Fachrichtung 6.1 Mathematik

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1 Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 89 Three dimensional fully localised solitary gravity-capillary waves in fluid interfaces Previous version Dominic Breit and Mark D. Groves Saarbrücken 010

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3 Fachrichtung 6.1 Mathematik Preprint No. 89 Universität des Saarlandes submitted: January 0, 011 Three dimensional fully localised solitary gravity-capillary waves in fluid interfaces Previous version Dominic Breit Saarland University Department of Mathematics P.O. Box Saarbrücken Germany Mark D. Groves Saarland University Department of Mathematics P.O. Box Saarbrücken Germany

4 Edited by FR 6.1 Mathematik Universität des Saarlandes Postfach Saarbrücken Germany Fax: preprint@math.uni-sb.de WWW:

5 1 Introduction The problem concerns two finite, immiscible, perfect fluids separated by an interface {y = ηx, z)}; the fluid motion is three-dimensional and the densities of the upper and lower fluid are ρ and ρ with ρ < ρ. Furthermore we define the fluid-domains Sη) := { x, y, z) : x, y R, ηx, z) y h }, Sη) := {x, y, z) : x, y R, h y ηx, z)}. Here h and h are two positive numbers describing the depth of the upper resp. lower fluid. Within each fluid domain the evolution is given by potential flow, so that u = ϕ, ϕ = 0 within Sη), u = ϕ, ϕ = 0 within Sη). The fluid interface obey the kinematic equations t η = ϕ y + η x ϕ x + η z ϕ z = ϕ N, t η = ϕ y η x ϕ x η z ϕ z = ϕ N. At the bond of Sη) Sη) one imposes Neumann boundary conditions on confining vertical walls, so that ϕ y x, h, z) = 0 = ϕ y x, h, z) for all x, z R. The Bernoulli-condition reads as )) ρ t ϕ + 1 η ϕ + gη σ div 1 + η )) = ρ t ϕ + 1 η ϕ + gη σ div. 1 + η Here g is the acceleration due to gravity and σ > 0 is the coefficient of surface tension. The kinetic energy of the system in each fluid domain is given by the Dirichlet integrals K = 1 ρ ϕ dxdydz, K = 1 ρ ϕ dxdydz. Sη) Sη) and the potential energy of the system is V = V 1 + V = 1 gρ ρ)η dxdz + 1 σρ ρ) 1 + η 1) dxdz. R R The Hamilton function is the total energy H = K + K + V 1 + V. 1

6 In order to obtain dimensionless variables we define x, y, z ) := 1 x, y, z) + h 1, 1, 1), h + h h + h ) 1 g t := t, h + h η x, z, t ) := 1 ηx, z, t), h + h ϕ x, y, z, t 1 ) := ϕx, y, z, t), h + h) 3 g 1 ϕ x, y, z, t 1 ) := ϕx, y, z, t). h + h) 3 g 1 Hence we receive the equations dropping the primes for notational simplicity) ϕ = 0, 0 < y < η + h 1.1) ϕ = 0, η + h < y < 1, 1.) were we have abbreviated h := h/h + h) with boundary conditions t η = ϕ y η x ϕ x η z ϕ z, y = η + h, 1.3) t η = ϕ y + η x ϕ x + η z ϕ z, y = η + h 1.4) ϕ y = 0, y = 0, 1.5) ϕ y = 0, y = 1, 1.6) )) ρ t ϕ + 1 η ϕ + η β div 1 + η ) = t ϕ + 1 η ϕ + η β div, y = η + h. 1 + η 1.7) Here we have ρ := ρ/ρ 0, 1) and β := σ/gh + h)) > 0. The kinetic and potential energies now reads as K = 1 ρ ϕ dxdydz, 1.8) Sη) K = 1 ϕ dxdydz, 1.9) Sη) V 1 = 1 1 ρ)η dxdz, 1.10) R

7 V = 1 β1 ρ) 1 + η 1) dxdz. 1.11) R Here we have changed the meaning of the fluid domains, i.e., Sη) := {x, y, z) : x, y R, 0 y η + h}, Sη) := {x, y, z) : x, y R, η + h y 1}. Steady waves are water waves which travel in a distinguish horizontal direction with constant speed and without change of shape; without loss of generality we assume that the waves propagate in the x-direction with speed c, so that ηx, z, t) = ηx ct, z), ϕx, y, z, t) = ϕx ct, y, z) and ϕx, y, z, t) = ϕx ct, y, z). Now we have to minimize the functional Eη, ϕ, ϕ) := Kη, ϕ) + Kη, ϕ) + V 1 η) + V η). 1.1) We deenote the boundary values of the velocity potentials by Φx, z) := ϕx, z, ηx, z)) and Φx, z) := ϕx, z, ηx, z)). Following Benjamin and Bridges [BB] we set ξx) := Φx) ρφ, and the natural choice of canonical variables is η, ξ) compare [CG]). Similarly to [CG] and [BGS], section 1.) we define Dirichlet-Neumann operators Gη) and Gη) which maps for a given η) Dirichlet boundary-data of solution of the Laplace-equation to the Neumann boundary-data, i.e. If we define Gη)Φx, z) : = 1 + η ) 1 ϕ NSη) )x, z), Gη)Φx, z) : = 1 + η ) 1 ϕ NSη) )x, z). Bη) := Gη) + ρgη), we obtain the Hamilton following the lines of [CG], p. 4) Hη, ξ) = 1 ξgη)bη) 1 Gη)ξ dxdz ρ)η dxdz R R + 1 β1 ρ) 1 + η 1) dxdz. R 1.13) The key of our existence theory is minimizing H subject to the constraint of a fixed value for the momentum of a wave in the x-direction Iη, ξ) := η x ξ dxdz. 1.14) R We tackle the problem of finding minimizers of H under Iη, ξ) = κρ, µ)µ, where in two steps. κρ, µ) = 1 ρ) 1 h + ρ ) 1, 1 h 3

