Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment

In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.


Introduction and results
Dune and megaripple generation and dynamics, on the seabed over a continental shelf, are the results of interaction between the seabed and water currents. The study of the physical processes allowing for the generation of dunes, or governing their evolution or stability involves modeling and numerical simulation. Roughly speaking, the models in use essentially couple an equation for the fluid fields (Navier-Stokes or shallow water equations) to an equation describing sand transport on the seabed. Those methods were used with success in DeVriend [10], Engelund and Hansen [11], Kennedy [19], Blondeau [6], Dawson, Johns and Soulsby [9], Johns, Soulsby and Chesher [18], Idier [16] and Idier, Astruc and Hulsher [17]. A careful watch reveals that the use of numerical simulation for the understanding of dune dynamics within tide-influenced environment is essentially not efficient. The reason why is that tide oscillation generally prompts a coming and going of large sand volumes having a very weak resulting effect on dune evolution. As a consequence, questions concerning dune morphodynamics or stability have to be considered over large periods of time, making the computation cost expensive. Since many dune fields are present in strong tide region (English Channel, Celtic Sea, Irish Sea, North Sea, etc.) the setting out of methods to tackle dune morphodynamics in tide influenced environments is an important challenge. The aim of this paper is to carry out modeling methods and asymptotical methods for this. More precisely, we focus on linear models for seabed evolution and on methods which allow the removal of the explicit presence of the tide oscillations from them.
As will be seen in section 2, for a small parameter ǫ and constants a, b and c, equation is a relevant model for the short-term dynamics of dunes. In equation (1.1), z ǫ = z ǫ (x, t) where, for a given constant T, t ∈ [0, T ), stands for the dimensionless time and x = (x 1 , x 2 ) ∈ T 2 , T 2 being the two dimensional torus R 2 / Z 2 , is the dimensionless position variable, is the dimensionless seabed altitude at time t and in position x. Operators ∇ and ∇· refer to gradient and divergence. Functions g a and g c are regular on R + and satisfy                g a ≥ g c ≥ 0, g c (0) = g ′ c (0) = 0, ∃d ≥ 0, sup u∈R + |g a (u)| + sup u∈R + |g ′ a (u)| ≤ d, sup u∈R + |g c (u)| + sup u∈R + |g ′ c (u)| ≤ d, ∃U thr ≥ 0, ∃G thr > 0, such that u ≥ U thr =⇒ g a (u) ≥ G thr .
θ, x) = 0 and ∇U(t, θ, x) = 0, (1.4) Remark 1.1 The last two assumptions in (1.4) are necessary when g a may vanish. In the case where g a (u) ≥ G thr for any u ≥ 0, then U thr = 0 and the last two assumptions of (1.4) are automatically satisfied by any U.
The following equation, for constants a, b and c with condition (1.2) on g a and g c and with u and m given by is a relevant model for mean-term dune dynamics. For mathematical reasons, we assume where U = U(t, θ, x) and U 1 = U 1 (t, τ, θ, x) are regular. We also assume that M = M(t, τ, θ, x) is regular and (1.8) A relevant model for long-term dune dynamics is the following equation where a, b and c are constants, where g a and g c satisfy assumption (1.2), and where z is defined on the same space as before. It is also relevant to assume Equations (1.1), (1.5) or (1.9) need to be provided with an initial condition 12) giving the shape of the seabed at the initial time.
We will now state the main results of the paper. The first concerns existence and uniqueness for the short and mean-term models.
there exists a unique function z ǫ ∈ L ∞ ([0, T ), L 2 (T 2 )), solution to equation (1.1) or (1.5) provided with initial condition (1.12). Moreover, for any t ∈ [0, T ], z ǫ satisfy for a constant γ not depending on ǫ and The proof of this theorem is done in section 3, except equality (1.15) which is directly gotten by integrating (1.1) or (1.5) with respect to x over T 2 .
