A note on a degenerate elliptic equation with applications for lakes and seas

Electronic Journal of Differential Equations, Vol. 2004(2004), No. 42, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A NOTE ON A DEGENERATE ELLIPTIC EQUATION WITH APPLICATIONS FOR LAKES AND SEAS DIDIER BRESCH, J´ ERˆ OME LEMOINE, & FRANCISCO GU´ ILLEN-GONZALEZ Abstract. In this paper, we give an intermediate regularity result on a degenerate elliptic equation with a weight blowing up on the boundary. This kind of equations is encountoured when modelling some phenomena linked to seas or lakes. We give some examples where such regularity is useful. 1. Introduction This paper is devoted to a degenerate elliptic equation that we can find in several models in oceanography when we consider a domain with a depth vanishing on the shore. A lot of mathematical studies in oceanography assume a domain with a strictly positive depth in order to prevent the study in weighted spaces. Few papers have been devoted to coefficients with degenerated behavior. Regularity results in weighted Sobolev spaces on degenerate elliptic equations have been studied for instance in [2, 3] with a vanishing weight on the boundary that implies no boundary condition on the unknown. Here we study a degenerate elliptic equation with a weight with a blowing up comportment on the shore. A H2 regularity in weighted spaces is proved allowing to consider general weights. We will obtain such regularity by a careful study of the weight adapting the standard method of translation, see [8]. For example, we use and adapt some results on weighted Sobolev spaces that have been studied in [7]. Section 2 is devoted to the regularity result related to the degenerate elliptic equation. Then in Section 3, we explain why this kind of equation is important in oceanography. In the last section we describe precise examples where such regularity is used. At first we give some examples where such regularity result is used for existence result or error estimates that means the planetary-geostrophic equations and the vertical geostrophic equations. Then we give an example where it is used in a splitting projection method. We also mention that such equation is obtained from the Great Lake equation. We remark that this degenerate elliptic equation may be found in an other field such as in electromagnetism with the maxwell’s system, see [18]. Similar degenerate elliptic equation may also be encountoured for a problem 2000 Mathematics Subject Classification. 35Q30, 35B40, 76D05. Key words and phrases. Regularity result, degenerate elliptic equation, geophysics, weighted Sobolev spaces, splitting projection method. c 2004 Texas State University - San Marcos. Submitted June 15, 2002. Published March 23, 2004. 1 2 D. BRESCH, J. LEMOINE, & F. GUILLEN-G. EJDE-2004/42 related to Saint-Venant’s equations if we want to apply Babuska-Brezzi’s Inf-Sup Lemma in weighted spaces, see [1]. 