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Preliminary section
This section is composed of two paragraphs. The rst one is devoted to give a quick review on the derivation of the model of viscous lake equations. In the second one, we introduce the functional spaces that will be used in order to prove the well posedness of the model. However, if we see the Appendix section, we observe that considering Muckenhoupt weights, most of the results in the unweighted case still hold true for the weighted one. However, the weighted context also causes di culties. A big di culty concerns Sobolev-like embedding theorems. Compared to the unweighted case, the known embedding theorems cause a greater loss of regularity. This problem becomes important when dealing with the Navier-Stokes equations since embedding theorems are crucial for estimating the nonlinear term.
Formal derivation of the model
The geometry under consideration is depicted in Figure 1. The horizontal position coordinates are denoted x = (x1; x2) and range over the horizontal domain , so that the xed lateral boundaries are located at x 2 @ . The vertical position coordinate z ranges from the xed bottom at z = b(x) to z = h(x), the height of the uid over the mean free surface at z = 0. All quantities are assumed to vary horizontally on typical scales X and vertically on scales on the order of the mean depth z = B. Hence, the domain occupied by the uid, which will be denoted by , is denoted by := f(x; y; z) 2 R3; (x; y) 2 ; b(x) < z < h(x)g:
The domain is assumed to be bounded with smooth boundary @ : Because no uid enters or leaves the basin, the average level of the top surface will be independent of time.
We start from the 3D incompressible Navier-Stokes equations for homogeneous uids. The derivation of viscous shallow water equations follows from depth-integrating the Navier-Stokes equations in the case where the horizontal length scale is much greater than the vertical length scale, i.e., =XB 1:
Under this condition, conservation of mass implies that the vertical velocity of the uid is small. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface implying that the horizontal velocity eld is constant throughout the depth of the uid. Vertical integration allows us to remove the vertical velocity from the equations. The shallow water equations are thus derived. For more details, see [79], [25], [26].
After that, we consider the regime when the Froude number (the ratio of typical horizontal speeds to gravity wave speeds) is small and the wave amplitude is very small. In this limit, the evolution of the vertically averaged velocity is governed by the so-called viscous lake equations (2.3) (see [22]). Noticing that under the same scaling assump-tions but starting from three-dimensional incompressible Euler ow, the so-called lake equations, see [31], and great lake equations, see [32], have been derived. Just as in-compressible uid dynamics describes the large scale vertical motion of a uid while suppressing its acoustic waves, the lake equations describe the large scale currents in a body of shallow water while suppressing its gravity waves. Indeed, the well established validity of incompressible uid dynamics gives us con dence in the validity of not only the lake equations, but also the rigid lid equations.
Noting that the main point of discussion usually made in the physics literature when discussing the existence, rather the regularity of the velocity solution of a system des-cribing the motion of uid con ned to a shallow basin with varying bottom topography is the in uence of the bottom along the domain and the regularity of the domain . Owing to the great depth of oceanic basins, bottom stresses are often considered unim-portant in marine studies. In contrast, the shallowness of lake basins calls for careful treatment of bottom stresses. Nonetheless, we assume here that the domain is smooth enough. However, it remains to analyze the e ect of the bottom on the behavior of the velocity of the ow. To do so, at least formally, we give later two simple examples where we make a short comparison which explains what we are talking about. Simultaneously, it must not be forgotten that the situation here is more complicated because the depth vanishes at the boundary @ and hence it is very interesting to know how b varies close to the boundary before doing this analysis. In order to keep ideas clear and avoid unplea-sant technicalities, we wish here sheds a little light in two following situations. Indeed, we wish here to spotlight on the following two situations : let us start by trying to design two pictures illustrating the following situations : the rst one corresponds to the case when b = (x) with < 1: In this case, the vector rb grows to in nity close to the boundary. The second case is when > 1: In such case, the vector rb vanishes close to the boundary.
Nevertheless, in the rst case the curvature of b close to the boundary becomes as a smooth curve and the ow in this case slip and return smoothly. Together, in the mathematical literature, the previous analysis is also more logic because the velocity will be more singular when the parameter grows. In the second case, the domain presents a cusp and the mathematical analysis is out of reach actually.
Denoting by u the velocity of the ow con ned in the basin, we want to show at least formally, how the parameter a ects the behavior of u. In Figure 2-a which is the case considered in this paper, the domain is Lipschitz due to the slop of b close to the shore. Let us now explain in few words the e ect of the bathymetry behavior close to the shore on the possible regularity of the velocity eld. Indeed, for u = 1= (x) and b = (x) . It is easy to show that the velocity u belong to L2b( ) if < (1 + )=2 and then when becomes small, the velocity u becomes regular in the sense that the rate of explosion of u close to the boundary is less severity, while when grows, the rate becomes more severity.