In the previous theorem, L 2 (T 2 ) stands for the usual space of square integrable functions defined on the torus T 2 and L ∞ ([0, T ), L 2 (T 2 )) stands for the space of functions mapping [0, T ] to L 2 (T 2 ) and which are bounded. . L ∞ ([0,T ),L 2 (T 2 )) stands for the usual norm on this space.
Remark 1.2 As equations (1.1) and (1.5) are linear, almost parabolic equations, the proof of the existence of z ǫ over a time interval depending on ǫ is a straight forward consequence of adaptations of results from Ladyzenskaja, Solonnikov and Ural'Ceva [21] or Lions [22]. But here, since we want to follow an asymptotic process consisting in making ǫ → 0, we need a time interval which does not depend on ǫ. Because of the presence of 1 ǫ factor and the fact that the diffusion term may cancel, the proof of theorem 1.1 needs several steps. In a first step, we prove the existence of a solution, periodic in time and space of a parabolic equation. From this first existence result, we deduce existence of a solution, periodic in time and space of an ad-doc degenerate parabolic equation. Those two results are interesting by themselves and complete the theorem collection in the topic of time and space time-periodic solution to parabolic equation. Concerning this topic, we refer for instance to Barles and Souganidis [3], Berestycki, Hamel and Roques [4,5], Bostan [7], Hansbo [15], Kono [20], Nadin [25,24], Namah and Roquejoffre [26] and Pardoux [29]. Then, having on hand the existence of the space-time periodic solution to the ad-doc degenerate parabolic equation, we can deduce that the solution z ǫ which exists on ǫ-dependant time interval, remains close to it. This allows us to deduce a large time existence. Remark 1.3 Moreover, notice that theorem 1.1, theorems 3.16 and 3.17 also complete the theorem collection concerning the topic of large time behavior of parabolic equation (see Barles and Souganidis [3], Da Lio [8], Norris [28], Park and Tanabe [30], Pardoux [29], Petita [31] and Tanabe [32].
We now give a result concerning the asymptotic behavior as ǫ −→ 0 of the short-term model.
where A and C are given by In this theorem, L ∞ # (R, L 2 (T 2 )) stands for the space of functions depending on θ and x mapping R to L 2 (T 2 ) and which are periodic of period 1 with respect to θ and L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) stands for the space of functions mapping [0, T ] to L ∞ # (R, L 2 (T 2 )) and which are bounded. For the definition and results about two-scale convergence we refer to Nguetseng [27], Allaire [1] and Frénod Raviart and Sonnendrücker [13].
Finally, we give a corrector result for the short-term model under restrictive assumptions.

Theorem 1.3 Under the same assumptions as in theorem 1.1 and if moreover
where U is the solution to (1.16), the following estimate is satisfied: where α is a constant not depending on ǫ. Furthermore z ǫ − U ǫ ǫ two-scale converges to a profile U 1 ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))), (1.20) which is the unique solution to where A and C are given by (1.17) and where A 1 and C 1 are given by

Sand transport equation
The equation modeling sand transport is the following (see Van Rijn [33], Idier [16]): In this equation the fields depends on time t ∈ [0, T ), for T > 0, on the horizontal position x = (x 1 , x 2 ) ∈ Ω, where Ω is a regular open set of R 2 . The field z = z(t, x) is the height of the seabed in position x and at time t and q = q(x, t) is the sand volume flow in x and at t. The parameter p ∈ [0, 1) is called sand porosity. Equation (2.1) has to be coupled with a low linking the sand flow q with the seabed height variation and the velocity of the water near the seabed. Usually, such a law is written where q f stands for the water velocity induced sand flow on a flat seabed and where |q f | stands for its norm. The constant λ is the inverse value of the maximum slope of the sediment surface when the water velocity is 0. A generic way to write q f is where g is a non-negative regular function defined on R + and where χ is a regular function from R to R, being 0 on R − and increasing on R + . u is the water velocity near the seabed, g(u) is regular function of u ∈ R + and u c is the threshold under which the water velocity does not make the sand move. Every law encountered in the literature, for instance Meyer-Peter and Müller [23] formula, Bagnold and Gadd formula (see [2] and [14]) and Van Rijn [33] formula, is recovered by setting functions χ and g.