8 { 1. Fix η 0 and minimize Hη, ) over T µ = ξ H 1/ R ) : Iη, ξ) = } κρ, µ)µ. This problem of minimizing a quadratic functional over a linear manifold) admits a unique global minimizer ξ η.. Minimize J η) := Hη, ξ η ) over η U {0} with U := B M 0) H 3 R ). Because ξ η minimizes Hη, ) over T µ there exists a Lagrange multiplier λ η such that Hence Gη)Bη) 1 Gη)ξ η = λ η η x. ξ η = λ η [ Gη)Bη) 1 Gη) ] 1 ηx = λ η [ Nη) + ρnη) ] ηx, where Nη) = Gη) 1 and Nη) = Gη) 1 are the Neumann-Dirichlet operators. Furthermore we get κρ, h)µ λ η =, Lη) = 1 [ ] η x Nη) + ρnη) ηx dxdz. 1.15) Lη) R For J η) we get the representation where J µ η) = Kη) + κρ, µ)µ, 1.16) Lη) { 1 Kη) = 1 ρ) R η + β } 1 + ηx + ηz β dx dz, 1.17) Our paper is organized as follows: In section we prove analiticity of the Neumann- Dirichlet operator. A consideration of minimizing sequences is given in section 3. In the forth section we have a look at strict sub-additivity of the infimum of J µ with respect to µ, whereas in the last section we follow the main Theorems about the existence of minimizers and the stability of the set of minimizers. The functional-analytic setting.1 The Neumann-Dirichlet operator Our first task is to find suitable function spaces for the functionals H and I defined in equations 1.13), 1.14) and introduce rigorous definitions of the Dirichlet-Neumann operators and their inverses. Since the functional J µ to be minimised involves Gη) 1 and Gη) 1 we begin with the formal definition of this Neumann-Dirichlet operator Nη) and Nη): for fixed ξ = ξx, z) we solve the boundary-value problem ϕ = 0, 0 < y < h + η,.1) 4

9 and define ϕ y η x ϕ x η z ϕ z = ξ, y = h + η,.) ϕ y = 0, y = 0.3) Nη)ξ = ϕ y=h+η. Furthermore we solve the boundary-value problem and define ϕ = 0, h + η < y < 1,.4) -ϕ y + η x ϕ x + η z ϕ z = ξ, y = h + η,.5) ϕ y = 0, y = 1.6) Nη)ξ = ϕ y=h+η. We study this boundary-value problems by transforming them to equivalent problems in fixed domains. The change of variable in the first problem y = h h + η y, ux, y, z) = ϕx, y, z) transforms the variable domain {0 < y < h + ηx, z)} into the slab Σ = {x, y, z) R 0, h) R} and the boundary-value problem.1).3) into where u = x F 1 + z F + y F 3, 0 < y < h,.7) u y = F 3 + ξ, y = h,.8) u y = 0, y = 0,.9) F 1 = η h u x + y h η xu y, F = η h u z + y h η zu y, F 3 = η η + h u y + y h η xu x + y h η zu z 1 y ηx h η + h u y 1 y ηz h η + h u y and we have again dropped the primes for notational simplicity; the Neumann-Dirichlet operator is given by Nη)ξ = u y=h. In the second problem the transformation y = 1 h 1 h η 1 y), ux, y, z) = ϕx, y, z) converts the domain {h + ηx, z) < y < 1} into Σ = {x, y, z) R 0, 1 h) R} and the boundary-value problem.4).6) into u = x F 1 + z F + y F 3, 0 < y < 1 h,.10) 5

10 where F 1 = F = F 3 = u y = F 3 + ξ, y = 1 h,.11) u y = 0, y = 0,.1) η 1 h u x y 1 h η xu y, η 1 h u z y 1 h η zu y, η η + h 1 u y y 1 h η xu x y 1 h η zu z + 1 The Neumann-Dirichlet operator is given by Nη)ξ = u y=1 h. y ηx 1 h η + h 1 u y + 1 y ηz 1 h η + h 1 u y. To develop a convenient theory for weak solutions of the boundary-value problems.7).9) and.10).1) we follow the lines of [BGS] section.1). We define the completion H 1 Σ) of SΣ, R) = {u C Σ) : x, z) m α 1 x α z u is bounded for all m, α 1, α N 0 } with respect to the norm u := Σ u x + u y + u z) dy dx dz. Here we have Σ {Σ, Σ} and Σ denotes the closure of Σ. The corresponding space for the traces u y=h and u y=1 h is the completion H 1/ R ) of the inner product space X 1/ R ) constructed by equipping the Schwartz class SR, R) with the norm η,1/ := 1 + k ) 1/ k ˆη dk 1 dk ; R its dual H 1/ X 1/ R )) = R )) is the space { u S R, R) : sup{ u, η) : η X 1/ } R ), η,1/ < 1} <, where S R, R) is the class of two-dimensional, real-valued, tempered distributions. A more convenient description of H 1/ R )) is proven in Prop..1 in [BGS]: Proposition.1. Let H 1/ R ) be the completion of the inner product space X 1/ R ) constructed by equipping SR, R) with the norm η, 1/ := 1 + k ) 1/ k ˆη dk 1 dk, R where SR, R) is the subclass of SR, R) consisting of functions with zero mean. The space H 1/ R ) can be identified with H 1/ R )). 6

11 With these preparations we obtain similarly to Lemma.4, [BGS] LEMMA.1. For each ξ H 1/ R ) and η B 1/ 0) W 1, R ) the boundaryvalue problems.7).9) and.10).1) have unique weak solutions u H Σ) 1 and u H Σ). 1 Here weak solutions are defined in the sense of [BGS] Def..3). We conclude with a rigorous definition of the Neumann-Dirichlet operators. Definition.1. a) The Neumann-Dirichlet operator for the boundary-value problem.7).9) is the bounded linear operator Nη) : H 1/ R ) H 1/ R ) defined by Nη)ξ = u y=h, where u H 1 Σ) is the unique weak solution of.7).9). b) The Neumann-Dirichlet operator for the boundary-value problem.10).1) is the bounded linear operator Nη) : H 1/ R ) H 1/ R ) defined by Nη)ξ = u y=1 h, where u H 1 Σ) is the unique weak solution of.10).1). At the end of this section we present the following useful representations compare [BGS], Remark.6, for details) LEMMA.. We have for ξ H 1/ R ) ξnη)ξ dx dz R = Σ u x yη x h + η u y ξnη)ξ dx dz R = u x Σ ) huy + h + η ) yη x η + h 1 u y + ) + u z ) 1 h)uy + u z η + h 1 yη ) ) z η + h u η + h y dx dy dz; h ) ) yη z η + h 1 u 1 η h y dx dy dz. 1 h. Analyticity of the Neumann-Dirichlet operators In this section we establish that Nη) and Nη) are analytic functions of η in the above function spaces, which clearly implies analicity of Nη) + ρnη). We start with the definition of analyticity compare Buffoni & Toland [BT], Definition 4.3.1). 7

