Long term behaviour of singularly perturbed parabolic degenerated equation

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1 Long term behaviour of singularly perturbed parabolic degenerated equation Ibrahima Faye Université de Bambey,UFR S.A.T.I.C, BP 3 Bambey Sénégal, Ecole Doctorale de Mathématiques et Informatique. Laboratoire de Mathématiques de la Décision et d Analyse Numérique L.M.D.A.N F.A.S.E.G/F.S.T. Emmanuel Frénod Université Européenne de Bretagne, Lab-STICC UMR CNRS 39, Université de Bretagne-Sud, Centre Yves Coppens, Campus de Tohannic, F-567, Vannes Cedex, France ET Projet INRIA Calvi, Université de Strasbourg, IRMA, 7 rue René Descartes, F-6784 Strasbourg Cedex, France Diaraf SECK 3 Université Cheikn Anta Diop de Dakar, BP 6889 Dakar Fann, Ecole Doctorale de Mathématiques et Informatique. Laboratoire de Mathématiques de la Décision et d Analyse Numérique L.M.D.A.N F.A.S.E.G/F.S.T. ET UMMISCO, UMI 9, IRD, France Abstract In this paper we consider models built in [3] for short-term, mean-term and long-term morphodynamics of dunes and megariples. We give an existence and uniqueness result for long term dynamics of dunes. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the mean-term and long-term models are homogenized. Introduction and results In Faye, Frénod and Seck [3], based on works of Bagnold [], Gadd, Lavelle and Swift [5], Idier[6], Astruc and Hulcher [7], Meyer-Peter and Muller [] and Van Rijn [3], we set out that a relevant model for short term dynamics of dunes, i.e. for their dynamics over several months, is grandmbodj@hotmail.com emmanuel.frenod@univ-ubs.fr 3 diaraf.seck@ucad.edu.sn z a bmg a u z = c bmg c u u u,.

2 Introduction and results where z = z x, t, is the dimensionless seabed altitude at t and in x. For a given constant T, t [, T, stands for the dimensionless time and x = x, x T, T being the two dimensional torus R / Z, is the dimensionless position variable. Operators and refer to gradient and divergence. Functions g a and g c are regular on R + and satisfy g a g c, g c = g c =, d, sup u R + g a u + sup u R + g au d,. sup u R + g c u + sup u R + g cu d, U thr, G thr >, such that u U thr = g a u G thr. Fields u and m are dimensionless water velocity and height. They are given by ut, x = Ut, t, x, mt, x = Mt, t, x,.3 where U = Ut, θ, x and M = Mt, θ, x are regular functions on R + R T, θ U, M is periodic of period, U, U, U θ, U, M, M, M, M are bounded by d, θ t, θ, x R + R T, Ut, θ, x U thr = U =, M =, Mt, θ, x = and Ut, θ, x =, θ α < θ ω [, ] such that θ [θ α, θ ω ] = Ut, θ, x U thr..4 A relevant model for mean term, i.e. when dune dynamics is observed over a few years, is z a b mg a u z = c b mg c u u,.5 u with condition. on g a and g c and with u and m given by ut, x = Ũt, t, t, x, mt, x = Mt, t, t, x,.6 For mathematical reasons, we assumed Ũt, τ, θ, x = Ut, θ, x + U t, τ, θ, x,.7 where U = Ut, θ, x and U = U t, τ, θ, x are regular. We also assumed that M = Mt, τ, θ, x is

3 Introduction and results 3 regular and θ U, U, M is periodic of period, τ U, M is periodic of period, U, U, U θ, U, U, U, U τ, U θ, U, M, M, M θ, M, M are bounded by d, τ t, τ, θ, x R + R R T, Ũt, τ, θ, x U thr = Ũ t, τ, θ, x =, Ũ t, τ, θ, x =, Ũt, τ, θ, x =, τ M t, τ, θ, x =, M t, τ, θ, x = and Mt, τ, θ, x =, τ θ α < θ ω [, ] such that θ [θ α, θ ω ] = Ũt, τ, θ, x U thr..8 A relevant model for long-term dune dynamics is the following equation z a bmg a u z = c bmg c u u u,.9 where a, b and c are constants, where g a and g c satisfy assumption., and where z is defined on the same space as before. It is also relevant to assume ux, t = Ut, t, x = U t + U t, x + U t, t, x, mt, x = M t, x + M t, t, x,. where U = U θ, U = U θ, x, U = U t, θ, x, M = Mθ, x and M = M t, θ, x are regular and θ U, U, U, M, M is periodic of period, U, U θ, U, U θ, U, U, U, U θ, U, M, M θ, M, M, M, M θ, M are bounded by d, t, θ, x R + R T, U θ + U θ, x + U t, θ, x U thr = U t, θ, x =, U θ, x =, U t, θ, x =, M t, θ, x =, Mθ, x =, M t, θ, x =, θ α < θ ω [, ] such that θ R, θ [θ α, θ ω ] = U θ + U θ, x + U t, θ, x U thr.. Equations.,.5 or.9 need to be provided with an initial condition z t= = z,.