2. The degenerate elliptic equation Let Obe a two-dimensional domain. This section is devoted to a regularity result on the following degenerate elliptic equation: Given h:O → Rand g:θ→R, the problem is to find Ψ : O → Rsuch that −∇x·(1 h∇xΨ) = gin O,Ψ = 0 on ∂O.(1) Here, his a function data satisfying h∈W1,∞(O), h > 0 in O,(2) h(x) = ϕ(δ(x))in a neightbourhood of ∂O(3) with δ(x) = dist(x,O). Moreover we assume that ϕis a non decreasing Lipschitz function, ϕ(0) = 0,(4) ∃c > 0 such that: ∀s > 0,ϕ0(s) ϕ(s)≤c s(5) and for all c1, c2>0, there exist α1, α2>0 such that ∀s, r > 0, c1≤s r≤c2=⇒α1≤ϕ(s) ϕ(r)≤α2.(6) Remark Note that ϕ(s) = c sα, 0 < α < 1 and c > 0, satisfies the previous hypothesis. Now, we define the function space H(O) = {Ψ∈L2(O) : ∇xΨ h1/2∈(L2(O))2,Ψ = 0 on ∂O} (7) endowed with the norm kΨkH(O)=k∇xΨ h1/2k(L2(O))2. where his defined by (2)–(3). We remark that k · kH(O)is a norm since 1 h≥c > 0 and Ψ = 0 on ∂Oimplies that there exists c > 0 such that for all Ψ ∈H(O), kΨkL2(O)≤ck∇xΨ h1/2k(L2(O))2. Lemma 2.1. Let H(O)be defined by (7) with hsatisfying (2)–(6). Then D(O)is dense in H(O). The proof of this lemma is similar to the proof of [7, Theorem 11.2]. Therefore, we omit it. The main result of this paper is the following Theorem 2.2. Let Obe a (two-dimensional) bounded domain of class C3. Let gbe such that δh1/2g∈L2(O)with hsatisfying (2)–(6). There exists a unique solution Ψof (1) such that Ψ∈H(O)and kΨkH(O)≤ckδh1/2gkL2(O). Moreover, if h1/2g∈L2(O) (8) EJDE-2004/42 DEGENERATE ELLIPTIC EQUATIONS 3 then h1/2∇x(1 h∇xΨ) ∈(L2(O))4,(9) kh1/2∇x(1 h∇xΨ)k(L2(O))4≤ckh1/2gkL2(O)(10) with ca constant depending only on the domain. Proof. Weak solutions: The existence and uniqueness of weak solutions of (1) follows from the Lax-Milgram theorem, since kΨk2 H(O)=ZO gΨ≤ kδh1/2gkL2(O)kΨ δh1/2kL2(O) ≤ckδh1/2gkL2(O)k∇xΨ h1/2k(L2(O))2. In the previous estimate we have used Hardy’s inequality in weighted space which will be proved in Lemma 2.3. Regularity: We use the usual difference quotients (cf. Brezis [8]). The interior regularity is well known since h≥c(ω)>0 in each ωbO. To obtain the regularity result up to the boundary, we define a local diffeomorphism Twhich preserves the normal direction. More precisely we define the local diffeomorphism T:Q→V by T(x∗, r)=(x∗, α(x∗)) + r n(x∗, α(x∗)) for all (x∗, r)∈Qwhere ∂Ois locally the graph of a C3function α(see Figure 1). This property, combined with the hypothesis (3) of h, will be strongly used in the sequel. We also define a cut-off function θsuch that θ∈ C2in V, θ= 0 on R2\V, θ = 1 in V1, ∂θ ∂en= 0 on a neightbourhood of ∂O ∩ V, where V1bVand the extension enof the normal nis defined for all (x, y)∈Vby en(x, y) = n(T(x∗,0)) where (x, y) = T(x∗, r), (x∗, r)∈Q. remark Due to the C3regularity of ∂O,Tis a C2diffeormorphism from Qonto a neightbourhood of (x0, y0) denoted by V. Multiplying (1) by θ, and denoting ξ=θΨ, it follows that ξis the (unique) solution in H(V∩ O) of −∇x·(1 h∇xξ) = fin V∩ O, ξ|∂(V∩O)= 0, (11) where f=θg +∇xh·∇xθ h2Ψ−1 