12 Definition.. Let X and Y be Banach spaces, U be a non-empty, open subset of X and L k s X, Y ) be the space of bounded, k-linear symmetric operators X k Y with norm m := inf{c : m{f} k) ) Y c f k X for all f X}. A function F : U Y is analytic at a point x 0 U if there exist real numbers δ, r > 0 and a sequence {m k }, where m k L k s X, Y ), k = 0, 1,,..., with the properties that and F x) = m k {x x 0 } k) ), x B δ x 0 ) k=0 sup r k m k <. k 0 Our main task is to establish the following theorem. THEOREM.3. The mappings from W 1, R ) LH 1/ R ), H 1/ R )) given by η ξ u y=h ) and η ξ u y=1 h ), where u H Σ) 1 and u H Σ) 1 are the unique weak solution of.7).9) resp..10).1), are analytic at the origin. Proof: If we can show ux, y, z) = ux, y, z) = u n x, y, z),.13) n=0 u n x, y, z),.14) n=0 where u n and u n are functions of η and ξ which are homogeneous of degree n in η and linear in ξ, then the claim of Theorem.3 follows by the lines of [BGS] section.). We refer to Nicholls & Reitich who developed this technique for proving analicity of Dirichlet- Neumann operators. Substituting.13) into the equations of the nether fluid, one finds that and for n = 1,, 3,..., where u 0 = 0, 0 < y < h,.15) u 0 y = ξ, y = h,.16) u 0 y = 0, y = 0.17) u n = x F n 1 + z F n + y F n 3, 0 < y < h,.18) u n y = F n 3, y = h,.19) u n y = 0, y = 0.0) F n 1 = η h un 1 x + y h η xu n 1 y,.1) 8

13 F n = η h un 1 z + y h η zu n 1 y,.) F n 3 = η h n 1 h l η) l u n 1 l y l=0 l=0 + y h η xu n 1 x y n h η x + ηz) h l η) l uy n l. Substituting.14) into the equations of the upper fluid, one finds that and for n = 1,, 3,..., where + y h η zu n 1 z.3) u 0 = 0, 0 < y < 1 h,.4) u 0 y = ξ, y = 1 h,.5) u 0 y = 0, y = 0.6) u n = x F n 1 + z F n + y F n 3, 0 < y < 1 h,.7) u n y = F n 3, y = 1 h,.8) u n y = 0, y = 0.9) F n η 1 = 1 h un 1 x y 1 h η xuy n 1,.30) F n η = 1 h un 1 z y 1 h η zuy n 1,.31) F n 1 n 1 3 = η 1 h) l η l uy n 1 l y 1 h 1 h η xux n 1 y 1 h η zuz n 1.3) l=0 y n 1 h) η x + ηz) 1 h) l η l u n l l=0 y. From this expansion we can follow the claim of Theorem.3 by the lines of [BGS] section.1). Observe that the formula 1.15) defining L may be written as Lη) = 1 ηkη)η dx dz, R where Kη) = x [ Nη) + ρnη) ] x ), and we obtain similarly to [BGS] Thm..19) THEOREM.4. Suppose that s > 1. The operator K ) : H s+3/ R ) LH s+1 R ), H s R )) is analytic at the origin. 9

14 We remark that the Fourier transforms of the weak solutions u 0 and u 0 of.15).17) and.4).6) are given by û 0 = cosh k y k sinh k h ˆξ, û 0 = cosh k y k sinh k 1 h) ˆξ..33) Using Theorem.3 and the continuity of the trace operator H s+1/ Σ) H s R ), we find that the series representations of the operators H s+3/ R ) LH s+1 R ), H s R )) given by η ζ u x y=h ) and η ζ u x y=1 h ) are given by Kη) = [ K n η) + ρk n η) ], n=0 where K n η)ζ = u n x y=h, K n η)ζ = u n x y=1 h and ξ = ζ x..3 The functionals K, L and a special testfunction The following lemma, whose proof is similar to the arguments in [BGS] section.4), formally states the analyticity property of K examine the explicit formula for K) and L see Theorem.4). In particular this result implies that K, L belong to the class C U, R) and that equation 1.16) defines an operator J µ C U \{0}, R), where U = B M 0) H 3 R ) and M is chosen sufficiently small. LEMMA.5. Equations 1.17), 1.15) define functionals K : H s+1 R ) R, L : H s+3/ R ) R for s > 1 which are analytic at the origin and satisfy K0) = L0) = 0. We have the following representation for the gradients of K, L and L, where the last two functionals are defined in a suitable fashion such that L = L + ρl. LEMMA.6. The gradients K η), L η) and L η) in L R ) exist for each η U. They are given by the formulae ) ) K βη x βη η) = 1 ρ) 1 ρ) z + 1 ρ)η, 1 + η x + ηz 1 + η x + ηz L η) = h 0 { x 1 h u x + u z) y h u xu y ) x y y h u η x u ) y zu y ) z + hh + η) x } y u y u + hh + η) η x + ηz) y + h dy u h + η) x y=h, L η) = 1 h 0 { 1 1 h) u x + u z) + y 1 h u xu y ) x + y 1 h u zu y ) z z + y η z u y ) hh + η) z 10