4 Introduction and results 4 giving the shape of the seabed at the initial time. In [3], we then gave an existence and uniqueness result for short-term model. if hypotheses.,.3 and.4 are satisfied and for the mean term one.5, if hypotheses.,.6,.7 and.8 are satisfied. This result was built on a time-space periodic solution existence result for degenerated parabolic equation. Under the same assumptions, the asymptotic behaviour of z, as, solution of short term model. is also given by homogenization methods. Futhermore if moreover U thr =, a corrector result was set out, which gives a rigorous version of asymptotic expansion of the sequence z : where U and U are solutions to where à and C are given by and à and C are given by z t, x = Ut, t, x + U t, t, x +...,.3 U à U θ = C,.4 U θ à U = C + U + à U,.5 à = a g a Ut, θ, x and C = c g c Ut, θ, x à t, θ, x = abmt, θ, x g a Ut, θ, x, and C t, θ, x = cbmt, θ, x g c Ut, θ, x Ut, θ, x Ut, θ, x,.6 Ut, θ, x Ut, θ, x..7 In [3], we did not state neither any existence result for long term model.9 nor any asymptotic behaviour result for mean term and long term models. Stating those results is the subject of the present paper. We will now state those main results. The first one concerns existence and uniqueness for the long-term model. Theorem. For any T >, any a >, any real constants b and c and any >, under assumptions.,. and., if z L T,.8 there exists a unique function z L [, T, L T, solution to equation.9 provided with initial condition.. Moreover, for any t [, T ], z satisfies z L [,T,L T γ,.9 for a constant γ not depending on and d z t, x dx T =.. dt

5 Introduction and results 5 The proof of this theorem is done in section, except equality. which is directly gotten by integrating.9 with respect to x over T. We now give a result concerning the asymptotic behaviour as of the long term model. We notice that, since U and M + M do not depend on t and x when U U thr, we have the following property: θ [, ], t, x [, T T such that Ut, θ, x = or Mθ, x + M t, θ, x = = t, x [, T T, Ut, θ, x = and Mθ, x + M t, θ, x =,. and Moreover we denote {θ [, ], U, θ, = and Mθ, + M, θ, = } is an union of several intervals.. Θ = [, T {θ R, U, θ, = and Mθ, + M, θ, = } T,.3 and Θ thr = {t, θ, x [, T R T, Ut, θ, x < U thr }..4 Theorem. For any T >, under the same assumptions as in theorem., the sequence of solutions z to equation.9 given by theorem. two-scale converges to a profile U L [, T ], L # R, L T which is the unique solution to à U = C on [, T R T \Θ,.5 U θ = on Θ thr,.6 and U dθ dx = z dx,.7 T T where à and C are given by à = a g a Ut, θ, x and C = c g c Ut, θ, x Ut, θ, x Ut, θ, x..8 Above and in { the sequel, for all p and q, we denote by L p # R, Lq T = f : R L q T mesurable and periodic of period in θ such that } θ fθ L q T L p [, ]. Remark. Notice that [, T R T \Θ Θ thr is not empty. On this set < U < U thr. This contributes to make of.5,.6 a well posed problem.