h∆xθΨ−2 h∇xθ· ∇xΨ (since h1/2f∈L2(V∩ O)). For this, it suffices to check that ∇xh· ∇xθ h3/2Ψ∈L2(V∩ O). Indeed, since ∂θ/∂en= 0 on a neightbourhood of ∂O ∩ V, then ∇xh· ∇xθ=∂h ∂eτ ∂θ ∂eτ= 0 4 D. BRESCH, J. LEMOINE, & F. GUILLEN-G. EJDE-2004/42 T n V Q Q + (x ,y ) 0 0 α(x) Figure 1. The local diffeomorphism T on a neightbourhood of ∂O ∩ V. We recall that hdoesn’t depend on eτ(where eτis defined as en) using that his given by (3) and using the definition of T. Now we use the difference quotient technic on (11) to deduce the weight regularity announced in the theorem. Let ϕ:O → Rn(n≥1) be a function. We denote, for all (x∗, r)∈Q+, eϕ(x∗, r) = ϕ(T(x∗, r)), akl = 2 X j=1 ∂jT−1 k(T(x∗, r))∂jT−1 l(T(x∗, r))|Jac T(x∗, r)|, k, l = 1,2, ˆ k(x∗, r) = e f(x∗, r)|Jac T(x∗, r)|. Then we get that e ξis the unique solution in e H(Q+) of: ZQ+X k,l akl e h∂ke ξ ∂leϕdx∗dr =ZQ+ ˆ keϕdx∗dr (12) for all eϕ∈e H(Q+) where e H(Q+) = {eϕ∈L2(Q+) : e h−1/2∇xeϕ∈(L2(Q+))2,eϕ= 0 on ∂Q+}. We choose eϕ=D−τ(Dτe ξ) with τ=|τ|e1,Dτe ξ=e ξ(x+τ)−e ξ(x)/|τ|and |τ|small enough in order to obtain eϕ∈H(Q+). Using k1 e h1/2D−τ(Dτe ξ)kL2(Q+)≤ck1 e h1/2∇x(Dτe ξ)kL2(Q+) and ke h1/2ˆ kkL2(Q+)≤ckh1/2gkL2(O) EJDE-2004/42 DEGENERATE ELLIPTIC EQUATIONS 5 we get kˆ kD−τ(Dτe ξ)kL1(Q+)≤ckh1/2gkL2(O)k1 e h1/2∇x(Dτe ξ)kL2(Q+).(13) Moreover, denoting I=X k,l ZQ+ Dτ(1 e hakl∂ke ξ)∂l(Dτe ξ) since Dτ(akl/e h) = Dτ(akl)/e h, (recall that e hdoes not depend on τ) and T∈ C2, we have I≥ck1 e h1/2∇x(Dτe ξ)k2 (L2(Q+))2−ck1 e h1/2∇xe ξkL2k1 e h1/2∇x(Dτe ξ)k(L2(Q+))2 ≥ck1 e h1/2∇x(Dτe ξ)k2 (L2(Q+))2−ckh1/2gkL2(O)k1 e h1/2∇x(Dτe ξ)k(L2(Q+))2 (14) Using the variational formulation satisfied by e ξ, (13) and (14) we get k1 e h1/2∇x(Dτe ξ)k(L2(Q+))2≤ckh1/2gkL2(O).(15) Thus, by classical arguments, ∂2 1e ξ e h1/2∈L2(Q+) and ∂2∂1e ξ e h1/2∈L2(Q+),(16) and their respective norms are bounded by ckh1/2gkL2(O). In particular, e h1/2∂1(e h−1∂1e ξ)∈L2(Q+),e h1/2∂1(e h−1∂2e ξ)∈L2(Q+) (17) and their respective norms depend continuously on h1/2gin L2(O). We remark that contrary to the homogeneous case, that means the standard Laplacian operator, we have not yet the regularity e h1/2∂2(e h−1∂1e ξ)∈L2(Q+). We will obtain such regularity using the hypothesis (5) on h. Indeed, in the distribution sense, e h1/2∂2(1 e h∂1e ξ) = −∂2e h e h3/2∂1e ξ+1 e h1/2∂2∂1e ξ. (18) Since ∂1e ξ= 0 on ∂Q+then (16) yields ∂1e ξ∈H(Q+). Using the Hardy’s inequality (21) and Hypothesis (5), we get ZQ+∂2e h e h3/2∂1e ξ 2≤cZQ+ |∂1e ξ|2 e δ2e h≤cZQ+ |∂2∂1e ξ|2 e h. Thus, using the regularity (16), we get from (18) e h1/2∂2(1 e h∂1e ξ)∈L2(Q+),(19) ke h1/2∂2(∂1e ξ/e h)k(L2(Q+))2≤ckh1/2gkL2(O). Now we use the variational formulation (12) satisfied by e ξto obtain the regularity on e h1/2∂2(∂2e ξ/e h). We have ZQ+ a22 e h∂2e ξ∂2Φ≤ckh1/2gkL2(O)k1 e h1/2ΦkL2(Q+) 6 D. BRESCH, J. LEMOINE, & F. GUILLEN-G. EJDE-2004/42 α(x) z V O x Figure 2. The local coordinates for all Φ ∈ D(Q+). Using now the weak regularity of e ξand a22 ≥c > 0 in Q+, this gives e h1/2∂2(1 e h∂2e ξ)∈L2(Q+).