15 y η x u ) y y η z u ) y 1 h)η + h 1) x 1 h)η + h 1) z y u y 1 h)η + h 1) η x + ηz) 1 } h)u y dy u η + h 1) x y=1 h, and define functions K : H 3 R ) H 1 R ), L, L : H s+3/ R ) H s R ) for s > 1 which are analytic at the origin and satisfy K 0) = L 0) = L 0) = 0. Proof. The formula for K is given in [BGS] Lemma.7), whereas for L and L we differentiate the equations in Lemma.: in the first case we have for ξ H 1/ R ) ξnη)ξ dx dz R = u x yη ) ) x h + η u huy y + + u h + η z yη ) ) z η + h u η + h y. dx dy dz h Σ We obtain abbreviating w = du[η]ω) dl[η]ω) { = 1 h Σ h + η)w x u x + w z u z ) yη x w x u y yη x u x w y yη z w z u y yη z u z w y + y u y w y h + η η x + ηz) + h u y w y h + η + y u y h + η η xω x + η z ω z ) + ω u x + u z) yω x u x u y yω z u z u y } ωy u y h + η) η x + ηz) ωu yh h + η) dx dy dz..34) Since u is a weak solution of.1)-.3), letting ξ = η x, we get { h + η Σ h u xv x + u z v z ) y h η xv x u y y h η xu x v y y h η zv z u y y h η zu z v y } + y u y v y hh + η) η x + ηz) + hu yv y h + η = η x v y=h dx dz R dy dx dz for every v H Σ). 1 Differentiating this equation with respect to η, we find that { h + η h w xv x + w z v z ) y h η xw x v y y h η xv x w y y h η zw z v y y h η zv z w y Σ + y v y w y hh + η) η x + η z) + h v yw y h + η + ω h u xv x + u z v z ) y h ω xv x u y y h ω xu x v y 11

16 y h ω zv z u y y h ω zu z v y + y u y v y hh + η) η xω x + η z ω z ) y u y v y hh + η) η x + ηz)ω h u } yv y h + η) ω dy dx dz = ω x v y=h dx dz R for every v H 1 Σ); subtracting this equation with v = u from.34) yields dl[η]ω) { = ω Σ h u x + u z) + y h ω xu x u y + y h ω zu z u y y u y hh + η) η xω x + η z ω z ) } y u y ωu + hh + η) η x + ηz)ω y + h dy dx dz + ω h + η) x u y=h dx dz R { = 1 Σ h u x + u z) y h u xu y ) x y y h u η x u ) y y η z u ) y zu y ) z + + hh + η) x hh + η) z } y u y u + hh + η) η x + ηz) y + h ω dy dx dz ωu h + η) x y=h dx dz. R Furthermore we have hence dl[η]ω) = Σ ξnη)ξ dx dz R = u x + Σ u z ) yη x η + h 1 u y + ) ) yη z η + h 1 u y 1 η h 1 h ) 1 h)uy η + h 1 dx dy dz, { 1 η + h 1 h w xu x + w z u z ) + y 1 h η xw x u y + y 1 h η xu x w y + y 1 h η zw z u y + y 1 h η zu z w y + u y w y 1 h) η + h 1 y u y w y 1 h)η + h 1) η x + ηz) ω 1 h) u x u z) + y 1 h ω xu x u y + y 1 h ω y u y zu z u y 1 h)η + h 1) η xω x + η z ω z ) 1

17 y u y ωu + 1 h)η + h 1) η x + ηz)ω y + 1 h) η + h 1) } dy dx dz,.35) where w = du[η]ω). Since u is a weak solution of.4)-.6), we get { η + h 1 Σ 1 h u xv x + u z v z ) y 1 h η xv x u y y 1 h η xu x v y y 1 h η zv z u y y 1 h η zu z v y } y u y v y 1 h)η + h 1) η x + ηz) + 1 h)u yv y dy dx dz η + h 1 = η x v y=1 h dx dz R for every v H Σ). 1 Differentiating this equation with respect to η, we find that { η + h 1 1 h w xv x + w z v z ) y 1 h η xw x v y y 1 h η xv x w y y 1 h η zw z v y Σ y 1 h η y v y w y zv z w y + 1 h)η + h 1) η x + ηz) + 1 h)v yw y η + h 1 + ω 1 h u xv x + u z v z ) y 1 h ω xv x u y y 1 h ω xu x v y y 1 h ω zv z u y y 1 h ω y u y v y η x ω x zu z v y + 1 h)η + h 1) } y u y v y η z ω z 1 h)η + h 1) y u y v y 1 h)η + h 1) η x + ηz)ω 1 h)u yv y η + h 1) ω dy dx dz = ω x v y=1 h dx dz R for every v H 1 Σ); adding this equation with v = u to.35) yields dl[η]ω) { ω = Σ 1 h) u x + u z) y 1 h ω xu x u y y 1 h ω y u yη x ω x zu z u y + 1 h)η + h 1) y u yη z ω z + 1 h)η + h 1) y u yηx + ηz)ω 1 h)η + h 1) 1 } h)ωu y dy dx dz η + h 1) + ω x u y=1 h dx dz R { 1 = Σ 1 h) u x + u z) + y 1 h u xu y ) x + y 1 h u zu y ) z y η x u ) y y η z u ) y y u yηx + ηz) + 1 h)η + h 1) x 1 h)η + h 1) z 1 h)η + h 1) 1 } h)u y ω dy dx dz ωu η + h 1) x y=1 h dx dz. R 13

18 Corollary.7. The gradient J µη) in L R ) exists for each η U and defines a function J C H 3 R ), H 1 R )). Our final results are useful a priori estimates. Lemma.8 shows in particular that 1 inf J µ η) < κρ, h)µ, κρ, h) := 1 ρ) η U\{0} h + ρ ) 1. 1 h LEMMA.8. There exists η µ U \{0} with compact support and a positive constant c such that η µ 3 c µ 1/ and J µ η µ) < κρ, h)µ cµ 3. Proof: We follow the ideas of [BGS] Lemma.9) and consider η µx, z) = α Ψαx, α z), 0 < α 1 with an appropriate choice of Ψ C 0 [ 1, 1 ] ) and α = αµ). We choose Ψx, z) := ψ x x, z), where ψ also belongs to C0 [ 1, 1 ] ). We begin by computing the leading-order terms in the asymptotic expansions of Kη ) and Lη ) in powers of α. We quote from [BGS], 60), Kη ) = 1 ρ) α Furthermore we see Ψ dx dz + 1 ρ) α3 β R R Ψ x dx dz + Oα 5 )..36) Lη ) = L η ) + L 3 η ) + O η 1, η 3) = L η ) + L 3 η ) + Oα 5 ) = L η ) + L η ) + L 3 η ) + L 3 η ) + Oα 5 ). Applying the calculations of [BGS] Appendix B) and noting.33) we conclude that L η ) = α h R Ψ dx dz + hα3 6 as well as have a look at Lemma.6) On the other hand we have R Ψ x dx dz α3 k1 h R k 1 + α k ˆψ z dk 1 dk + Oα 5 ) L 3 η ) = α3 h L η ) α R = Ψ dx dz + 1 h) R Ψ 3 dx dz + Oα 4 ) h)α3 6 R Ψ x dx dz