6 Existence and estimates, proof of theorem. 6 Now we turn to mean term model for which we set out asymptotic behaviours. Theorem.3 Under assumptions.,.6,.7 and.8, for any T, not depending on, the sequence z of solutions to.5 built in [3] provided with initial condition. two-scale converges to the profile U L [, T ] R, L # R, L T solution to where à and C are given by U θ à U = C,.9 à = a g a Ut, τ, θ, x and C = c g c Ut, τ, θ, x Ut, τ, θ, x Ut, τ, θ, x..3 Finally, a corrector result for the mean-term model is given under restrictive assumptions. Theorem.4 Under assumptions.,.6,.7,.8 and if moreover U thr =, considering function z L [, T, L T, solution to.5 with initial condition. and function U L [, T, L T defined by U t t, x = Ut,, t, x,.3 where U is the solution to.9, the following estimate is satisfied: where α is a constant not depending on. Furthermore, z U L [,T,L T α,.3 z U two-scale converges to a profile U L [, T ] R, L # R, L T,.33 which is the unique solution to U à U θ = C + à U U τ.34 where à and C are given by.3 and where à and C are given by à t, τ, θ, x = abmt, τ, θ, x g a Ut, θ, τ, x, and C t, τ, θ, x = cbmt, τ, θ, x g c Ut, τ, θ, x Existence and estimates, proof of theorem. Ut, τ, θ, x Ut, τ, θ, x..35 Setting: and A t, x = Ãt, t, x,. C t, x = C t, t, x,.

7 Existence and estimates, proof of theorem. 7 where and à t, θ, x = a bmt, θ, x g a Ut, θ, x,.3 C t, θ, x = c bmt, θ, x g c Ut, θ, x equation.9 with initial condition. can be set in the form z A z = C, z t= = z. Ut, θ, x Ut, θ, x,.4 Because of hyptothesis.9 and under assumptions. and., à and C given by.3 and.4 satisfy the following hypotheses: θ Ã, C is periodic of period, and x Ã, C is defined on T, à γ, C à γ, γ, C à γ, γ, à θ γ, C θ γ, à γ, C γ, C γ, G thr, θ α < θ ω [, ] such that θ [θ α, θ ω ] = Ãt, θ, x G thr, à à t, θ, x G thr = t, θ, x =, Ãt, θ, x =, C t, θ, x =, C t, θ, x =, C γ Ã, C γ Ã, à γ Ã, à γ Ã, à γ Ã, C γ Ã, C γ Ã, C γ Ã. In this section we focus on existence and uniqueness of time-space periodic parabolic equations. From this, we then get existence of solution to equation.5. Existence of z over a time interval depending on, is a traightforward consequence of adaptations of results from LadyzensKaja, Solonnikov and Ural Ceva [8] or Lions [9]. Our aim is to proove that z solution to.5 is bounded indepently of. We are going to introduce the following regularized equations. We recall that the method used is similar to the one used in [3] S θ à t,, + S = C t,,,.9 and µsµ + S µ θ à t,, + Sµ = C t,,,. where µ and are positive parameters. We first prove existence of solutions Sµ of. and we give estimates of Sµ.

8 Existence and estimates, proof of theorem. 8 Theorem. Under assumptions.6,.7 and.8, for any µ > and any >, there exists a unique Sµ = Sµt, θ, x C L R T, periodic of period with respect to θ, solution to. and regular with respect to the parameter t. Moreover, the following estimates are satisfied sup Sµθ, xdx =,. θ R T S µ L # R,L T γ,. S µ L # R,L T γ S µ θ L # R,L T γ +,.3 γ γ +,.4 γ Sµ L # R,L T + γ γ +,.5 γ Sµ L # R,L T + γ γ +,.6 S µ 3 γ L # R,L T + γ..7 Proof. of Theorem.. The proof of this theorem is very similar to the one of Theorem 3.3 of Faye, Frénod and Seck [3]. The big difference is the presence of factors in.. Hence we only sketch the most similar arguments and focus on the management of those factors. Integrating equation. over T gives then which gives T µ S Sµ µdx + T θ dx à + Sµ dx = T C dx,.8 T µ Sµdx + d S T µdx =, T θ Sµθ, xdx = Sµ θ, xe µθ θ dx. T T Since S µ is periodic of period with respect to θ, T S µθ, xdx is also periodic of period. Then. is true. Multiplying equation. by S µ, integrating over T and from to with respect to θ gives µ S µ L # R,L T + d Sµ dθ + dθ T à + S µ dxdθ γ T S µ dxdθ. Since à + and taking into account that the above first term is positive and the second one equals zero, we have S µ dxdθ γ T S µ L # R,L T,