(20) Therefore, (17), (19) and (20) give the regularity (9).  Lemma 2.3 (Hardy’s inequality in weighted spaces).Let hsatisfy (2)–(6) and let Ψ∈H(O). Then kΨ δh1/2kL2(O)≤ck∇xΨ h1/2kL2(O)2(21) where cdepends only on O. Proof. The proof of this lemma is similar to the proof of the classical Hardy’s inequality (see for instance [14]) introducing the corresponding weight. By density, it suffices to consider Ψ ∈ D(O). The interior estimate is obvious. In the local EJDE-2004/42 DEGENERATE ELLIPTIC EQUATIONS 7 coordinates (see Figure 2), we write Zz α(x) |Ψ(x, y)|2dy |y−α(x)|2ϕ(y−α(x)) ≤2Zz α(x)Z+∞ y dt |t−α(x)|2ϕ(t−α(x))Ψ(x, y)∂yΨ(x, y)dy ≤2Zz α(x) |Ψ(x, y)||∂yΨ(x, y)| |y−α(x)|ϕ(y−α(x))dy ≤2Zz α(x) |Ψ(x, y)|2 |y−α(x)|2ϕ(y−α(x))dy1/2Zz α(x) |∂yΨ(x, y)|2 ϕ(y−α(x))dy1/2. Therefore, Zz α(x) |Ψ(x, y)|2 |y−α(x)|2ϕ(y−α(x)) dy ≤4Zz α(x) |∂yΨ(x, y)|2 ϕ(y−α(x)) dy. Thus, integrating with respect to x, we get ZV∩O |Ψ(x, z)|2 |z−α(x)|2ϕ(z−α(x)) dz dx ≤4ZV∩O |∂yΨ|2 ϕ(ξ−α(x)) dξ dx. Since αis smooth enough, there exists c > 1 such that, for all (x, z)∈V∩ O, δ(x, z)≤ |z−α(x)| ≤ cδ(x, z). Therefore, using (5)–(6), we get ZV∩O |Ψ|2 δ2ϕ(δ)≤cZV∩O |∂yΨ|2 ϕ(δ) and the result follows.  3. Importance of this degenerate equation Let us introduce the three-dimensional oceanographic domain Ω = {(x, z)∈R3: x = (x, y)∈ O,−h(x) < z < 0} with O ⊂ R2the surface domain and h:O → Rwith h > 0 in O, the bottom function. Moreover, Γs=O × {0}is the surface boundary and Γb=∂Ω\Γsthe bottom. We denote ∇= (∇x, ∂z) the three dimensional gradient vector (with ∇x= (∂x, ∂y) the vectorial horizontal part) and ∆ is the Laplace operator. We explain, in this section, why such degenerate equation naturally appears in different models issued from oceanography when hydrostatic pressure is assumed. In all these equations, the field u= (v, w) and the pressure psatisfy the equation Lv +∇xp=f, ∂zp= 0 in Ω, ∂zw=−div xvin Ω,(22) and at least one of the the boundary conditions (v, w)·n∂Ω= 0, v ·n∂O= 0 (23) where we use the notation v=R0 −hv dz. We note that Lis a certain operator (algebraic or differential), see (30), (32) or (37) for some examples. 8 D. BRESCH, J. LEMOINE, & F. GUILLEN-G. EJDE-2004/42 Remark Of course Boundary conditions (23) are not necessary or sufficient to solve System (22). We have to choose other boundary conditions following the choice of the operator L. Integrating the divergence free equation with respect to the vertical coordinate and using the boundary condition (23), part 1, we obtain ∇x·v= 0 in O.(24) If the domain is simply connected then, using (24), there exists a stream function Ψ such that v=∇⊥ xΨ in O,(25) where ∇⊥ xis the 2Dcurl operator, i.e., (−∂y, ∂x). The boundary condition (23), part 2, gives Ψ = 0 on∂O.(26) We assume that vmaybe formally written as v=A∇xp+g1in Ω.