TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS IN BOUNDED DOMAINS

Transcription

1 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS IN BOUNDED DOMAINS CHRISTOPHE LACAVE, TOAN T. NGUYEN, AND BENOIT PAUSADER Astract. We investigate the influence of the topography on the lake equations which descrie the two-dimensional horizontal velocity of a three-dimensional incompressile flow. We show that the lake equations are structurally stale under Hausdorff approximations of the fluid domain and L p perturations of the depth. As a yproduct, we otain the existence of a weak solution to the lake equations in the case of singular domains and rough ottoms. Our result thus extends earlier works y Bresch and Métivier treating the lake equations with a fixed topography and y Gérard-Varet and Lacave treating the Euler equations in singular domains. Contents. Introduction.. Weak formulations 3.2. Assumptions 5.3. Main results 5 2. Well-posedness of the lake equations for smooth lake Auxiliary elliptic prolems Existence of a gloal weak solution Well-posedness of a gloal weak solution 6 3. Proof of the convergence Vorticity estimates harmonic functions: Dirichlet case harmonic functions: constant circulation Estimates of αn k Kernel part with Dirichlet condition Convergence of αn k Passing to the limit in the lake equation Non-smooth lakes 23 Appendix A. Equivalence of the various weak formulation 25 Appendix B. γ-convergence of open sets 28 References 28. Introduction The lake equations are introduced in the physical literature as a two-dimensional geophysical model to descrie the evolution of the vertically averaged horizontal component of the three-dimensional velocity of an incompressile Euler flow; see for example [5, 2,, ] and the references therein for physical discussions and derivation of the model. Precisely, the lake equations with prescried initial Date: Novemer 2, 23.

2 2 C. LACAVE, T. NGUYEN, B. PAUSADER and oundary conditions are t (v) + div (v v) + p = for (t, x) R +, div (v) = for (t, x) R +, (v) ν = for (t, x) R +, v(, x) = v (x) for x. Here v = v(t, x) denotes the two-dimensional horizontal component of the fluid velocity, p = p(t, x) the pressure, = (x) the vertical depth which is assumed to e varying in x, R 2 is the spatial ounded domain of the fluid surface, and ν denotes the inward-pointing unit normal vector on. In case that is a constant, (.) simply ecomes the well-known two-dimensional Euler equations, and the well-posedness is widely known since the work of Woliner [2] or Yudovich [3]. When the depth varies ut is ounded away from zero, the well-posedness is estalished in Levermore, Oliver and Titi []. Most recently, Bresch and Métivier [] extended the work in [] y allowing the varying depth to vanish on the oundary of the spatial domain. In this latter situation, the corresponding equations for the stream function are degenerate near the oundary and the elliptic techniques for degenerate equations are needed to otain the well-posedness. In this paper, we are interested in staility and asymptotic ehavior of the solutions to the aove lake equations under perturations of the fluid domain or rather perturations of the geometry of the lake which is descried y the pair (, ). Our main result roughly asserts that the lake equations are persistent under these topography perturations. That is, if we let ( n, n ) e any sequence of lakes which converges to (, ) (in the sense of Definition.4), then the weak solutions to the lake equations on ( n, n ) converge to the weak solution on the limiting lake (, ). In particular, we otain strong convergence of velocity in L 2 and we allow the limiting domain to e very singular as long as it can e approximated y smooth domains n in the Hausdorff sense. The depth is merely assumed to e ounded. As a yproduct, we estalish the existence of gloal weak solutions of the equations (.) for very rough lakes (, ). Let us make our assumptions on the lake more precise. We assume that the (limiting) lake (, ) has a finite numer of islands, namely: (H) := ( N \ C k), where, C k are ounded simply connected susets of R 2, is open, and k= C k are disjoints and compact susets of. (.) We assume that the oundary is the only place where the depth can vanish, namely: (H2) There is a positive constant M such that < (x) M in and in addition, for any compact set K there exists positive numer θ K (x) θ K on K. such that In the case of smooth lakes, we add another hypothesis. Near each piece of oundary, we allow the shore to e either of non-vanishing or vanishing topography with constant slopes in the following sense: (H3) There are small neighorhoods O and O k of and C k respectively, such that, for k N, (x) = c(x) [d(x)] a k in O k, (.2) where c(x), d(x) are ounded C 3 functions in the neighorhood of the oundary, c(x) θ >, a k. Here the geometric function d(x) satisfies = {d > } and d on. In particular, around each ostacle C k, we have either Non-vanishing topography when a k =, in which case (x) θ or Vanishing topography if a k > in which case (x) as x C k. As (H3)

3 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 3 will e only considered for smooth lakes C 3, we note that up to a change of c, θ, we may take d(x) = dist(x, )... Weak formulations. As in the case of the 2D Euler equations, it is crucial to use the notion of generalized vorticity, which is defined y ω := curl v = ( v 2 2 v ). Indeed, taking the curl of the momentum equation, it follows that the vorticity formally verifies the following transport equation t (ω) + div (vω) =. (.3) Thanks to the condition div (v) =, we will show in Lemma 3. that the L p norm of p ω is a conserved quantity for any p [, ], which provides an important estimate on the solution. When is not regular, the divergence free condition and the oundary condition v ν = have to e understood in a weak sense: (x)v (x) h(x) dx =, (.4) for any test function h in the function space G() defined y } G() := {w L 2 () : w = p, for some p Hloc (). For v L 2 (), such a condition in (.4) is equivalent to where v H(), (.5) H() = the closure in L 2 of {ϕ C c () div ϕ = }. (.6) This equivalence can e found, for instance, in [3, Lemma III.2.]. Moreover, in [3] the author points out that if is a regular ounded domain and if v is a sufficiently smooth function, then v verifies (.4) if and only if div v = and v ν =. Similarly to (.4), the weak form of the divergence free and tangency conditions on v also reads: h Cc ([, + ); G()), (x)v(t, x) h(t, x) dxdt =. (.7) R + Next, we introduce several notions of gloal weak solutions to the lake equations. The first is in terms of the velocity. Definition.. Let v e a vector field such that div (v ) = in, v ν = on, weakly (in the sense of (.4)) and curl v L (). We say that v is a gloal weak solution of the velocity formulation of the lake equations (.) with initial velocity v if i) curl v L (R + ) and v L (R + ; L 2 ()); ii) div (v) = in and v ν = on in the sense of (.7); iii) the momentum equation in (.) is verified in the distriutional sense. That is, for all divergencefree vector test functions Φ Cc ([, ) ) tangent to the oundary, there holds that ( Φ ) Φ t v dxdt + (v v) : dxdt + Φ(, x) v (x) dx =. (.8)