9 Existence and estimates, proof of theorem. 9 then S µ L # R,L T γ S µ L # R,L T, which gives.. Multiplying. by S µ θ, integrating over T and integrating from to with respect to θ gives S µ θ L # R,L T γ à θ S µ dxdθ + = T T S µ L # R,L T + S µ L # R,L T which gives.4. Multiplying. by S µ, and integrating over T gives or µ Sµ dx + T Since for any real number U and V T S µ S µ dx + à S θ µ S µdx+ T à + Sµ dx = T C Sµdx, T µ Sµ + d Sµ + à + S dθ µ dx = T à S µ S µdx T C Sµdx. T using this formula with U = Sµ, V = à S µ, we have à S µ S µdx T T Taking U = S µ, V = C These two results give C Sµ T T T C θ S µdxdθ.9,. UV à + U + 4 à + V,. à + S 4 µ dx + T à + à Sµ dx. and using again. we obtain à + S 4 µ dx + T à + C dx. µ Sµ + d Sµ + à + S dθ µ dx T à + S µ dx + à S µ + C dx,. T à +

10 Existence and estimates, proof of theorem. or, using.6, µ S µ + d S µ dθ Ã + + T Sµ dx γ S µ dx +,.3 T and integrating from to with respect to θ, we have µ Sµ Ã + L # R,L T + S T µ dxdθ γ S µ dxdθ +. T From this last inequality, we deduce then S µ L # R,L T γ γ +, Sµ L # R,L T γ γ +, which gives.3. As Sµ L # R,L T is bounded by γ see., we can deduce that there exists a θ [, ] such that Sµθ, γ..4 From.3 we have d S µ dθ Integrating.5 from θ to an other θ [, ] gives S µθ, S µθ, γ γ S µ dx +..5 T θ θ Sµ dx + dθ T γ S µ L #R,LT +,.6 giving the sought bound on S µθ, L # R,L T for any θ or, in other words.5. Using Fourier expansion argument, because of., we have and then.6. We have that S µ µ S µ + Sµθ, Sµθ, γ + γ γ +,.7 is solution to S µ θ from which we deduce µ S µ + d S µ dθ Ã + S µ + T Ã + Sµ = C + Ã S µ,.8 dx = T Č S µ dx,.9

11 Existence and estimates, proof of theorem. where Č = C + Ã C S µ, Č = + Ã S µ..3 From.6,. and.3, we have Č L # R,L T γ + γ, L Č γ + γ # R,L T + γ γ +..3 Integrating.9 from to with respect to the variable θ, we obtain S µ L # R,L T γ + γ Sµ, L # R,L T then S µ γ L # R,L T 3 + γ. Using the Fourier expansion of S µ, we have for a given θ Thus, as previously, we get S µ L # R,L T S µ θ, 3 γ + γ. 3 γ + γ, S µ 3 γ L # R,L T + γ. Since the estimates of theorem. do not depend on µ, making the process µ allows us to deduce the following theorem. Theorem. Under assumptions.6,.7 and.8, for any >, there exists a unique S = S t, θ, x L R T, periodic of period with respect to θ solution to.9 and submitted to the constraint sup S θ, xdx =..3 θ R T Moreover, the following estimates are satisfied S θ L # R,L T γ γ +, S L # R,L T γ + γ γ +,.33 S L # R,L T γ + γ γ +, S 3 γ L # R,L T + γ..34 Proof. of Theorem.. As estimates of Theorem. do not depend on µ, to proof existence of S, it suffices to make µ tend to in.. Uniqueness is insured by.3, once noticed that, if S and S are two solutions of.9, with constraint.3, S S is solution to S S θ Ã + S S =,.35

12 Existence and estimates, proof of theorem. from which we can deduce that S S L # R,L T =,.36 and because of.3, and its consequence: S S L # R,L T S S L # R,L T,.37 that S = S..38 Now we get estimates on S which do not depend on. Theorem.3 Under the assumptions.6,.7 and.8, the solution S, of.9 given by theorem. satisfies the following properties à S γ,.39 θ α L # R,L T θ ω / S γ dxdθ,.4 T G thr S θ, S γ G thr, for a given θ [θ α, θ ω ],.4 S L # R,L T L # R,L T γ Gthr + γ 3,.4 γ + γ 3 + γ + γ G thr Proof. of Theorem.3. Multiplying.9 by S and integrating over T yields d S dx + à + S dx = C S dx..44 dθ T T T Integrating.44 in θ over [, ] gives then we obtain.39. Assuming.7, we have θ α T à + S dx γ θ ω / θ ω G thr S dxdθ T θ α T T à S dxdθ à S dx,.45 à S L # R,L T..46

