were V = H0 () and H = L 2 () suc tat, t u + E 0 a.e. in Q and v V, for a.e. t in ]0, T [, { t uv + a( t u + E) [u + τ t u] v} dx = 0, (3) togeter wit te initial condition u t=0 = u 0 a.e. in. 26

3 An original convection-diffusion equation In te sequel τ > 0, E : [0, T ] R is a non negative continuous function and a is a continuous function defined on R + tat satisfies: 0 a M, a(0) = 0 and a(x) > 0 if x > 0, in order to ave implicitly t u + E 0 a.e. in Q and A(x) = x a(s) ds. Moreover, and for tecnical reasons (te motivation is to 0 derive various ways of approximating te Heaviside grap by Yosida regularized functions), one assumes tat a A as got on R + a continuity modulus wose square is an Osgood function; for simplicity, we consider te case were a A is Hölder continuous wit exponent 2. Remark. Te problem can be written in te non-classical form: { t u τ A( t u + E) div(a( t u + E) u) = 0 and t u + E 0 in Q, u = 0 and t u = 0 in ]0, T [, u t=0 = u 0 in. 2. If by extension a(x) = 0 for any negative x, te suitable test function v = ( t u + E) leads to t u + E 0 a.e. in Q. So, one may assumes in te sequel suc an ypotesis on te function a. 3. If u 0 V, ten u and t u belong to L (0, T, V ). By considering ε > 0 and v = tu ds 0 a(s+e)+ε in te variational formulation, one gets: M tu 2 H + [u + τ t u] t u dx 0, (4) and M tu 2 L 2 (0,t,H) + 2 u(t) 2 H N + τ tu 2 L 2 (0,t,H N ) 2 u 0 2 H N. (5) Note tat tis is also true for any t since u C s ([0, T ], V ), i.e. u(t) exists in V for any t. Let us consider again inequality (4) to obtain wit te above estimates te assertion: M tu 2 H + τ 2 tu 2 H N C u 0 2 H N 3 Existence of a strong solution Te key for understanding compactness properties in te regularizing procedure is te following assertion: Lemma Let us consider κ given in H (), E a real number and b a bounded essentially nonnegative continuous function suc tat b B is Hölder continuous wit exponent 2, B = b. Ten, tere exists at most one solution w in V suc tat for any v in V, {wv + b(w + E) [κ + w] v} dx = 0. (6) PROOF. If one denotes by w and w 2 two admissible solutions, one gets {(w w 2 )v + [B(w + E) B(w 2 + E)] v + [b(w + E) b(w 2 + E)] κ v} dx = 0. For a given µ > 0, let us set p µ (t) = [µ,+ [ (t) + ln( e µ t) [ µ e,µ[ (t) and v = p µ (ξ) were ξ = B(w + E) B(w 2 + E); terefore, it comes (w w 2 )p µ (ξ) dx + { µ e ξ µ} 2ξ 2 ξ dx C 2 κ dx, { µ e ξ µ} and te solution is unique by considering te limits wen µ goes to

4 G. Gagneux and G. Vallet 3. Semi-discretized degenerated processes Let us consider > 0, u 0 in V, E 0 and for a given positive α, a α = max(α, a). We record for later use te following assertion tanks to Scauder-Tyconov fixed point teorem and lemma. Proposition Tere exists a unique u α in V suc tat, v V, { u α u 0 v + a α ( u α u 0 + E) [u α + τ u α u 0 ] v} dx = 0. (7) We derive from te proposition 7 and te lemma a version of te degenerated case. Proposition 2 Let us consider > 0, u 0 in V, ten tere exists a unique u in V suc tat, v V, { u u 0 v + a( u u 0 + E) [u + τ u u 0 u u 0 ] v} dx = 0, + E 0 a.e. in. (8) 3.2 Semi-discretized degenerated differential inclusion Proposition 3 Tere exists (u, λ) in V L () suc tat, λ H( u u0 + E) were H is te maximal monotone grap of te function of Heaviside, and v V, { u u0 v + λ [u + τ u u } 0 ] v dx = 0. (9) PROOF. Let us assume in tis section tat for any given positive ε, a = a ε = max(0, min(, ε Id)). Te corresponding solutions are denoted by u ε and tanks to te above inequality, a sub-sequence still indexed by ε can be extracted, suc tat u ε converges weakly in V towards u, strongly in H and a.e. in. Moreover, u u0 + E 0 a.e. in. Furtermore, A ε converges uniformly towards (.) + and ten A ε ( uε u0 + E) converges weakly in V towards u u0 + E. Up to a new sub-sequence, let us denote by λ te weak limit in L () of a ε ( uε u0 + E) and remark tat inevitably λ = a.e. in { u u0 + E > 0} i.e. λ H( u u0 + E). Passing to te limits in te variational relation stating u ɛ and since uε u0 = 0 in {λ }, one gets (9). 3.3 Existence of a strong solution Inductively, te following result can be proved: let us consider N N wit = T N, u 0 in V and E k 0 for any integer k. Proposition 4 For u 0 = u 0, tere exists a unique sequence (u k ) k in V suc tat, v V, { uk+ u k v + a( uk+ u k + E k ) [u k+ + τ uk+ u k ] v} dx = 0. Moreover, uk+ u k + E k 0 a.e. in, 28 uk+ u k 2 H + τ uk+ u k 2 V + M 2 [ uk+ 2 V + u k+ u k 2 V u k 2 V ] 0. (0) For any sequence (v k ) k H, let us note in te sequel v = N k=0 N v k+ [k,(k+)[ and ṽ = k=0 [ vk+ v k (t k) + v k ] [k,(k+)[.