4 4 C. LACAVE, T. NGUYEN, B. PAUSADER We emphasize that the test functions Φ are allowed to e in Cc ([, ) ) rather than in Cc ([, ) ). Namely, for any test functions Φ elonging to Cc ([, ) ), there exists T > such that Φ for any t > T, and such that Φ(t, ) C () for any t, in the sense that D k Φ(t, ) is ounded and uniformly continuous on for any k (see e.g. [3]). In the aove definition, it does not appear immediately clear how to make sense of (.8) for test functions supported up to the oundary due to the term Φ/ which would then low up at the oundary. For this reason, let us introduce a weak interior solution v of the velocity formulation to e the weak solution v as in Definition. with the test functions Φ in (.8) eing supported inside the domain, i.e. Φ Cc ([, ) ). For this weaker solution, (.8) then makes sense under the regularity (i) when W, loc () (ecause (H2) gives an estimate of locally in space). Later on in Appendix A, we show that (.8) indeed makes sense with the test functions supported up to the oundary when the lake is smooth, even in the case of vanishing topography. The second formulation of weak solutions is in terms of the vorticity and reads as follows. Definition.2. Let (v, ω ) e a pair such that and div (v ) = in, v ν = on weakly (in the sense of (.4))) (.9) ω L (), curl v = ω (in the distriutional sense). (.) We say that (v, ω) is a gloal weak solution of the vorticity formulation of the lake equations on (, ) with initial condition (v, ω ) if i) ω L (R + ) and v L (R + ; L 2 ()); ii) div (v) = in and v ν = on in the sense of (.7); iii) curl v = ω in the distriutional sense; iv) the transport equation (.3) is verified in the sense of distriution. That is, for all test functions ϕ Cc ([, ) ) such that τ ϕ (i.e. constant on each piece of oundary), there holds ϕ t ω dxdt + ϕ vω dxdt + ϕ(, x)ω (x) dx =. (.) We also introduce a weaker intermediate notion: weak interior solution of the vorticity formulation to e the weak solution (v, ω) as in Definition.2 with the test functions eing supported inside the domain: i.e. ϕ Cc ([, ) ). We will estalish the relations etween these definitions in Appendix A. For example, when the lake is smooth, all velocity and vorticity formulations are equivalent. Following the proof of Yudovich [3], Levermore, Oliver and Titi [] estalished existence and uniqueness of a gloal weak solution (with the vorticity formulation) in the case of non-vanishing topography, assuming the lake is smooth and simply connected. Recently, Bresch and Métivier [] extended the well-posedness to the case of vanishing topography. In oth of these works, is assumed to e simply connected, C 3, and C 3 (). The essential tool in estalishing the well-posedness is a Calderón-Zygmund type inequality. This inequality is highly non trivial to otain if the depth vanishes, and the proof requires to work with degenerate elliptic equations. In Section 2, we shall sketch the proof of the well-posedness of the lake equations under our current setting (H)-(H3): Theorem.3. Let (, ) e a lake verifying Assumptions (H)-(H3) and (, ) C 3 C 3 (). Then for any pair (v, ω ) such that curl v = ω L (), there exists a unique gloal weak solution (v, ω) to the lake equations that verifies oth the velocity and vorticity formulations. Furthermore, we have that ω C(R +, L r ()), v C(R +, W,r ()), v ν = on, for aritrary r in [, ) and the circulations of v around C k are conserved for any k =... N. When the domain is not simply connected, the vorticity alone is not sufficient to determine the velocity uniquely from (.9)-(.). We will then introduce in Section 2. the generalized circulation

5 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 5 for lake equations, derive the Biot-Savart law (the law which yields the velocity in term of the vorticity and circulations), and prove the Kelvin s theorem concerning conservation of the circulation..2. Assumptions. For each n, let ( n, n ) e a lake of either vanishing or non-vanishing or mixed-type topography as descried aove in (H)-(H3) with constants θ n, M n, a,n,..., a N,n and function d n (x). In what follows, we write (, ) = (, ), which will play the role of the limiting lake. We assume that these lakes have the same finite numer of islands N, namely for any n n := ( N n \ where n, C k n are simply connected susets of R 2, n is open, and C k n n are disjoint and compact. In addition, let D e a ig enough suset so that n D, n. Definition.4. Assume that ( n, n ) C 3 C 3 ( n ) for all n. We say that the sequence of lakes ( n, n ) converges to the lake (, ) as n if there hold k= n in the Hausdorff sense; C k n C k in the Hausdorff sense; n L ( n) is uniformly ounded and for any compact set K there exist positive θ K and sufficiently large n (K) such that n (x) θ K for all x K and n n (K); n in L loc (). Here n converges to in the Hausdorff sense if and only if the Hausdorff distance etween n and converges to zero. See for example [4, Appendix B] for more details aout the Hausdorff topology, in particular the Hausdorff convergence implies the following proposition: for any compact set K, there exists n K > such that K n for all n n K, which gives sense to the fourth item of the aove definition. Definition.4, allows in particular the limit a n a =, with a n introduced as in (H3). This means that the passage from a lake of the vanishing type in which the slope gets steeper and steeper to a lake of non-vanishing type is allowed. This appears to e complicated to deduce from the analysis in [], where the condition a > is crucial. Remarkaly, it turns out that uniform estimates of the velocity in W,p are not needed in order to pass to the limit. As will e shown, L 2 estimates are sufficient..3. Main results. As mentioned, a velocity field is uniquely determined y its vorticity and its circulation around each ostacle. We recall that when the velocity field v is continuous, the circulation around each ostacle C k is classically defined y C k n γ k cl := C k v ds. However, with a low regularity velocity field as in our definitions of weak solutions, such a path integral might not e well defined a priori. We are led to introduce the generalized circulation γ k (v) := div (χ k v ) dx where χ k is some smooth cut-off function that is equal to one in a neighorhood of C k and zero away from C k. Oserving that div (χ k v ) = χ k v χ k curl v, the generalized circulation is well defined for the weak solution v y condition (i) in Definitions. and.2 (indeed, (H2) implies that v elongs to L 2 (supp χ k )). The generalized circulation does not depend on the choice of the cutoff function (indeed, if χ k and χ k are equal to one in a neighorhood of C k and zero in a neighorhood of j k C j, then we can integrate y part to verify that div ((χk χ k )v )dx = ). Actually, we will show in the proof of Proposition A.4 that the notions of generalized circulation and classical circulation are equivalent for smooth lakes. Most importantly, the velocity field is uniquely determined Note that since n is uniformly ounded in L, we directly see that the convergence holds in L p, p <. ),