13 Existence and estimates, proof of theorem. 3 From.39 and this last inequality we get.4. Then, there exists a θ [θ α, θ ω ] such that S satisfies.4. Using the Fourier expansion of S and the relation.3 we get S θ, S θ, γ Gthr..47 Multiplying.9 by S, integrating over T we obtain d S θ, + à + S θ, dx = dθ T C S θ, dx T Applying formula. with V = C and U = S, we get d S θ, + [ à + S θ, à + dx S θ, + dθ T T 4 à + C ] dx, which gives d S θ, + dθ à + S θ, S θ, T 4 dx T à + C dx..48 Using Fourier expansion of S θ,, one can prove that the second term of the left hand side of.48 is positive, then we have d S θ, dθ T à + C dx..49 Using.6,.8 and integrating.49 from θ to θ [, ], we obtain then inequality.4 is satisfied. Using inequality.39 and from hypothesis.8 we get à S L # R,L T S θ, S θ, + γ 3,.5 γ à S L γ..5 # R,L T Multiplying.9 by S and integrating in x T we get d dθ S + à + S dx + à S S dx = T T C S dx..5 T Using. with U = S and V = à S.5 becomes d dθ S + à + S dx T and with U = S and V = C, the equality T [ à à + S + C ] dx,.53 à + which, integrating from to yields à S dxdθ γ à S dxdθ + γ γ γ + γ..54 T T

14 Existence and estimates, proof of theorem. 4 As we obtain à γ Ã,.55 à L S # R,L T Now we set out the equation to which S then S S θ is solution to S θ = S = θ à + S is solution. We have γ + γ..56 à S + à + S + Ã, = à S + C..57 Multiplying.57 by S and integrating in x T, we get d S + à + S dx à S S dx + dθ T T Using the fact that T C T C S dx..58 γ Ã, the second term of the right hand side of.58 satisfies C S T dx γ à S dx γ à S..59 In the same way, using.8 we deduce the following estimate for the first term of the right hand side of.58 à T S S dx à S à S γ à S à S..6 Using inequalities.59,.6 and.39 and integrating.58 in θ over [, ], we have à + S L # R,L T γ à S + γ 3 à S L # R,L T L # R,L T..6 From this last inequality, we deduce à S γ + γ 3,.6 L # R,L T

15 Existence and estimates, proof of theorem. 5 and then θω θ α S dθ γ + γ3..63 G thr From.63, we deduce that there exists a θ [θ α, θ ω ] such that S θ, γ + γ3,.64 G thr and, since the mean value of S θ, is zero, S θ, γ + γ3 G thr..65 To end the proof of the theorem it remains to show that S is bounded independently of in L # R, L T. For this we will estimate the right hand side of.58 by applying formula. with V = C S and U = to treat the second term of the right hand side of.58 and with V = Ã S and U = S to treat the first. It gives: S θ S dx C Ã S Ã + dx + dx γ + 3 γ Ã S T Ã + T dx,.66 Ã + + T T where we used hypothesis.8 to get the last inequality. Integrating this last formula in θ over [θ, σ] for any σ > θ, we obtain, always remembering.39, S σ, S θ, + γ + γ From inequality.67 we obtain directly the inequality of.43, using the periodicity of S. Estimates.4 and.43 given in theorem.3 do not depend on. Making, allows us to deduce that, up to a subsequence S S L # R, L T weak. Concerning the limit S we have the following theorem. Theorem.4 Under assumptions.6,.7,.8, there exists a unique function S = St, θ, x L # R, L T, periodic of period with respect to θ, solution to and satisfying, for any t, θ R + R S θ Ãt,, S = C t,,,.68 T St, θ, xdx =..69

16 Existence and estimates, proof of theorem. 6 Moreover it satisfies: S L # R,L T γ G thr + γ 3,.7 S γ + γ 3 + γ + γ 4..7 L # R,L T G thr Proof. of Theorem.4. Uniqueness of S is not gotten via the above evoked process, but directly comes from.68. Assuming that there are two solutions S and S to.68, we easily deduce that d S S + Ã S S dx =,.7 dθ T which gives, because of the non-negativity of Ã, d S S..73 dθ From.7 we deduce that either or, for any θ R, Ã S S,.74 S θ +, S θ +, < S θ, S θ,..75 As.75 is not possible because of the periodicity of S and S, we deduce that.74 is true. Using this last information, we deduce, for instance yielding, because of property.69, Injecting.74 in.7 yields and then S S θ ω,,.76 S S θ ω, S S θ ω,..77 d S S dθ =,.78 S S θ, =,.79 for any θ θ ω and consequently or any θ R. This ends the proof of theorem.4. With this theorem on hand we can get the following result concerning z solution of equation.5. Theorem.5 Under properties.6,.7,.8, for any T, not depending on, equation.5, with coefficients given by. coupled with.3 and. coupled with.4 has a unique solution z L [, T ]; L T. This solution satisfies: where γ is a constant which do not depend on. z L [,T ],L T γ.8