In the sequel of the present paper we shall restrict ourselves to laws of the Van Rijn type [33] which consists in writting where τ b is the shear stress density imposed by the water on the seabed. It is linked with u by where ρ is the water density, C is a constant defined by C = ln( 12d 3DG ), d being the water height above the seabed and D G being the sand speck diameter. The threshold τ c expresses as and χ is given by The order of magnitude of constant α is 100.

Scaling
Now, we will scale (2.9) to write a dimensionless version of it. We introduce a characteristic timet and a characteristic lengthL and we define the dimensionless variables t ′ and x ′ , makingt andL the units by We also definez the characteristic height of the dunes and the dimensionless seabed height Concerning coefficients of equation (2.9), we introduceū the characteristic velocity of the water, we consider the mean water height H andM the characteristic height variation due to the tide. Then we define u ′ being the dimensionless water height variation by Once those variables and fields are introduced, we first approximate C, taking into account thatM H is small.
we get from equation (2.9) the following equation (2.16) Having this dimensionless model on hand, we will now consider several situations in setting the characteristic values for short, mean and long-term dune evolution and for small and big sand specks. Moreover, the order of magnitude of coefficient λ 1−p is 1, then we get Now we detail the sizes of every characteristic value and of their concerned ratios in equation (2.14), (2.16) for every situation. Short-term dynamics of dunes made of a small sand specks Here, we shall consider thatt is an observation period of time. We take ast the order of magnitude of the smallest period of time during which dunes undergo significant evolution in a tide-submitted environment, i.e.t = 100 days ∼ 2400 hours ∼ 8.6 10 6 s. Introducingω the main tide frequency,t has to be compared with the main tide period 1 ω ∼ 13 hours ∼ 4.7 10 4 s. This leads to the definition of a small parameter ǫ: We consider that the sand speck diameter D G is 0.1mm = 10 −4 m. According to Flemming [12] and Idier [16], this gives rise to dunes being about 1 meter high, the wave length of which is about 10 meters. Then we setz = 1 m andL = 10 m. (2.20) We also consider that the critical velocity u c is small compared withū. In other words we set As the computations of the factors in (2.20) yields where we removed the '. Concerning fluid fields u and m, we assume that they are periodic functions, with modulated amplitude, and of period the tide period. In other words for functions U and M being regular, and such that θ −→ (U(t, θ, x), M(t, θ, x)) is periodic of period 1, with a null mean value.
Finally, as dunes of the considered kind are, in nature, gathered into dunes fields it is not completely unrealistic to set equation (2.23) in a periodic position space. As matter of the fact, considering equation (2.23) is appropriate for the study of short-term dynamics of dunes made of small sand specks with a mathematical point of view.
Short-term dynamics of dunes made of a big sand specks For this regime, we consider:t ∼ 100 days ∼ 2400 hours ∼ 8.6 10 6 s, Mean-term dynamics of dunes made of a small sand specks By mean-term we mean a period of time of 4.5 years ∼ 54 months ∼ 1.4 10 8 s. Then, we takē t = 1.4 10 8 s, which is compared with 1 ω ∼ 13 hours ∼ 4.7 10 4 s giving We also consider a second tide period which is the time for the earth, the moon and the sun to recover approximately the same relative positions. This period of time 1 ωc is about one month. So we have 1 We also take D G = 5 10 −5 m andz Computing the coefficients in equation (2.16) gives As was previously seen, it is reasonable to set this equation in a periodic domain and concerning the fluid fields we consider . (2.32) to take into account the two tide periods under consideration. In (2.32) we take U and M as regular are periodic of period 1.