6 6 C. LACAVE, T. NGUYEN, B. PAUSADER y the vorticity and the circulations; see Section 2. We refer to [7] for an alternative definition of weak circulation. Our assumptions on the convergence of the initial data are in terms of the vorticity and circulations. Precisely, we assume that the initial vorticity ω n is uniformly ounded: for some positive M, and there holds the convergence ω n L ( n) M, (.2) ω n ω weakly in L (D), (.3) as n. Here ω n is extended to e zero in D \ n. Concerning the circulations, we assume that the sequence γ n = {γ k n} k N R N converges to a given vector γ = {γ k } k N in the sense that N γn k γ k, (.4) k= as n. Then, for each n, we define the initial velocity field v n to e the unique solution of the following elliptic prolem in n : div ( n v n) =, ( n v n) ν n =, curl v n = n ω n, γ k n(v n) = γ k n k N. (.5) The existence and uniqueness of v n are estalished in Section 2. Our first main theorem is concerned with the staility of the lake equations: Theorem.5. Let (, ) e a lake satisfying Assumptions (H)-(H3) with (, ) C 3 C 3 (). Assume that there is a sequence of lakes ( n, n ) which converges to (, ) in the sense of Definition.4. Assume also that (ω n, γ n, v n) are as in (.2) (.5). Let (v n, ω n ) e the unique weak solution of the lake equations (.) on the lake ( n, n ) with initial velocity v n, n. Then, there exists a pair (v, ω) so that v n v strongly in L 2 loc (R +; L 2 (D)), ω n ω weak- in L (R + D). Furthermore, (v, ω) is the unique weak solution of the lake equations on the lake (, ) with initial vorticity ω and initial circulation γ R N. This theorem, whose proof will e given in Section 3, links together various results on the lake equations, namely the flat ottom case (Euler equations [3]), non-vanishing topography [] and vanishing topography []. Indeed, we allow the limit a n (passing from vanishing topography to non vanishing topography), or the limit θ n if n = + θ n where verifies (H2)-(H3) (passing from non vanishing topography to vanishing topography). The convergence of the solutions of the Euler equations when the domains converge in the Hausdorff topology is a recent result estalished y Gérard-Varet and Lacave [4], ased on the γ-convergence on open sets (a rief overview of this notion is given in Appendix B). The present paper can e regarded as a natural extension of [4] to the lake equations i.e. to the case of non-flat ottoms n when we consider a weak notion of convergence of n. The γ-convergence is an H theory on the stream function (or an L2 theory on the velocity). Bresch and Métivier have otained estimates in W 2,p for any 2 p < + (namely, the Calderón-Zygmund inequality) for the stream function, which is necessary for the uniqueness prolem or to give a sense to the velocity formulation. For our interest in the sequential staility of the lake solutions, it turns out that we can treat our prolem without having to derive uniform estimates in W 2,p, which appear hard to otain. In fact, we will first prove the convergence of a susequence of v n to v and show that the limiting function v is indeed a solution of the limiting lake equations. Since the Calderón-Zygmund inequality is verified for the solution of the limiting lake equations, the uniqueness yields that the whole sequence indeed converges to the unique solution in (, ). In addition, since the Calderón-Zygmund inequality is not used in the compactness argument, it follows that the existence of a weak solution to the lake equations with non-smooth domains or nonsmooth topography can e otained as a limit of solutions to the lake equations with smooth domains. Our second main theorem is concerned with non-smooth lakes which do not necessarily verify (H3).

7 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 7 Theorem.6. Let (, ) e a lake satisfying (H)-(H2). We assume that for every k N, C k has a positive Soolev H capacity. For any ω L () and γ R N, there exists a gloal weak solution (v, ω) of the lake equations in the vorticity formulation on the lake (, ) with initial vorticity ω and initial circulation γ R N. This solution enjoys a Biot-Savart decomposition and its circulations are conserved in time. If we assume in addition that W, loc () then (v, ω) is also a gloal weak interior solution in the velocity formulation. Let us mention that we do not assume any regularity of ; for instance, can e the Koch snowflake. To otain solutions for the vorticity formulation, we do not need any regularity on either; it might not even e continuous. But even in the case where we assume the ottom to e locally lipschitz, choosing n := + n we can consider a zero slope: (x) = e /d(x) or non constant: (x) = d(x) a(x) ; our theorem states that (v, ω) is a solution of the vorticity formulation and an interior solution of the velocity formulation. Such a result might appear surprising, ecause the known existence result requires that the lake domain is smooth, namely (, ) C 3 C 3 () and (H3). The Soolev H capacity of a compact set E R 2 is defined y cap(e) := inf{ v 2 H (R 2 ), v a.e. in a neighorhood of E}, with the convention that cap(e) = + when the set in the r.h.s. is empty. We refer to [6] for an extensive study of this notion (the asic properties are listed in [4, Appendix A], in particular we recall that a material point has a zero capacity whereas the capacity of a Jordan arc is positive). Apparently, in such non-smooth lake domains, the Calderón-Zygmund inequality is no longer valid, and hence the well-posedness is delicate. For existence, our construction of the solution follows y approximating the non-smooth lake y an increasing sequence of smooth domains in which the solutions are given from Theorem.3. Finally, we leave out the question of uniqueness in the case of non-smooth lakes. We refer to [8] for a uniqueness result for the 2D Euler equations in simply-connected domains with corners. In [8] the velocity is shown in general not to elong to W,p for all p (precisely, if there is a corner of angle α > π, then the velocity is no longer ounded in L p W,q, p > p α, q > q α with p α 4 and q α 4/3 as α 2π). 2. Well-posedness of the lake equations for smooth lake In this section, we sketch the proof of existence of the lake equations in a non-simply connected domain (Theorem.3). The proof can e outlined as follows: we first prove existence of a gloal weak interior solution in the vorticity formulation. The proof follows y adding an artificial viscosity (as was done in [9]) and otaining compactness for the vanishing viscosity prolem (Section 2.2); as the lake is smooth, we then use the Calderón-Zygmund inequality estalished in [], which in turn implies that for aritrary r, ω C(R +, L r ()), v C(R +, W,r ()), and v ν = on ; thanks to the regularity close to the oundary, we can show y a continuity argument that the vorticity equation (.) is indeed verified for test functions supported all the way to the oundary (Proposition A.5). The existence of a gloal weak solution in the vorticity formulation (with conserved circulations) is then estalished. The solution also verifies the velocity formulation due to the equivalence of the two formulations (Proposition A.4). finally, uniqueness of a gloal weak solution is shown in Section 2.3 y following the celerated method of Yudovich. Essentially, this outline of the proof was introduced y Yudovich in his study of two-dimensional Euler equations [3], and it was used in [, ] in the case of the lake equations. We shall provide the proof with more details as it will e crucial in our convergence proof later on. Throughout this section, we fix a smooth lake (, ) namely: (, ) satisfying Assumptions (H)-(H3) (see Section ) and (, ) C 3 C 3 (). (2.)