17 Existence and estimates, proof of theorem. 7 Proof of Theorem.. Theorem. is a direct consequence of theorem.5. Proof. of Theorem.5. To prove uniqueness, we consider z and z two solutions of.5. Their difference is then solution to z z à z z =,.8 z z t= =, and multiplying the first equation of.8 by z z and integrating with respect to x gives d z z,.8 dt yielding and giving uniqueness. z z =, for any t,.83 Existence of z is a straightforward of adaptations of results of Ladyzenskaja, Sollonnikov and Ural Ceva [8] or Lions [9] on a time interval of length. Then, let us consider the function Z = Z t, x = St, t, x where S is solution to.69. We obtain Z Using equation.68 we deduce that Z is solution to then we deduce that Z = S t, t, x + S θ t, t, x.84 à Z = C + S z Z à z Z = S z Z t= = z S,, x. Multiplying.86 by z Z and integrating over T, we have d z Z + dt T à z Z dx = which gives d z Z dt T S z Z dx.87 γ + γ 3 + γ + γ 4 z Z..88 G thr Then we have γ + γ z t, Z t, 3 z S,, x + γ + γ 4 T..89 G thr As S L # R,L T γ Gthr when, then.8 is true.

18 3 Homogenization for long term dynamics of dunes, proof of theorem. 8 3 Homogenization for long term dynamics of dunes, proof of theorem. We consider equation.5 where A and C are defined by formulas. coupled with.3 and. coupled with.4. Our aim consists in deducing the equations satisfied by the limit of z solution to.5 as. It is obvious that with A t, x two-scale converges to Ãt, θ, x L [, T ], L # R, L T and C t, x two-scale converges to Ct, θ, x, 3. Ãt, θ, x = a g a Ut, θ, x and Ct, θ, x = c g c Ut, θ, x Assumptions.3 and.4 have the following equivalence here: and Ut, θ, x Ut, θ, x. 3. Θ = [, T {θ R : Ã, θ, = } T, 3.3 Θ thr = {t, θ, x [, T R T such that Ãt, θ, x < G thr }. 3.4 Moreover, we notice that because of. and. We have the following theorem. Ãt, θ, x = if and only if t, θ, x Θ. 3.5 Theorem 3. Under assumptions.6,.7,.8, 3., 3. and 3.5, for any T, not depending on, the sequence z of solutions to.5, with coefficients given by. coupled with.3 and. coupled with.4, two-scale converges to the profile U L [, T ], L # R, L T solution to à U = C on [, T R T \Θ, 3.6 U θ = on Θ thr, 3.7 Udθ dx = z dx, 3.8 T T where à and C are given by 3.; Θ and Θ thr are given by 3.3 and 3.4. Proof. of Theorem.. Theorem. is a direct consequence of theorem 3. Proof. of Theorem 3.. Multiplying.5 by ψ t, x = ψt, t, x regular with compact support in [, T T and -periodic in θ, we obtain z xψ,, xdx T T T A z ψ dt dx = T T T T ψ z dt dx+ C ψ dx. 3.9