5 An original convection-diffusion equation Ten, from (0), one as tat (u ) and (ũ ) are bounded sequences in L (0, T, V ) and tat ( t ũ ) is a bounded sequence in L 2 (0, T, V ). Still coming from (0), one gets uk+ u k 2 H + τ uk+ u k 2 V M 2τ uk+ 2 V and ( t ũ ) is a bounded sequence in L (0, T, V ). In particular, (ũ ) is bounded in H (0, T, V ) and tere exists a sub-sequence, still indexed by suc tat for any t, ũ (t) u(t) in V. Moreover, if t [k, (k+)[, u (t) ũ (t) V = ũ (k) ũ (t) V (k+) k t ũ (s) V ds C. Ten, for any t, u (t) u(t) in V. Since t ũ is bounded in L (0, T, V ), it follows tat for any t of Z were Z [0, T ] is a measurable set suc tat L([0, T ]\Z) = 0, t ũ (t) is a bounded sequence in V. Terefore, up to a sub-sequence indexed by t t ũ t ξ(t) in V, strongly in L 2 and a.e. in wit ξ(t) + E(t) 0 a.e. in. Let us note tat for any t in [k, (k + )[, v V, { t ũ v + a( t ũ + E ) [u + τ t ũ ] v} dx = 0. () Given tat, a( t ũ t + E) v converges towards a(ξ(t) + E) v in L 2 () N and tat [u t + ε t ũ t ] converges weakly towards [u(t) + εξ(t)] in L 2 () N, ξ(t) is a solution to te problem: at time t, find w in V wit w + E(t) 0 a.e. in, suc tat for any v in V, {wv + a(w + E(t)) [u(t) + τw] v} dx = 0. (2) Tanks to Lemma wit b = τ a in R + and κ = τ u(t), te solution ξ(t) is unique and all te sequence t ũ (t) converges towards ξ(t) weakly in V. For any f in H (), since t < f, ξ(t) > is te limit of te sequence of measurable functions t < f, t ũ (t) >, it is a measurable function tanks to Pettis teorem (K. Yosida [6, p. 3]), since V is a separable set. For any v in L 2 (0, T, V ), ( t ũ (t), v(t)) converges a.e. in ]0, T [ towards (ξ(t), v(t)). Moreover, ( t ũ (t), v(t)) C v(t) V a.e. since te sequence ( t ũ ) is bounded in L (0, T, V ). Tus, te weak convergence in L 2 (0, T, V ) of t ũ towards ξ can be proved. And one gets tat ξ = t u. At last, for t a.e. in ]0, T [, passing to limits in () leads to te existence of a solution. 4 About te differential inclusion On te one and, one may remark tat te estimations (0) remain for te sequence of solutions given by (9). Moreover, te sequence of associated parameters (λ k ) is bounded in L () and ten, one gets: v L 2 (0, T, V ), { t ũ v + λ [u + τ t ũ ] v} dxdt = 0, (3) Q were te same a priori estimates old for te sequences (ũ ) and (u ) tan te one given in te existence section. Furtermore, (λ ) is bounded in L (Q) and λ H( t ũ + E ). On te oter and, if a = a ε = max(0, min(, ε Id)) and if u ε is a solution to te problem (3), te remarks to te presentation section lead to: (u ε ) (resp. (λ ε ) ε = (a ε ( t u ε + E)) ε ) is a bounded sequence in W, (0, T, V ) (resp. L (Q)) and v L 2 (0, T, V ), { t u ε v + λ ε [u ε + τ t u ε ] v} dxdt = 0. (4) Q In bot cases, eac accumulation point provides a mild solution in te sense of P. Bénilan and al. [4] but, as already mentioned in G. Gagneux and al. [8] te double weak convergence in te term λ ε u ε does not allow us to pass to limits. 29