8 8 C. LACAVE, T. NGUYEN, B. PAUSADER We allow the lake to have either vanishing or non-vanishing topography. We shall egin the section y deriving the Biot-Savart law. We then otain the well-posedness of the lake equations (.) in the sense of Definitions. and Auxiliary elliptic prolems. Let us introduce the function space X := { f H () : /2 f L 2 () }. We will sometimes write the function space as X instead of X to emphasize the dependence on. Clearly, (X, X ) is a Hilert space with inner product f, g X := /2 f, /2 g L 2 and norm f X := f, f /2 X. Our first remark is concerned with the density of C c () in X. Lemma 2.. Let (, ) e a smooth lake in the sense of (2.). Then Cc () is dense in X with respect to the norm X. The proof relies on a variant of the Hardy s inequality. As it was noted in the introduction, we can consider that d is the distance to the oundary in (.2). With the notation: we estalish the following Hardy type inequality: R := {x : d(x) R}, (2.2) Lemma 2.2. Let (, ) e a smooth lake in the sense of (2.). Then the following inequality holds uniformly for every f H () and any positive R: /2 (f/d) L 2 ( R ) /2 f L 2 ( R ). (2.3) Here in Lemma 2.2 and throughout the paper, the notation g h is used to mean a uniform ound g Ch, for some universal constant C that is independent of the underlying parameter (in (2.3), small R > and f). Proof of Lemma 2.2. We start with the following claim: for any f H () and any positive R, there holds that f(x) 2 dx R 2 f(x) 2 dx. (2.4) R d(x) 2R d(x) 2R The claim follows directly from the fundamental theorem of Calculus and the standard Hölder s inequality at least for smooth compactly supported functions. By density, it extends to H (). Next, y (2.4) and the classical Hardy s inequality, the lemma follows easily for functions f H () whose support is away from the set k:a k > O k. It suffices to consider functions f that are supported in the set O j for a j >. Again y (2.4), we can write /2 (f/d) 2 L 2 ( R ) = ( ) f(x) 2 dx k N R 2 k d(x) 2R d(x) (x) (R2 k ) (aj+2) f(x) 2 dx k N R 2 k d(x) 2R (R2 k ) aj f(x) 2 dx k N 2 k d(x) 2R (R2 k ) a j f(x) 2 dx. R k N : 2 k d(x) 2R Since the summation in the parentheses in the last line aove is ounded y, the integral on the righthand side is ounded y /2 f 2 L 2 ( R ). The lemma is thus proved.

9 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 9 Proof of Lemma 2.. Fix ε > and f X. It suffices to construct a cut-off function χ C c () such that ( χ)f X ε. (2.5) The lemma would then follow simply y approximating the compactly supported function χf with its Cc mollifier functions. Now since f X, there exists a positive R ɛ such that f(x) 2 dx Rɛ (x) ε2. (2.6) Let us introduce a cut-off function η C (R + ) such that η, η(z) if z and η(z) if z /2 and define χ(x) = η(d(x)/(r ɛ )). Clearly, χ Cc (). In addition, we note that [( χ)f] = ( χ) f f χ. It then follows y (2.6) that ( χ(x)) 2 f(x) 2 dx (x) f(x) 2 dx Rɛ (x) ε2. Meanwhile using the fact that and Lemma 2.2, we otain f χ = Rɛ fη (d(x)/r ɛ ) d(x) (f/d)(x) η L f(x) χ(x) 2 dx (x) f(x) η L Rɛ 2 d(x) 2 This yields (2.5) which completes the proof of the lemma. dx (x) f(x) 2 dx Rɛ (x) ε2. Next, we consider the following auxiliary elliptic prolem [ ] div ψ = f in, with ψ =. (2.7) Proposition 2.3. Let (, ) e a smooth lake in the sense of (2.). Given f L 2 (), there exists a unique (distriutional) solution ψ X of the prolem (2.7). Proof. Let us introduce the functional E(ψ) := ( 2 ψ 2 + fψ) dx. Since f L 2, the functional E( ) is well-defined on X. Let ψ k X e a minimizing sequence. Thanks to the Poincaré inequality and the fact that is ounded, ψ k is uniformly ounded in X. Up to a susequence, we assume that ψ k ψ weakly in X. By the lower semi-continuity of the norm, it follows that E(ψ) lim inf k E(ψ k). Hence, ψ X is indeed a minimizer. In addition, y minimization, the first variation of E(ψ) reads ( ) ϕ ψ + ϕf dx =, ϕ Cc (), (2.8) which shows that ψ is a solution of (2.7). We recall that the Dirichlet oundary condition is encoded in the function space X. For the uniqueness, let us assume that ψ X is a solution with f. Then, (2.8) simply reads ϕ, ψ X =, for aritrary ϕ Cc (). It follows y density (see Lemma 2.) that ψ X = and so ψ =. This proves the uniqueness as claimed.

10 C. LACAVE, T. NGUYEN, B. PAUSADER Definition 2.4. We say that Φ is a -harmonic function if Φ H ( ), /2 Φ L 2 (), where is as introduced in (H), so that Φ solves the prolem [ ] div Φ = in, and τ Φ = on. We denote y H the space of -harmonic functions. We remark that since a -harmonic function Φ elongs to H (), we can define its trace at the oundary, and so τ Φ = should e understood as its trace eing constant on each connected component of. Proposition 2.5. Let (, ) e a smooth lake in the sense of (2.). For k N, there exists a unique -harmonic function ϕ k such that ϕ k = on, ϕ k = δ ik on C i, i =... N. Moreover, the family {ϕ k } k=..n forms a asis for the set of -harmonic functions. Proof. Let δ = min i j{dist(c i, C j ), dist(c i, )}. For each k, we introduce a cut-off function χ k ( ) which is supported in a δ-neighorhood of C k and satisfies C c In particular, χ k (x) = if d(x, C k ) > δ, χ k (x) = if d(x, C k ) < δ/2. (2.9) χ k = δ ik in a neighorhood of C i, i =... N. By Proposition 2.3, there exists a unique solution ϕ k X to the prolem div [ ϕk] = div [ χk] in, ϕ k = on. Indeed, since χ k is smooth and vanishes near the oundaries, the right-hand side of the aove prolem clearly elongs to L 2 (). Now if we define ϕ k := ϕ k + χ k, (2.) the existence of a -harmonic function ϕ k follows at once as claimed. The uniqueness follows from the uniqueness result in Proposition 2.3: indeed, let ϕ and ϕ 2 e two -harmonic functions which have the same trace on each component of. Then, Φ := ϕ ϕ 2 satisfies (2.7) with f = and elongs to H () hence to Φ X, which is the function space where the uniqueness of solutions to (2.7) was proved. Finally, since any -harmonic function y definition is constant on each connected component of, it follows clearly that the family {ϕ k } k N forms a asis of H. To recognize the divergence free condition (.4), we need the following simple lemma: Lemma 2.6. Let (, ) e a lake satisfying assumptions (H)-(H2) (not necessarily smooth). ψ X, c k R and χ k C () as introduced in (2.9). Then the vector function satisfies v := (ψ + N k= c kχ k ) div (v) = in, v ν = on weakly (in the sense of (.4)). (2.) Conversely, let v e a vector field so that v L 2 () and (2.) holds. Then there exists ψ H ( ) such that v = ψ in and τ ψ = on. Let