19 3 Homogenization for long term dynamics of dunes, proof of theorem. 9 Using the Green formula and where ψ = ψ + ψ t, x = ψ t, t, x and ψ, 3. θ ψ t, x = ψ θ θ t, t, x, 3. we obtain T T ψ ψ + z dt dx + θ T + T C ψ dt dx = T T z A ψ dt dx T z xψ,, xdx 3. Multiplying by and using the two-scale convergence due to Nguetseng [], Allaire [], Frénod, Raviart and Sonnendrucker [4], as z is bounded in L, T, L T, there exists a profile Ut, θ, x, periodic of period with respect to θ, such that for all ψt, θ, x, regular with compact support with respect to t, x and periodic of period with respect to θ, we have then T T U à ψ dθ dt dx = T T C ψ dθ dt dx, 3.3 à U = C, 3.4 with à and C given by 3.. Since à and C vanish on Θ, we deduce 3.6 from 3.4. Moreover, because of.7, in points where Ãt, θ, x < G thr, t and x. Hence U depends only on θ. In other words, C = and à does not depend on Ut, θ, x = Uθ on Θ thr. 3.5 Taking now test functions ψ not depending on x in 3.9, the two last terms of the left hand side of 3. vanish. Then passing to the limit, we obtain the weak formulation of Ut, θ, xdx T = 3.6 θ which yields because of 3.5 U θ = on Θ thr. 3.7 Finally, taking test function ψ depending only on t we obtain ending the proof of the theorem. Ut, θ, xdθ dx = T z xdx, T 3.8

20 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.4 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.4 Making the same as in the begining of section, setting: A t, x = Ãt, t, t, x, 4. and C t, x = C t, t, t, x, 4. where and à t, τ, θ, x = a b Mt, τ, θ, x g a Ut, τ, θ, x, 4.3 C t, τ, θ, x = c b Mt, τ, θ, x g c Ut, τ, θ, x equation.5 with initial condition. can be set in the form z A z = C, z t= = z. Ut, τ, θ, x Ut, τ, θ, x, 4.4 Under assumptions. and.8, à and C given by 4.3 and 4.4 satisfy the following hypotheses: τ Ã, C is periodic of period, θ Ã, C is periodic of period, x Ã, C defined on T, à γ, C à γ, γ, C 4.6 γ, à γ. à θ γ, C θ γ, à γ, C γ, C γ, G thr, θ α < θ ω [, ] such that θ [θ α, θ ω ] = Ãt, τ, θ, x G thr, à à à t, τ, θ, x G t, τ, θ, x = thr = τ t, τ, θ, x =, Ãt, τ, θ, x =, 4.7 C t, τ, θ, x = C τ t, τ, θ, x =, C t, τ, θ, x =, C γ Ã, C γ Ã, à γ Ã, à γ Ã, à γ Ã, C γ Ã, C γ Ã, C γ à 4.8 à γ Ã, à γ Ã. τ τ For 4.5, if hypotheses 4.6, 4.7 and 4.8 are satisfied, an existence and uniqueness result is given in [3]. 4.5

21 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.4 4. Homogenization Let us consider equation 4.5 with A and C given by 4. and 4.; θ Ã, C is periodic of period, τ Ã, C is periodic of period, 4.9 A t, x 3-scale converges to Ãt, τ, θ, x L [, T ] R, L # R, L T and C t, x 3-scale converges to Ct, τ, θ, x, 4. with Ãt, τ, θ, x = a g a Ut, τ, θ, x and Ct, τ, θ, x = c g c Ut, τ, θ, x Ut, τ, θ, x Ut, τ, θ, x. 4. Theorem 4. Under assumptions 4.6, 4.7, 4.8,4.9, 4. and 4., for any T, not depending on, the sequence z of solutions to 4.5, with coefficients given by 4. coupled with 4.3 and 4. coupled with 4.4, 3-scale converges to the profile U L [, T ] R, L # R, L T solution to U θ à U = C, 4. where à and C are given by 4.. Proof. of Theorem.3. Theorem.3 is a direct consequence of theorem 4.. Proof. of Theorem 4.. Considering test functions ψ t t, x = ψt,, t, x for all ψt, τ, θ, x regular with compact support on [, T T and periodic of period with respect to τ and θ. ψ = ψ + ψ τ + ψ, 4.3 θ where ψ t, ψ x = t, t, t, x, ψ ψ = τ τ t, t, t, x, Multiplying 4.5 by ψ t, t, t, x and integrating on [, T T, we get Replacing ψ T z xψ,,, xdx T T = T T by the relation 4.3, we have T z [ ψ ψ z dt dx T C ψ dt dx. T T ψ ψ ] + + dt dx + T τ θ T + T C ψ dt dx = z xψ,,, xdx. T T ψ ψ = θ θ t, t, t, x. z A ψ dt dx z A ψ dt dx 4.4

