N-Modal Steady Water Waves with Vorticity

J. Math. Fluid Mech. 20 (2018), 853–867 c 2017 The Author(s).  This article is an open access publication 1422-6928/18/020853-15 https://doi.org/10.1007/s00021-017-0346-1 Journal of Mathematical Fluid Mechanics N-Modal Steady Water Waves with Vorticity Vladimir Kozlov and Evgeniy Lokharu Communicated by A. Constantin Abstract. Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation. 1. Introduction The existence theory for rotational steady water waves of finite depth goes back to the paper [6] of DubreilJacotin published in 1934. However only after the paper [5] of Constantin and Strauss appeared in 2004 the theory of rotational steady waves has attracted a lot of attention from the mathematical community. The latter paper was devoted to large amplitude Stokes waves with an arbitrary vorticity distribution. Stokes waves are periodic travelling waves with exactly one crest and one trough in every minimal period. A natural question can be raised here: are there periodic waves with more complicated geometry? To simplify the discussion we will restrict this question to the case of small-amplitude periodic waves without surface tension. In the class of periodic uniderectional waves with vorticity only Stokes waves exist (see [11,17] and [16]). Thus, to answer the question affirmatively one shall consider a wider class of waves allowing internal stagnation or critical layers. The first positive result in this direction was obtained (see [9]) in 2011 by Ehrnström, Escher and Wahlén. They prove existence of bimodal travelling waves of small amplitude (see also [1]). A bimodal wave has several crests per period and the first approximation of the surface profile is given by a combination of two simple modes: cosine functions with different wavelengths. Later, in 2015 Ehrnström and Wahlén proved (see [10]) the existence of trimodal waves for which the first approximation is given by a combination of three basic modes. Both papers use a similar technique based on a bifurcation argument. The crucial role there is the choice of the bifurcation parameters. The main difficulty for the higher order bifurcation is the fact that the number of natural parameters of the problem is limited. Normally, one may consider only the Bernoulli constant or the wavelength as a bifurcation parameters. In the case of a linear vorticity, the derivative of the vorticity function (which is a constant in this case) is another possible parameter. The main novelty of our approach is that an arbitrary number of the bifurcation parameters are used. These parameters are the roots of the dispersion equation while the corresponding vorticity is a perturbation of a linear function. Our argument is different to the one used in [10] to construct trimodal waves. In [10] the authors use a Lyapunov–Schmidt reduction directly to the initial infinite-dimensional problem. This complicates calculation of the determinant of the matrix of the reduced system which is essential for the proof of the existence. In contrast, we first reduce the problem to a finite dimensional system for which the linear part is given by an ordinary differential operators and then use Lyapunov–Schmidt reduction. After that the corresponding matrix is diagonal and then we can quite straightforward prove the existence of symmetric and periodic waves with an arbitrary number of basic modes. 854 V. Kozlov and E. Lokharu JMFM The paper is organized as follows. In the beginning of Sect. 2 we formulate the problem in a suitable way. In Sect. 2.1 and 2.2 we discuss stream solutions and the linear approximation for the initial problem. The Sect. 3 is an essential part of the proof and concerns to the inverse spectral problem related to the dispersion equation. Here we prove Theorem 3.1 allowing to use roots of the dispersion equation as bifurcation parameters. Next, in Sect. 4, we first formulate our main result. The rest of the paper is dedicated to the proof of the main theorem. We start by writing the problem in an operator form and then in Sect. 4.4 we reduce it to a finite-dimensional system of equations. The rest of the proof is contained in Sect. 4.5, where we perform Lyapunov–Schmidt reduction to the reduced system. In contrast to the classical bifurcation