Long-term dynamics of dunes made of small sand specks We take heret ∼ 16 years ∼ 1.4 10 5 hours ∼ 5 10 9 s. We compare this period of time with the second tide period 1 ωc ∼ 1 month ∼ 2.6 10 6 s. Then, we define ǫ by with those values equation (2.16) yields As, at the second tide period scale the tide phenomena may almost be considered as really periodic we set is periodic of period 1 and such that 3 Existence and estimates, proof of theorem 1.1 Defining and where In the same way, setting and and defining A ǫ and C ǫ from A ǫ and C ǫ by (3.1) on R + × R × R × T 2 , for a constant γ depending only on a, b, c and d and not on ǫ.
Concerning (3.9) in the case where A ǫ and C ǫ are defined by (3.6) and (3.7), it reduces to and there exists a constant G thr depending only on a, b, d and G thr and two numbers θ α and θ ω in (3.13) We also have the following inequalities | C ǫ | ≤ γ| A ǫ |, (3.14) possibly changing the value of γ and making it also depend on G thr . Inequality (3.14) is a direct consequence of (1.2). Since for small values of | C ǫ |, | C ǫ | 2 ≤ | C ǫ | and since C ǫ and A ǫ are bounded, inequality (3.15) follows from (3.14). When | A ǫ | ≤ G thr , then ∇ A ǫ = 0. Hence (3.16) is realized. When | A ǫ | ≥ G thr , since A ǫ and ∇ A ǫ are bounded, (3.16) is obviously realized. Hence (3.16) is true. With a similar argument (3.17)-(3.20) may be obtained. In order to obtain (3.21), we just have to notice that, when | A ǫ | ≤ G thr , ∇ · C ǫ = 0. Hence we can give the same argument as above. In the same manner, the last two inequalities may be obtained.
As estimate (3.25) depends on ν, letting ν go towards 0 we obtain the following corollary.

Corollary 3.2 For any
, and under assumptions (3.11), then for any ǫ > 0, there exists a unique solution z ǫ ∈ L ∞ ([0, T ); L 2 ∩ L ∞ (T 2 )) to (3.5). Moreover it satisfies The uniqueness of the solution to (3.5) is a direct consequence of the linearity of this equation.
As we want to study the asymptotic behavior of z ǫ as ǫ goes to 0, estimates (3.26) and (3.25) are not enough. We need estimates which do not depend on ǫ. For this, we first consider the following problems, which consists in finding S ν = S ν (t, τ, θ, x) and S ν µ = S ν µ (t, τ, θ, x) being periodic of period 1 with respect to θ, solutions to and In equations (3.27) and (3.28) t and τ are only parameters. The method to get the desired estimates which do not depend on ǫ is shared in several steps. In the first, we set out the existence of periodic solution S ν µ to equation (3.28). We, moreover, set out that sequence S ν µ is bounded independently of µ and ǫ. We also show that S ν µ is differentiable with respect to t and τ. In a second step, letting µ go to 0, we get existence of S ν , with the same properties as S ν µ . The third step consists in finding estimates on S ν , ∂S ν ∂t and ∂S ν ∂τ which are independent of ν to be able to make the process ν −→ 0 and to obtain the existence of a solution S to (3.27) with ν = 0 and consequently a periodic solution Z ǫ to an equation close to (3.5) in a fourth step. The fifth step consists in noticing that the solution z ǫ of (3.5) is not far from Z ǫ . The framework of periodic solutions to parabolic equations is widely studied in both linear and nonlinear cases. We refer for instance to Barles[3], Berestycki-Hamel and Roques [5,4], Bostan [7], Hansbo [15], Kono [20], Nadin [25,24], Namah and Roquejoffre [26] and Pardoux [29] for a revue of the results on this topic that our result (see theorem 3.3 and 3.10) completes. Inspired by ideas that may be found in those references, concerning equation (3.28) we can state the following theorem. Theorem 3.3 Under assumptions (3.8), (3.9) and (3.11), for any µ > 0 and any ν > 0, there exists a unique S ν µ = S ν µ (t, τ, θ, x) ∈ C 0 ∩ L 2 (R × T 2 ), periodic of period 1 with respect to θ, solution to (3.28) and regular with respect to the parameters t and τ . Moreover, there exists a constant γ 3 , which depends only on γ and ν such that The following estimates with respect to the parameters t and τ are also true In the above theorem, the norms are defined by Proof of theorem 3.3. The point of departure to prove theorem 3.3 consists in considering, for ξ ∈ L 2 (T 2 ), the solution ξ ν µ to (3.36) whose existence and uniqueness on any finite interval is a direct consequence of Ladyzenskaja, Solonnikov and Ural'Ceva [21] or Lions [22]. We also consider the application and for it we can prove the following lemma.