11 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS Proof. As ψ X H (), we can easily check that ψ elongs to H() (see (.6)) and then satisfies (.4) (y the equivalence proved in [3, Lemma III.2.]). Moreover, since χ k is smooth and vanishes in a neighorhood of the oundary, an integration y parts implies that χ k verifies the oundary condition (.4). This gives (2.). The second one is a classical statement which does not depend on the regularity of. Indeed, as (.4) is equivalent to (.5) when v L 2 (), we can find a sequence of divergence-free vector fields v n Cc (), such that v n v in L 2 (). Then v n is supported in a smooth set, and we can use the classical Hodge-De Rham theorem: v n = ψ n where ψ n is constant near the oundaries. Choosing ψ n such that ψ n (x) in a neighorhood of, we then infer y Poincaré inequality that ψ n is a Cauchy sequence in H ( ) (where we have extending ψ n in C k y constant values). Passing to the limit, we infer that v = ψ for some ψ H ( ) such that τ ψ on. Remark 2.7. Let (, ) e a smooth lake in the sense of (2.). If ψ X and Φ is a simili harmonic function, then v := (ψ + Φ) verifies div (v) = in, v ν = on, weakly (in the sense of (.4)). Indeed, Proposition 2.5 states that there exists c k such that Φ N k= c kϕ k, so using (2.), we can decompose v as v = ( ψ + N k= c kχ k ) with ψ X. Then Lemma 2.6 can e applied. With the regularity considered in Definition 2.4, it is not clear that C k Φ ˆτ ds is well defined. Using χ k defined as in (2.9), we introduce the generalized circulation of a vector field v around C k y [ γ k (v) := div χ k v ] [ ] ( ) dx = curl χ k v dx = χ k v + χ k curl v dx. (2.2) Lemma 2.8. Let (, ) e a smooth lake in the sense of (2.). If ψ is a -harmonic function such that the generalized circulation of the vector field ψ around each C k is equal to zero for all k, then ψ must e identically zero. Proof. Set v := ψ. We egin the proof with the following claim: there exists a function f such that We oserve that From Remark 2.7, we get that v = f. (2.3) curl (v) = div ( ψ) =. (2.4) div (v) = in, v ν = on, weakly (in the sense of (.4)). As is regular, y local elliptic regularity we deduce from (2.4) that v is a continuous function in. Thus we can define the classical circulation v τds along any closed path and we infer y the curl free property that this circulation does not depend on the homotopy class of the path. Next, choose c k a closed curve supported in the region where χ k = so that c k is homotopic to C k (see (2.9) for the definition of χ k ). Let c k e another homotopic path supported in {x, χk (x) = }. We let A k e the region ounded y c k and c k. Using (2.2), we then compute that ( ) v τds = v τds = (χ k v) τds = curl (χ k v) dx = curl (χ k v) dx =, c k A k c k where we have used the fact that vanishes outside of A k. c k c k curl (χ k v) = χ k curl (v) v χ k

12 2 C. LACAVE, T. NGUYEN, B. PAUSADER Since any smooth loop can e decomposed into a finite numer of loops which are either homotopic to the trivial loop or homotopic to one of the c k s, we have that the circulation of v along any closed curve vanishes. Therefore, fixing P an aritrary point in and letting f(q) = v τds γ P Q where γ P Q is any (smooth) path from P to Q, we otain (2.3). As ψ is a -harmonic function, we have that ψ = v = f elongs to L 2 (), hence, v 2 ψ dx = f dx. (2.5) Moreover, y Proposition 2.5 and (2.) we can decompose ψ as ψ = N c k ϕ k = ψ + k= N c k χ k where ψ X. By density of Cc () in X (see Lemma 2.) we have ψ f ψ dx = lim n f dx = lim n n k= ψ n f dx = where we have integrated y parts and used that ψ n Cc (). Moreover, as χ k is smooth and χ k vanishes close to the oundary, we also have y an integration y parts: χ k f dx = χ k f dx = for all k. Putting together these two relations, (2.5) implies that v is equal to zero, from which we conclude that ψ is constant in. Since ψ H ( ), ψ vanishes in as claimed. Proposition 2.9. Let (, ) e a smooth lake in the sense of (2.). There exists a asis {ψ k } N k= of H which satisfies γ i( ψ k ) = δ ik, i, k. Proof. Consider the linear mapping Ψ : H R N, Ψ(g) = (γ,..., γ N ), γ i := γ i( g Lemma 2.8 states that Ψ is one-to-one and Proposition 2.5 implies that dim H = N. Consequently, Ψ is ijective and we can define ψ i = Ψ (e i ) where e i is the i-th vector in the canonical asis of R N. Proposition 2. (Decomposition). Let (, ) e a smooth lake in the sense of (2.). Let ω L () and γ = (γ,, γn ) RN. Then there exists a unique vector field v such that v L 2 (), and div (v) = in, v ν = on, weakly (in the sense of (.4)) (2.6) Moreover, we have the following Biot-Savart formula: curl (v) = ω in D (), γ i (v) = γ i. (2.7) v = ψ + ). N α i ψ i, (2.8) i=