(27) where Ais a matrix function (see the examples below). The purpose is to obtain some regularity result on v. Integrating (27) with respect to z(taking into account that ∂zp= 0 in Ω), we obtain v=A∇xp+g1, where A=R0 −hAand g1=R0 −hg1=. Therefore, using (25), we obtain ∇⊥ xΨ = A∇xp+g1 and thus, assuming Ainvertible ∇xp=B(∇⊥ xΨ−g1) (28) where B= (A)−1. Taking the horizontal curl operator of (28), using that ∇⊥ x·∇x= 0, we get ∇⊥ x·(B(∇⊥ xΨ−g1)) = 0 in O,Ψ = 0 on ∂=O.(29) On the other-hand, (27) and (28) yield ∇xv=∇xAB(∇⊥ xΨ−g1)+∇xg1. Thus the regularity of ∇xvdepends on the regularity of Ψ and g1. Theorem 2.2 may be extended easily to more general degenerate elliptic equations including for instance (29). Now assume that A= Id then we get B= 1/h Id and therefore AB(∇⊥ xΨ)= ∇⊥Ψ/h. Then the regularity of ∇xvin (L2(Ω))4is given by the regularity of Ψ h1/2∇x(h−1∇xΨ) ∈(L2(O))4 deduced from Theorem 2.2. Let us give now some applications of such regularity results on the stream function. 4. Some applications for lakes and seas We consider again, in the three first examples, the three-dimensional domain Ω = {(x, z)∈R3: x = (x, y)∈ O,−h(x) < z < 0}. EJDE-2004/42 DEGENERATE ELLIPTIC EQUATIONS 9 4.1. Planetary geostrophic equation. Let us consider the hyperviscous parametrization for the planetary equations given by εHv+fv⊥+∇xp= 0, ∂zp= +T= 0, div xv+∂zw= 0,(v, w)·n∂Ω= 0, v ·n∂O= 0, ∂tT−Kh∆xT−Kv∂2 zT+v· ∇xT+w∂zT+γDT=Q, (30) with f= (1 + βy) where βis a constant, with suitable bounda= ry and initial conditions and Da suitable fourth order differential operator. This system has been studied in [6] using some relations between Tand u= (v, w) jointly with the regularity result proved in Theorem 2.2. Here, we only recall how to get the estimate on divxvwhich is necessary to obtain the existence result for (30) (the regularity result proved in Theorem 2.2 is used there). Let (v, p, T) be an approximate sequence of (30) built for instance using a Galerkin method. Then we get the following estimate. Theorem 4.1. Let Obe a bounded simply connected domain of class C3and h satisfied the hypothesis of Theorem 2. Then kdivxvkL2(Ω) ≤ckD1/2TkL2(Ω) +k∇xTk(L2(Ω))2. where cdepends only on Ω. This estimate allows to pass to the limit on the approximate solutions and to get an existence result of weak solutions for (30). Proof. The hydrostatic equation ∂zp+T= 0 reads p=ps+R0 zTwith ps=ps(x). Then, f Mv +∇xps=∇xZ0 z T(31) with f M=εH−f f εH. Using the same procedure than in Section 3, we prove that there exists Ψ such that v=∇⊥ xΨ. The stream function Ψ satisfies a degenerate elliptic equations similar to (1) and therefore it is possible to prove that h1/2∇x(1 h∇xΨ) ∈(L2(O))4 using Theorem 2.2 and the weak regularity satisfied by T. This regularity result implies the required estimate on divxvusing (31) and the relation −∇xps=f M h∇⊥ xΨ + 1 hZ0 −h ∇x(Z0 z T) coming from the hydrostatic approximation. The suitable fourth order differential operator is obtained using the relation between vand T. 4.2. Vertical-geostrophic equations. We recall here the equations studied in [5] in domains with vanishing depth. These equations may also be obtained in