19 and α3 k1 1 h) R k 1 + α k ˆψ z dk 1 dk + Oα 5 ) L 3 η ) = α 3 Ψ 3 dx dz + Oα 4 ). 1 h) R Combining the above results shows that Lη ) = α 1 h + ρ ) R Ψ dx dz + α3 1 h 6 α3 1 h ρ ) Ψ 3 dx dz 1 h R α3 1 h + ρ ) 1 h Let α be the solution of the equation µ = Lη ) then h + ρ1 h)) Ψ x dx dz R k1 R k 1 + α k ˆψ z dk 1 dk + Oα 4 )..37) κρ, h)µ Lη ) κρ, h)µ = κρ, h)lη ) = 1 h + ρ ) 1 1 ρ)lη ). 1 h This means we have α = ch, ρ)µ/ Ψ 0 + oµ). Hence one finds abbreviating that 1 C 1 ρ, h) := h + ρ ) 1 h + ρ1 h)) 1 h 1 C ρ, h) := h + ρ ) h h ρ ) 1 h J ηµ) κρ, h)µ 1 = Kηµ) h + ρ ) 1 1 ρ)lη ) 1 h = α3 1 ρ) ) β C 1ρ,h) )Ψ 3 x + C ρ, h)ψ 3 dx dz + α3 1 ρ) R k1 R k 1 + α k Finally, let us choose ψ C 0 [ 1, 1 ] ) such that and set ψ = A ψ; it follows that J µ η µ) κρ, h)µ R ψ3 x dx dz < 0 15 ˆψ z dk 1 dk + Oα 4 ).

20 = α3 1 ρ) ) [A β C 1ρ,h) ψ 3 xx dx dz + A k1 ˆ ψz dk R R k 1 + α k 1 dk ] + A 3 C ρ, h) ψ3 x dx dz + Oα 4 )< 0 for sufficiently large values of A. R 3 Minimising sequences 3.1 The penalised minimisation problem In this section we study the functional J ρ,µ : H 3 R ) R { } defined by Kη) + µ Lη) + ρ η 3), u U \{0}, J ρ,µ η) =, η U \{0}, in which ρ : [0, M ) R is a smooth, increasing penalisation function such that ρt) = 0 for 0 t M and ρt) as t M. The number M 0, M) is chosen so that M > c + κρ, h)d)µ see below), and the following analysis is valid for every such choice of M, which in particular may be chosen arbitrarily close to M. The following two lemma collects some properties of minimizing sequences of J ρ,µ LEMMA 3.1. Every minimising sequence {η n } for J ρ,µ has the properties that J ρ,µ η n ) < κρ, h)µ, Lη n ) > µ, L η n ) cµ, M µ η n ) cµ 3, η n 1, cµ 3 for each n N, where M µ η) = J ρ,µ η) K η) K η) = 1 ρ) R L η) = L η) + ρl η). κρ, h)µ, L η) { β η x + β η z + η } dx dz, The last two terms are the quadratic parts in the expansions of L und L. Proof. Only part four needs a comment, for the rest we refer to [BGS] Lemma 3.). Observe that remember h 0, 1)) L η) = ηu 0 x y=h dx dz R 16

21 = = k coth hk ˆη dk 1 dk 1 k1 h R k ˆη dk 1 dk + 1 k 3h 1 ˆη dk 1 dk R + 1 k1 h R k hk coth hk hk ) ˆη dk 1 dk 1 ˆη dk 1 dk + 1 h R h β k ˆη dk 1 dk R h1 ρ) K η), R k 1 in which we have used.33) and estimated β > 1/3, k coth k k 0. Analogously we receive It follows that Hence we obtain and L η) 1 h)1 ρ) K η). 1 K η) 1 ρ) h + ρ ) 1 L η) = κh, ρ)l η). 1 h K η) + κρ, µ)µ L η) µ κρ, µ)k η) L η) M µ η n ) J ρ,µ η n ) κρ, h)µ cµ 3 using the arguments from [BGS] proof of Lemma 3.4). κh, ρ)µ 3. Application of the concentration-compactness principle The next step is to apply the concentration-compactness principle Lions [Li1,]) in order to show strong convergence of a subsequence to a minimizer of the functional J ρ,µ which do not touch the boundary of U. THEOREM 3.. Any sequence {u n } L 1 R ) of non-negative functions with the property that lim u n x, z) dx dz = l > 0 n R admits a subsequence for which one of the following phenomena occurs. Vanishing: For each R > 0 one has that lim n sup x, z) R B R x, z) u n x, z) dx dz 17 ) = 0.

22 Concentration: There is a sequence {x n, z n )} R with the property that for each ε > 0 there exists a positive real number R with u n x + x n, z + z n ) dx dz l ε for each n N. B R 0) Dichotomy: There are sequences {x n, z n )} R, {M n }, {N n } R and a real number λ 0, l) with the properties that M n, N n, M n /N n 0, u n x + x n, z + z n ) dx dz λ, as n. Furthermore B Mn 0) B Nn 0) lim n u n x + x n, z + z n ) dx dz λ, sup x, z) R B r x, z) u n x, z) dx dz for each r > 0, and for each ε > 0 there is a positive, real number R such that u n x + x n, z + z n ) dx dz λ ε for each n N. B R 0) We apply Theorem 3. to the sequence {u n } defined by ) λ u n = η nxx + η nxz + η nzz + η nx + η nz + η n, 3.1) so that u n L 1 R ) = η n 3. Quoting the arguments from [BGS] section 3.) on can easily deduce from Lemma 3.1 part 5 LEMMA 3.3. The sequence {u n } does not have the vanishing property. Let us now investigate the consequences of concentration and dichotomy, replacing {u n } by the subsequence identified by the relevant clause in Theorem 3. and, with a slight abuse of notation, abbreviating the sequences {u n + x n, + z n )} and {η n + x n, + z n )} to respectively {u n } and {η n }. The fact that { η n 3 } is bounded implies that {η n } is weakly convergent in H 3 R ); we denote its weak limit by η 1). Lemma 3.4 deals with the concentration case; which is proved by an argument given by Groves & Sun [GS]. We refer to [BGS] Prop. 3.7 and Lemma 3.8) for details, note that in our situation the constants depends on ρ and h, too. 18