13 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 3 where ψ i H is the function defined as in Proposition 2.9 aove, α i = γ i + ωϕi dx, ϕ i defined as in Proposition 2.5, and ψ X the unique solution (see Proposition 2.3) of the prolem div ( ψ ) = ω in D (), ψ X. Proof. We egin y showing that the vector field defined as N ( ) u := ψ + γ i + ωϕ i dx ψ i i= verifies (2.6)-(2.7). By definition, u L 2 (). The curl condition in (2.7) is ovious from the definitions of ψ and the -harmonic functions. Condition (2.6) comes from Remark 2.7. The hardest part is to compute the circulation of ψ. By the definition (2.) of ϕ i, we use that χ i = ϕ i ϕ i with ϕ i X, to get: ( ) [ γ i ( ψ ) = ϕ i ψ + ϕ i ω dx + div ϕ i ψ ] dx. (2.9) Now, for any Φ C c (), we note that [ div Φ ψ ] dx =, and as ϕ i elongs to X and Cc is dense in X, we deduce from the fact that oth /2 ψ and ω elong to L 2 () that [ div ϕ i ψ ] [ ] dx = /2 ( ϕ i Φ) /2 ψ ( ϕ i Φ)ω dx =. (2.2) Concerning the first term on the right hand side of (2.9), as ψ X we can again use the density of Cc in X to state that the fact that ϕ i is a -harmonic function reads as ϕ i ψ dx = ϕ i ψ dx = /2 ϕ i /2 ( ψ Φ) dx + ϕ i Φ dx =. Therefore, putting these last two equalities together with (2.9) gives that γ i (u) = ( ωϕi dx + γ i + ) ωϕi dx = γ i, which shows that u verifies (2.6)-(2.7). To prove the uniqueness of u, let v e another vector field such that v L 2 () and v satisfies (2.6)-(2.7). By Lemma 2.6, there exists ψ H ( ) such that So v = ψ in D (), ( Ψ := ψ ψ + N ( γ i + i= τ ψ = on. ωϕ i dx )ψ i) is a -harmonic function such that the circulation of Ψ around each C k is equal to zero. Lemma 2.8 gives that v = u, which ends the proof Existence of a gloal weak solution. In this susection we prove the existence of a gloal weak interior solution for the vorticity formulation: Lemma 2.. Let (, ) e a smooth lake in the sense of (2.). Let ω L () and {γ i } i N fixed numers and define v y (2.8). Then there exists a gloal weak interior solution to the lake equations on (, ) in the vorticity formulation (see Definition.2). Moreover, the circulations of this solution are conserved.

14 4 C. LACAVE, T. NGUYEN, B. PAUSADER The original idea comes from Yudovich: we introduce an artificial viscosity εdiv ( ω ε ) in the vorticity equations, assuming the Dirichlet condition for the vorticity at the oundary. This viscosity is artificial, ecause of the oundary condition: in the (physically relevant) Navier-Stokes equations, the Dirichlet condition is given on v, which does not imply the Dirichlet condition on the vorticity. Although the inviscid prolem in Navier-Stokes equations is a hard issue, the limit prolem ε with the artificial viscosity is possile to achieve. Actually, to use directly a result from [9], we consider ε := +ε ε >, and we approximate ω y ωε Cc such that ωε L 2 ω L. As ε is strictly positive, standard arguments for Navier-Stokes equations [9] gives the existence and uniqueness of a gloal solution ω ε C([, ); H ()) L 2 loc ([, ); H2 ()) of the prolem (in the sense of distriution) t ( ε ω ε ) + ε v ε ω ε εdiv ( ε ω ε ) = for (t, x) R +, v ε = N ψ γ i ε[ω ε ] + + εω ε ϕ i ε dx ψ ε ε, i for (t, x) R +, ε i= ω ε = for (t, x) R +, ω ε (, x) = ωε(x) for x, (2.2) where ε > is aritrary and γ i are given independently of ε and t. The aove system is exactly the prolem studied in [9]. Indeed the authors work in non-simply connected domains, and Lemma 5 therein is similar to our decomposition (Proposition 2.). In their case, the tangential part v τ is clearly defined (as ε > ) so their definition of the circulation as an integral along C k is the same as our generalized circulation. In this work, the test functions are compactly supported in : ϕ Cc ([, ) ). Indeed, for Navier-Stokes equations, the general framework is of H to H () and test functions in Cc ([, ) ) are sufficient ecause the Dirichlet oundary condition is already encoded y the fact that the velocity (here the vorticity) elongs to H. Moreover, we have the energy relation : t ε ω ε (t, ) 2 L 2 () + ε ε ω ε (s, ) 2 L 2 () ds ε ωε 2 L 2 (), t. (2.22) Next, the idea is to pass to the limit ε. Let us perform this limit as follows: y integration y parts and Poincaré inequality on we have that (thanks to the tangency condition of v ε ): ε v ε 2 L 2 () εω ε L 2 () ψ ε L 2 () C 2 M + ε ε ω ε L 2 () ψ ε L 2 () 2C 2 2 (M + ) 3/2 ω L () ε v ε L 2 (), where ψ ε is the stream function vanishing on associated to ε v ε : N ψ ε := ψε[ω ε ] + (γ i + ε ω ε ϕ i ε dx)ψε. i Hence ε v ε is ounded in L (R + ; L 2 ()), uniformly in ε: i= ε v ε (t) L 2. (2.23) for ε fixed, since ε ɛ >, we oserve that t ω ε L 2 loc (R +; H ()) and also that ω ε C(R + ; L 2 ()) L 2 loc (R +; H ()). Hence one can multiply the vorticity equation y some power of ω ε to get for all time, and for all p, ( ε ) p ω ε (t, ) L p ( ε ) p ω ε L p (M + ) p ω ε L p 2[(M + )( + )] p ω L. As the constant at the right hand side is uniform in p, we infer that ω ε (t, ) L 2 ω L. (2.24)

15 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 5 Therefore, Banach-Alaoglu theorem implies that, up to a susequence, ω ε converges weak- to ω in L (R + ), and ε v ε converges weak- to v in L (R + ; L 2 ()), as ε. This weak convergence is sufficient to get (i), (ii) and (iii) in Definition.2. Moreover, y construction (see (2.2)), γ i (v ε (t)) = γ i for all t R +, i =... N. Hence, the weak limit is also sufficient to pass to the limit in the circulation definition (2.2) which implies that the circulations of v is conserved. To get (iv), we will pass to the limit in equation (2.2), ut we need a strong convergence of the velocity. Without loss of generality, we may restrict ourselves to O a smooth simply connected open suset of such that O. We introduce the Leray projector P O from L 2 (O) to H(O) (see (.6) for the definition), i.e. P O is the unique operator such that v ε = P O v ε + q ε, div(p O v ε ) =, (P O v ε ) ν O =. (2.25) All the details aout the Leray projector can e found e.g. in [3]. In particular, it is known that such a projector is orthogonal in L 2, and so P O v ε 2 L 2 (O) + q ε 2 L 2 (O) v ε 2 L 2 (O) θ O ε v ε 2 L 2 () for some positive constant θ O (see (H2) for the last inequality, ecause O is a compact suset of ). This together with (2.23) yields that, up to a susequence, q ε and P O v ε converge weak- in L ([, T ]; L 2 (O)) to q and P O v, with v = P O v+ q, as ε. Besides, since curl(p O v ε ) = curl(v ε ) = ε ω ε is uniformly ounded in L and using (2.25), we see that {P O v ε (t)} always remains inside a compact set of L 2 (O) (indeed, y the standard Calderón-Zygmund theorem on O, j jp O v ε (t) L 2 curl(v ε (t)) L 2). As (2.2) is verified for test functions ϕ Cc (O) with O a simply connected smooth domain, like in Proposition A.2 in Appendix A we infer that we have a velocity type equation: Φ t v ε dxdt + ω ε ε v ε Φ dxdt ε ε ω ε Φ dxdt = for all divergence-free Φ Cc (O). For such a test function, using (2.22), (2.23) and (2.24), we otain that P O v ε (t 2 ), Φ P O v ε (t ), Φ = v ε (t 2 ), Φ v ε (t ), Φ t2 t2 = ε ω ε v ε Φ dxdt ε ε Φ ω ε dxdt t Φ L 2[ M + ε v ε L L2 ω ε t L t,x t t 2 + ε(m + ) ] ε ε ω ε L 2 t,x t t 2 2 [ Φ L 2C(ω ) t t 2 + ] ε t t 2 2. By density, we note that the aove estimates remain valid for Φ H(O). However, P O is an orthogonal projector, hence for any Φ L 2 (O), we can write that P O v ε (t), Φ = P O v ε (t), P O Φ. Thus, for any Φ L 2 (O), since P O Φ H(O), there holds [ P O v ε (t 2 ), Φ P O v ε (t ), Φ P O Φ L 2C(ω ) t t 2 + ] ε t t 2 2 [ Φ L 2C(ω ) t t 2 + ] ε t t 2 2 which implies that the family {P O v ε } is equicontinuous in L 2 (O). Since we have seen that it takes values in a compact set, the Arzela-Ascoli theorem gives us the precompactness of {P O v ε } in C([, T ]; L 2 (O)). Finally, we can now pass to the limit in (2.2). We recall that for any ϕ Cc ([, T ) O), the first equation in (2.2) reads [ ] = ϕ t ε ω ε + ε ω ε v ε ϕ ε ε ω ε ϕ dxdt + ϕ(, x) ε ω ε (x) dx. (2.26) t