Functional spaces
We prove that the viscous lake model (2.3) is well posed with these choices of viscosity term cited in the introduction. We use Sobolev spaces with weight b. For this reason, let us introduce the following space : the space of in nitely di erentiable and compactly supported functions, which satis es the weighted incompressibility condition :
Vb( ) = fu 2 (C1( ))2; div(bu) = 0 in ; bu n = 0 on @ g:
Also, we de ne the spaces
Hb = fu; u 2 Lb2( ); div(bu) = 0; bu n = 0 on @ g;
Vb = fu; u 2 Hb1( ); div(bu) = 0; bu n = 0 on @ g:
The de nition of weighted Lebesgue and Sobolev spaces can be found in the Appendix section. Nevertheless, we will shed here a little light on the the de nition of the trace operator since it will be necessary to understand the di culties caused by the presence of boundary conditions. Similar di culties can be also appears when we proceed to study the existence of weak solution for compressible Navier-Stokes equations in suitably bounded domains when the viscosity coe cients vanish on vacuum, see for instance [17]. However, for a weight of Muckenhoupt type, the de nition of trace operator is well
de ned. The reader at this stage can consult the works of A. Frohlich in [50], [52] for more explanation about the de nition of the trace of weighted Sobolev spaces with Muckenhoupt weights. In this work, we restrict ourselves mostly to such weight whose expression is given by b = (x); 0 < < 1 (or 1=2); (x) = dist(x; @ ) for x 2 V (@ ); where V (@ ) is a neighborhood of the boundary @ : Before giving a de nition of the trace operator in this situation, let us just make our assumptions on the domain more precise.
(I) For an integer m, 1 m 2; we set Qm = (0; 1)m: We suppose that there exists a bilipschitz mapping B : Q2 ! ; such that B(Q1) = @
We are able now to state the result proved by A. Nekvinda in [92].
Theorem 2.1. ([92], Theorem 2.8). Suppose the Hypothesis (I) holds. Then for b = (x); 1 < < q 1, there exists a unique bounded linear operator
1;q 1;q ( )!W 1 1+ ;q (@ );
Tb (@ ) : Wb q
such that
Tb1;q(@ )(u) = uj@
Remark 2.1. Notice that, one can nd a lot of domains whose satisfy the condition (I). In fact, because of Riemann Theorem’s, we know that if is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from onto the open unit disk. More information about this theorem and its application can be founded in the the book of C. Pommerenke [97].
As a consequence of Theorem 2.1, we remark that if f 2 H1 ( ); 0 < < 1 the trace of f is well de ned and belong to L2(@ ) ,! L2b(@ ): This consequence helps us to de ne the boundary integrals coming from the Navier boundary conditions (see the boundary integrals in (2.7) for example). Throughout the paper (except the cases when we mention the regularity of ), we assume that the domain satis es Hypothesis (I).
Remark about the notation : Let us x some notations which will be used throughout in the sequel :
Du : Dv = Dui;jDvi;j:
i;j=1
rb u := (@jb ui)1 i;j 2:
We shall say that u is a b-divergence free or u satis es the b-incompressibility condition if div(bu) = 0
If E is a Banach space and R2 is an open domain, we denote by Cu( ; E) the set of all uniformly continuous functions from into E.
A function f 2 C([0; T ]; Hb weak) if and only if tlimt0 jhf(t) f(t0); gibj = 0; 8g 2 Hb; 8 t0 2 [0; T ]: !
On the coercivity of a bilinear form
We consider in this section the degenerate Stokes problem below where we prove a weighted version of second Korn inequality for all functions in Hb1( ) satisfying the « b-incompressibilty condition », namely div(bu) = 0. In the proof, we exclude the case when the domain is a disk, b is radial and b is not identically zero at the boundary @ . The reason will be explained in details in the proof and it is reformulated in Remark 2.2. However, these conditions will be eliminated when we study the nonstationary case because the time-derivative term gives us further estimate on the velocity u (as in Lions-Tartar Theorem, [[103], Theorem 7.7]) which can help us to prove such an inequality without assuming the above conditions.
The importance of showing this inequality is illustrated by the fact that it allows us to establish (2.27) which is necessary in the proof of Theorem 2.2. Nevertheless, for Ab given in (i), we consider the following System,
> div(2bD(u) + 2b div u I) + brp = bf; in ;
bu n = 0; on @ ;
8 div(bu) = 0; in ;
< (2.6)
> 2b(D(u) n + div u I n) + b u
> = 0; on @ ;
where n and means the unit normal and tangential vector at the boundary respectively, = (x; ) 0 is a bounded turbulent boundary drag coe cient de ned on @
The Navier boundary condition, which has been rstly used by Navier in 1872, means that there is a stagnant layer of uid close to the wall allowing the uid to slip, and the slip velocity is proportional to the shear stress.