Since A ǫ + ν > 0, we can deduce that giving the lemma.
Since A ǫ and C ǫ are periodic with respect to θ we deduce that the sought periodic solution S ν µ of (3.28) is nothing but the solution of (3.36) associated with initial condition ζ such that (ζ) = ζ. Hence we proved the existence of S ν µ claimed in theorem 3.3. The fact is that S ν µ is continuous comes from the fact that it is a solution of a parabolic equation with regular coefficients. We now turn to the properties of the function S ν µ .
Proof of lemma 3.9. Obviously, we get that ∂S ν µ ∂t is a solution to (3.68) From the previous estimates, we can deduce that there exists a constant γ 5 , which only depends on γ and ν, such that (3.69) Proceeding in the same way as when proving lemma 3.6, we get . (3.71) Proceeding now as in the proof of lemma 3.7, we get and using (3.70), from which we can deduce for a given θ 0 , we finally obtain Since the mean value of ∂S ν µ ∂t (θ, ·) is 0 for any θ ∈ R, using the same argument as in lemma 3.8, we conclude that the first estimate of (3.33) is true. We can do the same for the second estimate, to end the proof of the lemma and also of theorem 3.3.
Since the estimates in theorem 3.3 do not depend on µ, making the process µ −→ 0 allows us to deduce the following theorem. Theorem 3.10 Under the assumptions (3.8), (3.9) and (3.11), for any ν > 0, there exists a unique S ν = S ν (t, τ, θ, x) ∈ L 2 (R×T 2 ), periodic of period 1 with respect to θ solution to (3.27) and submitted to the constraint sup Moreover, there exists a constant γ 3 which depends only on γ and ν such that Proof of theorem 3.10. As previously said, existence of S ν follows from making µ tend to 0 in (3.28). Formulas (3.75)-(3.77) directly come from estimates (3.29)-(3.32). Uniqueness is insured by (3.75), once noticed that, if S ν and S ν are two solutions of (3.27), with constraint (3.75), S ν − S ν is solution to from which we can deduce that and because of (3.75), and its consequence: This ends the proof of theorem 3.10.
Having on hand theorem 3.10, we will set out the properties of S ν which will allow us to make the process ν −→ 0.