16 6 C. LACAVE, T. NGUYEN, B. PAUSADER Clearly, thanks to (2.22), we can pass to the limit as ε in all the (linear) terms except the nonlinear term: ε ω ε v ε ϕ. For the remaining term, using the relation (A.), we get ε ω ε v ε ϕ dxdt = (curl v ε )vε ϕ dxdt [ = div ( ε v ε v ε ) ϕ ] ε 2 v ε 2 ϕ dxdt = div ( ε v ε v ε ) ϕ dxdt. ε In addition, we can write ε v ε v ε = ε P O v ε v ε + ε q ε P O v ε + ε q ε q ε, in which P O is the Leray projector defined as aove. The integration involving the first two terms on the right hand side converges to its limit y taking integration y parts and using a weak-strong convergence argument. For the last term, we further compute: div [ ε q ε q ε ] = ε 2 ( q ε 2 ) + ( ε q ε + ε q ε ) q ε. Here we note from (2.25) that div v ε = q ε, and as div ε v ε =, we get that q ε = ε ε v ε. Hence, we have This yields div ( ε v ε v ε ) ϕ ε dxdt = div [ ε q ε q ε ] = ε 2 ( q ε 2 ) ( ε P O v ε ) q ε. [ 2 ( q ε 2 ) ϕ + ( ε P O v ε ) q ε ϕ ε ] dxdt in which the first integral vanishes, whereas the second integral passes to the limit again y a weakstrong convergence argument. By putting these into (2.26) and using the same algera as just performed, it follows in the limit that ϕ t ω dxdt + ϕ vω dxdt + ϕ(, x)ω (x) dx = (2.27) for all ϕ Cc ([, T ) O). Recall that O was an aritrary smooth simply connected domain in. This proves that the identity (2.27) holds for all ϕ Cc ([, ) ). To conclude, we have shown that (v, ω) is an interior weak solution of the lake equations in the vorticity formulation, which completes the proof of Lemma Well-posedness of a gloal weak solution. In this susection, we use the Calderón-Zygmund inequality (2.28), elow (see also Bresch and Métivier []), to upgrade our solution (v, ω) to a weak solution in the vorticity formulation, which is then equivalent to a weak solution in the velocity formulation. Using again the Calderón-Zygmund inequality, we prove that weak solutions in the velocity formulation are unique, which ends the proof of Theorem.3. Gain of regularity for smooth lakes. First, we recall the main result of Bresch and Métivier in []: if the lake is smooth with constant slopes, then we have a Calderón-Zygmund type inequality. Namely: Proposition 2.2 ([, Theorem 2.3]). Let (, ) e a smooth lake in the sense of (2.). Let f L p () for p > 4 and v L 2 (). If the triplet (, v, f) verifies the following elliptic prolem and div (v) = in, v ν = on, weakly(in the sense of (.4)) curl v = f in D (),

17 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 7 then v C 2 p () and v L p (). Moreover, there exists a constant C which depends only on and so that for any p > 4 ( ) v L p () Cp f L p () + v L 2 (). (2.28) In addition, v ν = on. This inequality is well known in the case of non-degenerate topography ( θ > ) and it was extended y Bresch and Métivier in the case of a depth which vanishes at the shore like d(x) a for a >. The authors decompose the domain in two pieces: one which is far from the oundary where they use classical elliptic estimates, and one near the oundary. As for the latter piece, they flatten the oundary and reduce the prolem to study a degenerate elliptic equation with coefficients vanishing at the oundary of a half-plane. This decomposition in several sudomains explains why we have the terms v L 2 in the right hand side part of the Calderón-Zygmund inequality (2.28), coming from the support of the gradient of some cut-off functions. We remark also that we can easily have some islands with vanishing (where a k can e different from a ) or non vanishing topography, which gives a lake where the Calderón-Zygmund inequality holds true. By Lemma 2. there exists (v, ω) verifying the elliptic prolem ii)-iii) in Definition.2. Then, Proposition 2.2 states that v elongs to L p for any p > 4. This estimate is crucial to prove that (v, ω) is actually a gloal weak solution to the vorticity formulation (Proposition A.5), which is also a gloal weak solution to the velocity formulation (Proposition A.4), ecause the circulations are conserved. The Calderón-Zygmund inequality will e also the key for the uniqueness. Let us also note that we deduce in the proof of Proposition A.4 that the regularity of v implies the equivalence etween the notions of generalized circulation and classical circulation. By using the renormalized solutions in the sense of DiPerna-Lions, it follows that ω C([, ); L p ()) and v C([, ); W,p ()) for any p > 4 (see the proof of Lemma 3. for details aout the renormalized theory). Uniqueness. The uniqueness part now follows from the celerated proof of Yudovich [3]. Let v and v 2 e two weak gloal solutions for the same initial v. We introduce ṽ := v v 2. As ṽ elongs to W,p for any p (4, ), we get from the velocity formulation some estimates for t ṽ. This allows us to replace the test function y ṽ = P (ṽ) C ([, T ]; L 5 4 ()). As ṽ C ( R +, L 5 (R 2 ) ), we get for all T R + T T ṽ(t ) 2 L 2 () = 2 t (ṽ), ṽ 5 ds 2 L 4 ṽ(s, x) v L 5 2 (s, x) ṽ(s, x) dxds where we have used that div v = div ṽ =. Next, we use the Calderón-Zygmund inequality (2.28) on v 2 to infer y interpolation that ṽ(t, ) 2 L 2 2Cp Together with a Gronwall-like argument, this implies T ṽ 2 2/p dt. L 2 ṽ(t, ) 2 L 2 (2CT ) p, p 2. Letting p tend to infinity, we conclude that ṽ(t, ) L 2 = for all T < /(2C). Finally, we consider the maximal interval of [, ) on which ṽ(t, ) L 2, which is closed y continuity of ṽ(t, ) L 2. If it is not equal to the whole of [, ), we may repeat the aove proof, which leads to a contradiction y maximality. Therefore uniqueness holds on [, ), and this concludes the proof of well-posedness. Constant circulation. If the domain is not simply connected, we have proved in the first susection that the vorticity alone is not sufficient to determine the velocity uniquely, and that we need to fix