Therefore, we de ne the bilinear form a(u; v) : Vb Vb ! R by a(u; v) = (T u; v) = 2 Z D(u) : D(v) b dx + 2 Z div u div v b dx + Z@ u vb ds (2.7) and the linear form L : Vb ! R by L(v) = Z f v b dx:
We introduce the weak elliptic Stokes problem : nd u 2 Vb such that a(u; v) = L(v) 8v 2 Vb: (2.8)
Proposition 2.1. Suppose that b 2 A2 and (x) is de ned as above. We exclude here the case when the domain is a disk, b is radial and not identically zero on the boundary @ . Then for f 2 L2b( ), there exists a unique solution u 2 Vb solving the weak elliptic Stokes problem (2.8).
Proof. The proof is based on the Lax-Milgram theorem. The di culty is located in the proof of the coercivity of the bilinear form a: Remarkably, since b vanishes at the boundary, then the boundary integral @ juj2b ds does not have any e ect on the proof of this coercivity. A key tool employs here is the « b-incompressibilty » condition, div(bu) = 0: So, let us just sketch the proof.
Analysis of system (2.3) with di usion (i)
Our goal in this section is to establish the existence and uniqueness of global weak so-lutions of the viscous lake equations in the case when Ab = div(2bD(u )+2b div u I): Unfortunately, we leave aside here the question concerning the regularity of weak solu-tion since it needs an adequate analysis of the weights and the study of the regularity of solutions of some degenerate elliptic and parabolic equations. This subject will be discussed in a forthcoming paper. In the rst subsection, we prove the global existence of weak solutions and the second subsection is devoted to prove the uniqueness of the weak solutions.
Remark 2.3. Remark that unlike the case of classical 2D Navier-Stokes equations, the function u here is not continuous in time. As we will show later, the function bu is continuous in time and this fact make since to take the solution u as a test function.
Global solution of the viscous lake equation
This subsection is devoted to prove the existence and uniqueness of global weak solutions. Our proof di ers from the classical proof of the well posedness of the 2D Navier-Stokes equations by using the weighted Sobolev spaces. In fact, due to the presence of b in the di usion operator and the « b-incompressibilty condition » in (2.3), weighted Sobolev spaces are the natural ambient spaces. In this subsection, we will omit the indices in u for the sake of simplicity.
As we have mentioned in the beginning of Section 2, the embedding theorems in weighted spaces cause a greater loss of regularity. In the next Lemma when we wish to establish the a priori estimates concerning the nonlinear term, we will see that we need b 2 A3=2: This fact is strongly related to the regularity of the pressure proved in the end of Proposition 2.2. The details are contained in the following Lemma.
Lemma 2.1. Let be an open bounded Lipschitz domain of R2; and b 2 Aq with 1 < q 3=2 and u 2 L2(0; T ; Vb); v 2 Vb: Then
u 2 L2(0; T ; (L6b( ))2);
(u r)u 2 L1(0; T ; (L3=2( ))2);
Table of contents :
1 Introduction
1.1 Contexte général
1.2 Mécanique des fluides et théorie cinétique
1.2.1 Description macroscopique
1.2.2 Approche des milieux continus
1.2.3 Description cinétique
1.2.4 Limite hydrodynamique
1.3 Un modèle asymptotique pour les lacs
1.3.1 Le modèle de Saint-Venant
1.3.2 Le modèle des lacs
1.3.3 Nos résultats
1.3.4 Dificulté relatives aux équations à coécients dégénérées
1.4 Dérivation du modèle de ghost effect
1.4.1 Modèle de Navier-Stokes dispersif
1.4.2 Nos résultats sur le modèle de ghost effect
1.5 Nouvelle inégalité fonctionnelle
2 Degenerate lake equations
2.1 Introduction
2.2 Preliminary section
2.2.1 Formal derivation of the model
2.2.2 Functional spaces
2.3 On the coercivity of a bilinear form
2.4 Analysis of system (2.3) with diffusion (i)
2.4.1 Global solution of the viscous lake equation
2.4.2 Weighted Gagliardo-Nirenberg inequality
2.4.3 Uniqueness of weak solution
2.5 Analysis of the model (2.3) with diusion (ii)
2.5.1 Weak solution
2.6 Vanishing viscosity limits
2.7 Appendix
2.7.1 Appendix A
2.7.2 Appendix B
3 Ghost effect system
3.1 Introduction
3.2 Reformulation of the system and definitions of weak solution
3.2.1 Useful equalities
3.2.2 Reformulation of the System
3.2.3 Weak solution
3.3 Main Theorem
3.4 Proof of Theorem 3.1
3.4.1 A priori estimates
3.4.2 Construction of solution
3.5 Appendix
4 Logarithmic Sobolev inequalities
4.1 Introduction
4.2 Main results
4.3 Application to fluid dynamics systems
5 Conclusion et perspectives
5.1 Conclusion
5.2 Perspectives
Bibliographie