6 G. Gagneux and G. Vallet 5 Conclusion and open problems A solution as been found in Lip([0, T ], H0 ()) to te degenerated problem t u τ A( t u + E) div(a( t u + E) u) = 0 in Q, t u + E 0 in Q, u = 0 and t u = 0 in ]0, T [, u t=0 = u 0 in. In order to conclude tat tis problem is well-posed in te sense of Hadamard, one still as to prove tat suc a solution is unique. Tis is still an open problem, mainly due to a beaviour of ysteresis type of te equation. Let us cite a recent paper of Z. Wang and al. [5] were te uniqueness of te solution to a weakly similar equation as been proved. Te metod is based on a Holmgren approac. Anoter possibility consists in getting information on te localization and te finite speed propagation of ground via te metods of S.N. Antontsev and al. s book []. Te above solution is a solution to a perturbation of te real problem since a as to be te grap of te Heaviside function in order to satisfy te condition (2). Tis problem is open too as mentioned in te section: About te differential inclusion. References [] Antontsev S.N., Díaz J.I. and Smarev S. (2002) Energy metods for free boundary problems. Applications to nonlinear PDEs and fluid mecanics. Progress in Nonlinear Diff. Equ. and teir Appl. 48. Basel, Birkäuser. [2] Antontsev S.N., Gagneux G. and Vallet G. (2003) On some stratigrapic control problems, Journal of Applied Mecanics and Tecnical Pysics (New York), 44(6) [3] Bainov D. and Simeonov P. (992) Integral inequalities and applications, 57, Kluwer Acad. Pub., Dordrect. [4] Bénilan P., Crandall M.G. and Pazy A. (988) Bonnes solutions d un problème d évolution semi-linéaire, C. R. Acad. Sci. Paris, Sér. I, 306, [5] Duvaut G. and Lions J.L. (972) Les inéquations en mécanique et en pysique, Dunod, Paris. [6] Etienne D., Gagneux G. and Vallet G. (2005) Nouveaux problèmes unilatéraux en sédimentologie, publication interne du Laboratoire de Matématiques Appliquées, University of Pau, to appear. [7] Eymard R., Gallouët T., Granjeon D., Masson R. and Tran Q.H. (2004) Multi-litology stratigrapic model under maximum erosion rate constraint, Internat. J. Numer. Metods Eng. 60(2) [8] Gagneux G., Luce R. and Vallet G. (2004) A non-standard free-boundary problem arising from stratigrapy, publication interne du Laboratoire de Matématiques Appliquées n 2004/33, University of Pau. [9] Gagneux G. and Vallet G. (2002) Sur des problèmes d asservissements stratigrapiques, Control, Optimisation and Calculus of Variations, 8, [0] Granjeon D., Huy Tran Q., Masson R. and Glowinski R. (2000) Modèle stratigrapique multilitologique sous contrainte de taux d érosion maximum, Institut Français du Pétrole. Rapport interne. [] Lions J.L. (969) Quelques métodes de résolution des problèmes aux limites non linéaires, Dunod, Paris. [2] Mualem Y. and Dagan G. (975) Dependent domain model of capillary ysteresis, W. Res. Rec., (3), [3] Poulovassilis A. and Cilds E. C. (97) Te ysteresis of pore water: te non-independence of domains, Soil Sci., 2(5), [4] Vallet G. (2003) Sur une loi de conservation issue de la géologie, C. R. Acad. Sci. Paris, Sér. I, 337,

7 An original convection-diffusion equation [5] Wang Z. and Yin J. (2004) Uniqueness of solutions to a viscous diffusion equation, Appl. Mat. Letters, 7(2) [6] Yosida K. (965) Functional analysis, Berlin, Springer-Verlag. XI. G. Gagneux G. Vallet Lab. de Matématiques Appliquées Lab. de Matématiques Appliquées UMR CNRS 542 UMR CNRS 542 IPRA BP 55 IPRA BP Pau Cedex (FRANCE) 6403 Pau Cedex (FRANCE) 3