18 8 C. LACAVE, T. NGUYEN, B. PAUSADER the generalized circulation to derive the Biot-Savart law. In the following section, the main idea is to prove compactness in each terms in this Biot-Savart law. Therefore, it is crucial to estalish Kelvin s theorem in our case, namely that the generalized circulations are conserved. Fortunately, this is valid in great generality following Proposition 2.3 as follows. Proposition 2.3. Let (, ) e a lake satisfying (H)-(H2) with W, loc (). Let v e a gloal interior weak solution of the velocity formulation and a gloal weak solution of the vorticity formulation. Then for each k =,, N, the generalized circulation γ k defined as in (2.2) is independent of t. Proof. Let l(t) Cc ([, )), note that since χ k in a neighorhood of the oundary, then l(t) χ k (x) is a test function for which the velocity equation (.8) is verified. As χ k is constant in each neighorhood of the oundary, l(t)χ k (x) is a test function for which the vorticity equation (.) holds. Then, we can compute γ k (t) d R dt l(t)dt d [ γk ()l() = l(t) χ k] d [ v dxdt l(t)χ k] ω dxdt γ k ()l() R dt R dt { [ ] } = (v v) : R χ k + χ k v curl(v) l(t) dxdt { } + l() v χ k + χ k curl(v ) dx γ k ()l() { [ ] } = (v v) : χ k + χ k v curl(v) l(t) dxdt. R Using the fact that div(v) = and χ k in a neighorhood of the oundary, we may integrate y parts and use (A.) to have γ k (t) d R dt l(t)dt γk ()l() = l(t) χ k v 2 R 2 dxdt. Now, we let χ k e a smooth function, compactly supported inside and such that χ k χ k = χ k. Integrating y parts, we then find that ] χ k v 2 2 dx = χ k [ χ k v 2 dx = div { χ k} χ k v 2 dx =. 2 2 This finishes the proof. 3. Proof of the convergence In this section, we shall prove our main result (Theorem.5). Here, we recall our main assumption that ( n, n ) converges to the lake (, ) as n in the sense of Definition.4. Let us denote y D a large open all such that D contains and n, and extend the ottom functions and n to zero on the sets D \ and D \ n, respectively. We prove the main theorem via several steps. First, from the velocity equation, it is relatively easy to otain an a priori ound on n v n in L (R + ; L 2 ) (here one needs uniform estimates on n vn in L 2 ( n )). Unfortunately, such a ound is too weak to give any reasonale information on the possile limiting velocity solution v. To otain sufficient compactness, we derive estimates on the stream function ψ n, defined y v n = n ψ n. (3.) The Biot-Savart law (2.8) which is estalished in Proposition 2. gives ψ n (t, x) = ψ n(t, x) + N αn(t)ψ k n(x), k (3.2) k=

19 TOPOGRAPHY INFLUENCE ON THE LAKE EQUATIONS 9 where for each n, ψn solves div ( n ψn) = n ω n with the Dirichlet oundary condition on n, and the so-called -harmonic functions ψn k solve div ( n ψn) k = and have their circulations equal to δ jk around each island Cn, j j =,, N. The real numers αn(t) k are given y αn(t) k = γn k + n (x)ω n (t, x)ϕ k n(x) dx where γ k n = γ k (v n ) is the circulation of v n around each C k n introduced as in (2.2), which is constant in time (see Proposition 2.3). 3.. Vorticity estimates. We egin y deriving some asic estimates on the vorticity ω n. Lemma 3.. For each n, the L p norm of /p n ω n is conserved in time and uniformly ounded for all p, that is, /p n ω n (t) L p ( n) = /p n ωn L p ( n) ωn L, t. In addition, ω n is ounded in L x,t, uniformly in n. Proof. We recall that the vorticity ω n solves (.3) in the distriutional sense and elongs to L (R + n ). Thanks to Proposition 2.2 we deduce that the velocity is regular enough to apply the renormalized theory in the sense of DiPerna-Lions: let f : R R e a smooth function such that f (s) C( + s p ), t R, for some p, then f(ω n ) is a solution of the transport equation (.3) (in the sense of distriution) with initial datum f(ω ). By smooth approximation of s s p for p <, the renormalized solutions yields d dt ( n ω n p ) = n v n ω n p = div ( n v n ω n p ). Integrating this identity over n and using the Stokes theorem, we get d dt n (x) ω n p (t, x) dx = n v n ν ω n p dσ n (x), n n where the oundary term vanishes due to the oundary condition on the velocity (see (.)). The lemma is proved for p <. The case p = is easily otained y taking f a function vanishing on the interval [ ω L, ω L ] and strictly positive elsewhere. Indeed, it shows that the L norm cannot increase, and y time reversiility that it is constant. Lemma 3. in particular yields that the vorticity ω n is ounded in L ( n ) and, after extending ω n y in D \ n, y the Banach-Alaoglu theorem, we can extract a susequence such that /p n ω n /p ω weakly- in L (R + ; L p (D)) ω n ω weakly- in L (R + D) harmonic functions: Dirichlet case. We now derive estimates for the -harmonic solutions ϕ k n, k =,, N. We recall that ϕ k n vanishes on the outer oundary n and solves [ ] div ϕ k n =, in n n (3.3) ϕ k n = δ jk, on Cn, j j =,, N. The existence and uniqueness of ϕ k n was estalished in Proposition 2.5. We otain the following. Lemma 3.2. The sequence /2 n ϕ k n converges strongly to /2 ϕ k in L 2 (D). In particular, ϕ k n is uniformly ounded in H (D) and /2 n ϕ k n is uniformly ounded in L 2 (D). In this statement and in all the sequel, /2 n ϕ k n is extended y zero on D \ n.