Lemma 3.14 Under assumptions (3.8), (3.9), (3.11), (3.12), (3.13) and (3.14) -(3.23) the solution S ν to (3.27) given by theorem 3.10 satisfies Proof of lemma 3.14. In a first step, remembering inequality (3.84) proved in the beginning of the proof of lemma 3.11, and using (3.18), we deduce In a second step, following the way to prove lemma 3.7, we multiply (3.27) by −∆S ν and we integrate the resulting equality in x ∈ T 2 to get Now using (3.16), we deduce In the third step, we set out the equation to which ∂S ν ∂t is a solution. In a way similar to the one followed in the proof of lemma 3.9, we obtain (3.100) Multiplying equation (3.99) by ∂S ν ∂t and integrating in x ∈ T 2 , in the same spirit as in the proofs of lemma 3.6 and 3.9, we deduce To estimate the right hand side of (3.101), we first notice that, applying (3.23), we have (3.102) Then using (3.17) we deduce As a consequence of (3.102), (3.103) and (3.84), integrating (3.101) in θ over [0, 1] yields , (3.104) where once again (3.84) is also used. From this last inequality, we deduce and then The fourth step consists in deducing from (3.106) that there exists a θ 0 ∈ [θ α , θ ω ] such that and, since the mean value of ∂S ν ∂t (θ 0 , ·) is zero, (3.108) The fifth and last step, consists in going back to (3.101). Applying formula (3.53) with U = | ∂ Cǫ ∂t | and V = |∇ ∂S ν ∂t | to treat the first term of the right hand side of (3.101) and with U = | ∂ Aǫ ∂t ||∇S ν | and V = |∇ ∂S ν ∂t | to treat the second, we get where we used (3.23) and (3.17) to find the last inequality. Integrating this last formula in θ over [θ 0 , σ] for any σ > θ 0 , we obtain, always remembering (3.84), Inequality (3.110) yields directly the first inequality of (3.93), using the periodicity of S ν . The proof of the second inequality of (3.93) is done in a similar way. This ends the proof of lemma 3.14.
As neither estimate (3.87) nor estimate (3.93) depend on ν, we can deduce that, extracting a subsequence, as ν −→ 0, S ν −→ S in L ∞ # (R, L 2 (T 2 )) weak-*, and that the limit satisfies estimates looking like (3.87) and (3.93), but also a property of the type (3.75), and that it is solution to equation (3.27) with ν = 0. In other words, we deduce the following theorem.
Proof of theorem 3.16. To prove uniqueness, we consider z ǫ 1 and z ǫ 2 two solutions of (3.5). Their difference is then solution to and multiplying the first equation of (3.125) by (z ǫ Multiplying (3.130) by z ǫ − Z ǫ and integrating in x, using estimates (3.114) and (3.115) gives: Coupling local existence and estimate (3.124) yield global existence and then theorem 3.16 is true.
As a consequence theorem 1.1 is also true.
As a by-product of theorem 3.10, using a way similar of the one used to prove theorem 3.16 we can obtain a theorem giving long-term existence of space-periodic solution to parabolic equation.
for an extracted subsequence. As A ǫ and C ǫ are bounded (see (3.8)) and ψ ǫ is a regular function, A ǫ ∇ψ ǫ and ∇ψ ǫ can be considered as test functions. Using (4.1) we have and T 2 T 0 C ǫ · ∇ψ ǫ dtdx two scale converges to From this we obtain from (4.11) the equation satisfied by U : which is equation (4.3).
Existence and uniqueness of equation (4.3) is given by theorem 3.15 (applied with ǫ-independent coefficient and right hand side). From this uniqueness, we can deduce that the whole sequence (z ǫ ) converges.
Proof of theorem 5.1. Using (5.1) and (5.2) equations (3.5) becomes (5.14) Multiplying equation (5.14) by 1 ǫ and using the fact that z ǫ = z ǫ − U ǫ + U ǫ in the right hand side of equation (5.14), z ǫ −U ǫ ǫ is solution to: Our aim here is to prove that z ǫ −U ǫ ǫ is bounded by a constant α not depending on ǫ. For this let us use that A ǫ , A ǫ 1 , C ǫ and C ǫ 1 are regular and bounded coefficients (see (5.6)) and that A ǫ ≥ G thr (see (5.7)). Thus, ∇ · C ǫ 1 is bounded, ∇ · ( A ǫ 1 ∇U ǫ ) is also bounded. Since U ǫ is solution to (5.13),  [21], ∂U ∂t is regular and bounded . Then the coefficients of equations (5.15) are regular and bounded. Then, using the same arguments as in the proof of theorem 1.1 we obtain that z ǫ −U ǫ ǫ is bounded, that it two-scale converges to a profile U 1 ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) and that this profile U 1 satisfies equation (1.21).