23 LEMMA 3.4. Suppose that {u n } has the concentration property. The sequence {η n } admits a subsequence which satisfies lim η n 3 M n and converges in H r R ) for r [0, 3) to η 1). The function η 1) satisfies the estimate η 1) 3 DKη 1) ) < Dµ, minimises J ρ,µ and minimises J µ over Ũ \{0}, where Ũ = {η H3 R ) : η 3 < M}. Now we have to exclude the dichotomy -case. Therefore we can follow the ideas of [BGS]. A cruical tool are the pseudo-local properties of the operator L, which means we have LEMMA 3.5. Consider two sequences {v m 1) }, {v m ) } with sup v m 1) + v m 1) 3 < M and supp v m 1) B R 0), supp v m ) R \B Sm 0), where R > 0 and {S m } is an increasing, unbounded sequence of positive real numbers. Clearly as m. Lv m 1) + v m ) ) Lv m 1) ) Lv m ) ) 0, L v m 1) + v m ) ) L v m 1) ) L v m ) ) 0, L v m ) ), v m 1) 0 0 For the proof we refer the reader to [BGS] Appendix D). The arguements which are presented there have to be applied seperately to the funcitonals L and L. Note that Lemma 3.5 clearly stays true, if we replace L by K, since K and K are local operators. This finally shows LEMMA 3.6. The sequence {u n } does not have the dichotomy property. 4 Strict sub-additivity The goal of this section is to establish that the quantity c µ = inf J µ η) η U\{0} is a strictly sub-homogeneous function of µ, that is c aµ < ac µ, a > 1. Its strict sub-homogeneity implies that c µ also has the strict sub-additivity property that c µ1 +µ < c µ1 + c µ, µ 1, µ > 0 4.1) see Buffoni [B]); inequality 4.1 plays a crucial role in the variational theory for the stability theory below. Applying the arguments from [BGS] Thm. 4.1) to our problem we obtain 19

24 THEOREM 4.1. There exists a minimising sequence { η n } for J µ over U \{0} with the properties that η n 3 cµ for each n N and lim J µ η n ) = c ρ,µ, n lim J n µ η n ) 1 = 0. The strict sub-homogeneity of c µ follows from the existence of a minimising sequence {η n } for J on U \{0} with the property that the function a a 5/ M a µaη n ), a [1, ] 4.) is decreasing and strictly negative see Lemma 4.5). Suppose that L η) > cµ, Lη) > cµ and observe that 1 M µ η) = K nl η) + µ Lη) 1 ) L η) = K 4 η) + K 6 η) K m1 η) + R m1 η) ) + µ N 1 η) + N 0 η) + N 1 η) N m 1η) + S m η) 4.3), where N j η) and J j η) are homogeneous of degree j and R m1 η), R m 1 η), η 0 = O η m 1 3 ), S m η), S m η), η 0 = O η m 3 ) for integers m 1 and m 0. For this purpose we construct a wighted norm on H 3 R ). Due to the structure of our problem with parameters ρ, h 0, 1) it is not possible to use the approach from [BGS], hence we proceed as in [GW]. The idea mentioned there is more natural to the problem itself. Firstly we define gk) = gk 1, k ) := 1 ρ)1 + β k ) ) k κh, ρ) 1 k k coth hk + ρ k 1 k coth 1 h)k k = 1 ρ)1 + β k ) κh, ρ) k coth hk + ρ k coth 1 h)k ) ) k + κh, ρ) k k coth hk + ρ k k coth 1 h)k k =: g 1 k) + g k) and for µ > 0 and α, 1) η α := 1 + µ 11 α g 11 4 k)) η dk R a norm on H 3 R ). For the norm α we obtain the following properties. Proposition 4.1. a) The function g behaves like k + k k = r + sin θ. 0

25 b) There is a constant c, independent of the value of µ, such that we have for all η H 3 R ) i) η cµ 3α η α; ii) η cµ 5α η α. Proof: For the lower bound we have ] 1 g 1 k) 1 ρ)1 + β k ) κh, ρ) [1 + hk + ρ ]) 1 h)k [1 + h 3 1 h 3 1 = 1 ρ) κh, ρ) h + ρ ) ) κh, ρ) + 1 ρ)β [h + ρ1 h)] k 1 h 3 = 1 ρ) β 1 ) κh, ρ) [h + ρ1 h)] k, 3 1 ρ) where the term in brackets is strictly positive since κh, ρ) 1 [h + ρ1 h)] = 1 ρ) h + ρ ) 1 [h + ρ1 h)] < 1 1 h and β > 1. Furthermore we have 3 1 k g k) κh, ρ) h k + ρ ) k 1 h k = 1 ρ) k k, where we used t coth t 1 for t 0. Both together proves the lower bound, whereas the upper bound is a consequence of 1 g 1 k) 1 ρ)1 + β k ) κh, ρ) h + ρ ) = 1 ρ)β k 1 h and ] 1 k g k) κh, ρ) [1 + hk + ρ k [1 + h k 3 1 h k 1 ρ) k κh, ρ) + h + ρ1 h)) k. k 3 Let P ) be a Fourier-multiplier-operator. Then P )η P k) η L 1 R P k) 1 + µ 11 α gk) 11 4 ]) 1 h)k 3 dk ) η α. In the case P k) = 1 we can bound the term in brackets in the following way using part a) R P k) dk C dk 1 + µ 11 α gk) 11 4 R µ 11 4 α k 11 + µ 11 1 α k 11 k 11