22 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.4 Multiplying by we have T z [ ψ ψ ψ A ψ ] dt dx+ T τ θ T C ψ dt dx = z xψ,,, xdx. T T The functions ψ, ψ τ and ψ θ are periodic with respect to the two variables τ, θ. Here we use the 3-scales convergence see []. Taking the limit as, using the 3-scales convergence, we have T T [,] U ψ θ + U Ã ψ T dτ dθ dt dx = C ψ dτ dθ dt dx. T [,] Then, the limit U of z solution to.5 satisfies the following equation U Ã U θ = C. 4.5 There is indeed existence and uniqueness of the equation 4.5 according to the application of the theorem 3.5 of [3]; thus 4.5 is the homogenized equation. In 4.5, τ and t are only parameters. 4. A corrector result Considering equation 4.5 with coefficients 4. and 4. and hypothesis 4. leads to A t, x = Ã t, x + Ã t, x + Ã t, x, 4.6 where C t, x = C t, x + C t, x + C t, x 4.7 Ã t, x = Ãt, t, t, x, C t, x = Ct, t, t, x 4.8 Ã t, x = Ãt, Ã t, x = Ãt, t, t, x, t, t, x, C t, x = C t, C t, x = C t, t, t, x 4.9 t, t, x 4. Because of hypotheses 4.6, 4.7 and 4.8, Ã, Ã, Ã, Ã, Ã, Ã, C, C, C, C, C and C are regular and bounded coefficients. Theorem 4. Under assumptions 4.6, 4.7, 4.8,4.9, 4. and 4., considering function z L [, T, L T, solution to.5 and function U L [, T ], L # R, L T defined by U t t, x = Ut,, t, x, where U is the solution to 4., the following estimate is satisfied: z U L [,T,L T α, 4.

23 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.43 where α is a constant not depending on. Furthermore z U 3-scale converges to a profile U L [, T ] R, L # R, L T, 4. which is the unique solution to U à U θ = C + à U U τ. 4.3 Proof. of Theorem.4. Theorem.4 is a direct consequence of theorem 4.. Proof. of Theorem 4.. Using the relations 4.8, 4.9 and 4., equation 4.5 becomes z à z = C + C + C + à z + à z. 4.4 As U is solution to 4. and taking into account that U = U + U τ + U, 4.5 θ we obtain the following equation U à U = C + Considering equation 4.4 and 4.6, z U is solution to z U U τ U à + z U Ã+à = C + C + à U + à U U U. 4.7 τ Using the fact that U solution to 4. belongs to L [, T ] R, L # R, L T, U is solution to 4.6 and a results of Ladyzenskaja, Solonnikov and Ural Ceva [8], all the terms U τ, U are bounded. The terms Ã, Ã, C and C are also bounded by hypotheses and then so are C, C and à U, à U. Using the same arguments as in the proof of Theorem. in [3] we obtain that z U converges to a profile U L [, T ] R, L # R, L T solution to U à U θ = C + à U U τ. 4.8 References [] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 3 99,

24 4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem.3 and.44 [] R.A. Bagnold, The movement of desert sand, Proceedings of the Royal Society of London A , [3] I. Faye, E. Frénod, D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment,discrete and Continuous Dynamical Systems, Vol 9 N o 3 March, pp -3. [4] E. Frénod, P. A. Raviart, and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field, J. Math. Pures et Appl. 8, [5] P.E. Gadd, W. Lavelle, and D.J.P. Swift, Estimates of sand transport on the New York shelf using near-bottom current meter observations, J. Sed. Petrol , [6] D. Idier, Dunes et bancs de sables du plateau continental: observations in-situ et modélisation numérique, Ph.D. thesis, INP Toulouse, France. [7] D. Idier, D. Astruc, and S.J.M.H. Hulcher, Influence of bed roughness on dune and megaripple generation, Geophysical Research Letters 3 4, 5. [8] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural ceva, Linear and quasi-linear equations of parabolic type, AMS Translation of Mathematical Monographs [9] J. L. Lions, Remarques sur les équations différentielles ordinaires, Osaka Math. J , 3 4. [] J. L. Lions, D. Lukkassen, L. E. Persson, Reiterated homogenization of monotone operators C. R. Acad. Sci. Paris, t. 33, Srie I, p , Equations aux dérivées partielles/partial Differential Equations [] E. Meyer-Peter and R. Müller, Formulas for bed-load transport., The Second Meeting of the International Association for Hydraulic Structures, Appendix, 948, [] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 989, [3] L. C. Van Rijn, Handbook on sediment transport by current and waves, Tech. Report H46:..7, Delft Hydraulics, 989.