26 = C = C 0 0 π R Cµ 3α 0 π r dθ dr 1 + µ 11 α r 11 + µ 11 α sin 11 θ r 1 + µ 11 α r 11 + µ 11 α θ 11 s R 1 + s 11 + φ 11 dφ ds dθ dr and the claim of b) i) follows since the remaining integral is finite. If P k) = ik we obtain with the same arguments R P k) dk C r 3 dθ dr 1 + µ 11 α gk) R 1 + µ 11 α r 11 + µ 11 α θ 11 Cµ 5α. Now we come to the main tool in this section. THEOREM 4.. The minimising sequence { η n } for J over U\{0} has the property that η n α cµ 1/ for each n N provided we have α < 5. Proof: We start with the following equality κh, ρ)l η n ) K η n ) = K η n ) K η n ) J µ η κh, ρ)µ n ) L η n ) L η n ) + κh, ρ) {L η n ) L η n )} + κh, ρ)l η n ) = K nl η n ) κh, ρ)l nl η n ) + κh, ρ)sη)l η) J µ η n ) =: RHS, 4.4) where we abbreviate } S η n ) := {1 µ. L η n ) From the calculation 4.4) we obtain R g 11 4 k) η n dk = gk) 3 4 gk) η n, gk) η n 0 = gk) 3 4 RHS, RHS 0 By Young s inequality we obtain c k 3 RHS, RHS 0 + c RHS, RHS 0. k 3 RHS, RHS 0 = k RHS, k 1 RHS 0 k RHS, k RHS 0 + k 1 RHS, k 1 RHS 0 ; k 1 RHS, k 1 RHS 0 = k RHS, RHS 0 k RHS, k RHS 0 + RHS, RHS 0

27 and consequently R g 11 4 k) η n dk c RHS ) Hence we have to estimate the r.h.s. of 4.4). A Taylor expansion for K nl shows that the leading term is where O is to understand in terms of 0) K 4 η 1 ρ)β ) n ) = [ ηn ) x + η n ) z) η n ) x ] x + [ η n ) x + η n ) z) η n ) z ] z = O η n 4 η n 0). It follows K 4 η n ) c η n η n + η n η n 3 η n + η n 3 η n ) and by Proposition 4.1 b) ii) hence K nl η n ) 0 c η n 4 + η n 4 ) η n 3 cµ 6α η n 4 α η n 3 cµ 6α+1 η n 4 α, K nl η n ) 1 cµ 6α+1 η n 4 α. 4.6) In the estimation of L nl we just have to calculate L 3 using the following formulae compare the calculations in [BGS], 6) and 63) and Lemma.6) L 3 η n ) = 1 u0 x) 1 u0 z) + 1 u0 y) y=h + K 1 η n ) η n ; L 3 η n ) = 1 u0 x) + 1 u0 z) 1 u0 y) y=1 h + K 1 η n ) η n. We have u 0 x) y=h 0 u 0 x y=h u 0 x y=h 0. For the first norm we get on account of Proposition 4.1 b) y=h c η n + η n ) cµ 3α η n. u 0 x u 0 x Moreover we obtain y=h = c R 0 R k 1 k η n F 1 [ k 1 k coth hk η n ) dx dz + c R ]) dx dz k 1 k hk coth hk 1) η n 3 ) dx dz

28 from which we deduce on account of t coth t 1 = Ot) u 0 x) y=h 0 cµ 3α η n η n 3 cµ 3α+1 η n. Furthermore we have to calculate u 0 x) y=h = 4 u 0 x u 0 x y=h 4 We observe u 0 x y=h = 0 R ) k c 1 R k η n dx dz + c c η n 3 cµ. 0 0 u 0 x y=h F 1 [ k 1 coth hk η n ]) dx dz Plugging all together we have shown u 0 x) y=h cµ 3α+1 η n α. Using the ame arguments we obtain u 0 z) y=h cµ 3α+1 η n α. R 1 1 u 0 x y=h. ) k 1 k hk coth hk 1) η n dx dz Furthermore we obtain u 0 y) y=h = 4 η n ) x η n ) x 0 c η n η n 3 cµ 5α+1 η n α using again Proposition 4.1 b) ii), as well as 0 u 0 y) y=h 0 = ηn ) x c η 0 n η n 3 cµ 5α+1 η n α. Finally we get K 1 η n ) η n u 1 = 1 y=h x c F F ) F This follows exactly as in [BGS], Lemma A.6. We only show how to bound F 1 3 1, the other norms can be calculated in the same fashion: F 1 3 = η n u 0 y + η n ) x u 0 x + η n ) z u 0 z, hence we have to estimate ) ) ηn + η n u 0 x 1 + u 0 z 1 + u 0 y

29 If we can show u 0 y 1 η n 3 4.7) we receive the final estimation not that the calculations for u 0 x 1 and u 0 z 1 follows similar) K 1 η n ) η n 1 cµ3α+1 η n α. Now we have a look at 4.7): from.33) it follows u 0 y 0 = h 0 c R R [ F 1 cosh k y k 1 sinh k h η n k1 k η n ) dx dz + c Plugging all together we arrive at R ]) dx dz dy h R k 1 coth k h) η n ) dx dz ) k1 k hk coth hk 1) η n dx dz c η n 3. L nl η n ) 1 cµ 3α+1 η n α. Since exactly the same arguments are applicable for estimating L nl η n ) we receive L nl η n ) 1 cµ 3α+1 η n α. 4.8) Replacing η n ) by a subsequence if necessary, we may assume J µ η n ) 1 cµ N 4.9) for a N such that N 11 α 1 compare [BGS], Thm. 4.1, for details). In order to estimate S η n ) we apply the arguments from [BGS], 73) and 74), to obtain in which µ L η n ) 1 + R η n), Rη) = J µη), η 0 κh, ρ)µ K nlη) κh, ρ)µ + K nl η), η 0 4κh, ρ)µ µ L nl η), η 0 4Lη)L η) + µl nlη) Lη)L η) + µl nlη) Lη) + µl nlη) L nl η), η 0. 4Lη) L η) Using the arguments already mentioned, as well as L η n ) > µ and L η n ) > cµ, we can conclude J µη), η 0 κh, ρ)µ 1 K nl η) cµn ; κh, ρ)µ, K nl η), η 0 4κh, ρ)µ cµ5α+1 η n α 5

30 µ L nl η), η 0 4Lη)L η), µl nl η) Lη)L η), µl nl η) Lη) µl nl η) L nl η), η 0 4Lη) L η) cµ3α η n α. Combining this inequalities shows Finally from 4.4)-4.10) we can follow cµ 3α ηn α ; S η n )L η n ) 0 cµ 3α+1 η n α. 4.10) η n α cµ 1 5 α η n α + µ N 11 α + µ), 4.11) and consequently the claim, choosing µ small enough to satisfy µ 1 5 α 1 c possible on account of α <. 5 which is The following proposition shows how to estimate the terms in 4.3). Proposition 4.3. Every function η H 3 R ) with η α cµ 1/, η 3 cµ 1/ and L η) > cµ satisfies the inequalities K 4 η), K 6 η) = Oµ 5α+ ) and If α > 1 3 N 1 η) = Oµ 3 α 1 ); N0 η), N 1 η), N η) = Oµ 3α ). we obtain for an δ > 0 K 4 η), K 6 η) = Oµ 3+δ ) and N 1 η) = Oµ δ ); N 0 η), N 1 η), N η) = Oµ 1+δ ). Proof. Suppose that η α cµ 1/. It follows from the formulae K 4 η) = β ηx + η 8 z) dx dz, K 6 η) = β ηx + η R 16 z) 3 dx dz R and Proposition 4.1 b) ii) that K 4 η) c η η 3 cµ 5α η α η 3 cµ 5α+, K 6 η) c η 4 η 3 cµ 10α η 4 α η 3 cµ 10α+3, from which the first inequalities are a direct consequence. The calculation L j η) c η j 1, η 3 cµ 3 αj ) η j α η 3 cµ 3 αj )+ 1 j )+1 6

31 shows that in particular L 3 η) = Oµ 3 α+ 3 ), L4 η), L 5 η), L 6 η) = Oµ 3α+ ), and combining these estimates with the explicit formulae we find that N η) = 1 L η), N 1 η) = L 3η) L η), N 0 η) = L 4η) L η) + L 3η) L η) 3, N 1 η) = L 5η) L η) + L 3η)L 4 η) L η) 3 L 3η) 3 L η) 4, N η) = L 6η) L η) + L 4η L η) 3 3L 3η) L 4 η) L η) 4 + L 3η) 4 L η) 5, N 0 η), N 1 η), N η) = Oµ 3α ). Now, we find that Proposition 4.4. The function is decreasing and strictly negative, if α > 1 3. a a 5/ M a µa η n ), a [1, ] By Proposition 4.3 above we can quote the claim from [BGS], Proposition 4.9. The final result in this section follows directly from Proposition 4.4 compare [BGS], Lemma 4.10) LEMMA 4.5. The strict sub-homogeneity property holds for each a > 1. 5 Conclusion c aµ < ac µ In this section we present the results of the paper, the existence and stability theorem. The first one is a consequece of the theory from section 3 and 4, whereas the stability theorem follows from this theorem details are given in [BGS], section 5). The following theorem, which is proved using the results of Sections 3 and 4, is our final result concerning the set of minimisers of J µ over U \{0}. 7

32 THEOREM 5.1. i) The set C µ of minimisers of J µ over U \{0} is non-empty. ii) Suppose that {η n } is a minimising sequence for J µ on U \{0} which satisfies sup η n 3 < M. 5.1) n N There exists a sequence {x n, z n )} R with the property that a subsequence of {η n x n +, z n + )} converges in H r R ), 0 r < 3 to a function η C µ. The next step is to relate the above result to our original problem finding minimisers of Eη, Φ) subject to the constraint Iη, Φ) = κρ, h)µ, where E and I are defined in equations 1.1) and 1.14). THEOREM 5.. i) The set D µ of minimisers of E on the set is non-empty. S µ = {η, Φ) U H 1/ R ) : Iη, Φ) = κρ, h)µ} ii) Suppose that {η n, Φ n )} S µ is a minimising sequence for E with the property that sup η n 3 < M. k N There exists a sequence {x n, z n )} R with the property that a subsequence of {η n x n +, z n + ), Φ n x n +, z n + )} converges in H r R ) H 1/ R ), 0 r < 3 to a function in D µ. It is also possible to obtain a bound on the speed of the waves described by functions in D µ. LEMMA 5.3. The fully localised solitary wave corresponding to η, Φ) D µ is subcritical, that is its dimensionless speed is less than unity. Our stability result Theorem 5.4 below) is obtained from Theorem 5. under the following assumption concerning the well-posedness of the hydrodynamic problem with small initial data. Well-posedness assumption) There exists a subset S of U H 1/ R ) with the following properties. i) The closure of S\D µ in L R ) has a non-empty intersection with D µ. 8

33 ii) For each η 0, Φ 0 ) S there exists T > 0 and a continuous function t ηt), Φt)) U H 1/ R ), t [0, T ] such that η0), Φ0)) = η 0, Φ 0 ), Eηt), Φt)) = Eη 0, Φ 0 ), Iηt), Φt)) = Iη 0, Φ 0 ), t [0, T ] and sup ηt) 3 < M. t [0,T ] THEOREM 5.4. Choose r [0, 3). For each ε > 0 there exists δ > 0 such that η 0, Φ 0 ) S, distη 0, Φ 0 ), D µ ) < δ distηt), Φt)), D µ ) < ε, for t [0, T ], where dist denotes the distance in H r R ) H 1/ R ). References [B] [BGS] [BT] [CG] [GW] [GS] [Li1] [Li] [NR] Buffoni, B., Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Rat. Mech. Anal. 173, ) Buffoni, B, Groves, M. D., Sun, M. S.: Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capilary water waves Buffoni, B, Toland, J.F.: Analytic theory of global bifurcation. Princeton, N.J.: Princeton University Press 003). Craig, W., Groves, M.D.: Normal forms for wave motion in fluid interfaces. Wave motion 31, ). Groves, M. D., Wahlen, E., On the existence and conditional energetic stability of solitary water waves with weak surface tension, Groves, M. D. & Sun, S.-M. 008 Fully localised solitary-wave solutions of the threedimensional gravity-capillary water-wave problem. Arch. Rat. Mech. Anal. 188, Lions, P. L The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, Lions, P. L The concentration-compactness principle in the calculus of variations. The locally compact case, part. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, Nicholls, D.P., Reitich, F.: A new approach to analicity of Dirichlet-Neumann operators. Proc. Roy. Soc. Edin. A 131, ). 9