theory, we find solutions only for the values of small parameters t = (t1 , ..., tN ) such that |t|2 < ǫ|tj | for some small ǫ. 2. Statement of the Problem Let an open channel of a uniform rectangular cross-section be bounded below by a horizontal rigid bottom and let water occupying the channel be bounded above by a free surface not touching the bottom. In an appropriate Cartesian coordinates (x, y), the bottom coincides with the x-axis and gravity acts in the negative y-direction. The steady water motion is supposed to be two-dimensional and rotational; the surface tension is neglected on the free surface of the water, where the pressure is constant. These assumptions and the fact that the water is incompressible allow us to seek the velocity field in the form (ψy , −ψx ), where ψ(x, y) is referred to as the stream function. The vorticity distribution ω is supposed to be a prescribed smooth function depending only on the values of ψ. We choose the frame of reference so that the velocity field is time-independent as well as the unknown free-surface profile. The latter is assumed to be the graph of a function y = η(x), x ∈ R, where η is a positive continuous function, and so the longitudinal section of the water domain is D = {x ∈ R, 0 < y < η(x)}. We use the non-dimensional variables proposed by Keady and Norbury [13]. Namely, lengths and velocities are scaled to (Q2 /g)1/3 and (Qg)1/3 respectively. Here Q and g are the dimensional quantities for the rate of flow and the gravity acceleration respectively, whereas (Q2 /g)1/3 is the depth of the critical uniform stream in the irrotational case. This scaling leads to to an equivalent problem, provided the mass flux is not zero. The following free-boundary problem for ψ and η has long been known (cf. [13]): ψxx + ψyy + ω(ψ) = 0, (x, y) ∈ D; ψ(x, 0) = 0, x ∈ R; ψ(x, η(x)) = 1, x ∈ R; |∇ψ(x, η(x))|2 + 2η(x) = 3r, x ∈ R. (2.1) (2.2) (2.3) (2.4) In the condition (2.4) (Bernoulli’s equation), r is a constant considered as a problem’s parameter and referred to as Bernoulli’s constant/the total head. The problem (2.1)–(2.4) describes two-dimensional steady water waves with a nonzero mass flux. In what follows, we will assume that ω ∈ C ∞ (IR). For the further analysis of the problem it is convenient to rectify the domain D by scaling the vertical variable to z=y d , η(x) while the horizontal coordinate remains unchanged. The parameter d > 0, which by meaning is the depth of a uniform stream to be determined later, can be chosen arbitrarily. In other words, for a given d > 0 we will find a vorticity function and a stream solution for which we prove the existence of multi-modal waves of small amplitude bifurcating from the uniform stream. Vol. 20 (2018) N-Modal Steady Water Waves 855 Thus, the domain D transforms into the strip S = IR × (0, d). Next, we introduce a new unknown function Φ̂(x, z) on S̄ by  z  Φ̂(x, z) = ψ x, η(x) . d A direct calculation shows that the problem (2.1)–(2.3) reads in new variables as     2  d zηx zηx zηx Φ̂x − Φ̂x − Φ̂zz + ω(Φ̂) = 0; Φ̂z − Φ̂z + (2.5) η η η η x z Φ̂(x, 0) = 0, Φ̂(x, d) = 1, x ∈ IR; while the Bernoulli equation (2.4) becomes Φ̂2z = η2 d2  3r − 2η 1 + ηx2 (2.6)  . (2.7) In what follows by a solution of (2.5)–(2.7) we mean a pair (Φ̂, η) ∈ C 2,α (S̄) × C 1,α (IR) satisfying these equations. 2.1. Stream Solutions A pair (u(y), d), where u : [0, d] → IR and d > 0 is called a stream solution corresponding to the vorticity function ω and the depth d, if u′′ + ω(u) = 0, u(0) = 0, u(d) = 1. (2.8) The corresponding Bernoulli constant r is calculated by 3r = [u′ (d)]2 + 2d. Note that in our definition of a stream solution the Bernoulli constant is not fixed. In this paper we will be interested in the case when the vorticity ω is a small perturbation of a linear vorticity ω0 (p) = bp, where b is a positive constant: ω(p) = ωδ (p) = bp + δ1 ω1 (p) + · · · + δN ωN (p). (2.9) Here the coefficients δj are small and compactly supported functions ωj ∈ C0∞ (0, 1) will be chosen later. The choice of the sign of the constant b is crucial because negative values of b give rise only to unidirectional uniform streams, for which the dispersion equation has at most one root and all periodic small-amplitude waves are Stokes waves. We will require the constants b and d to satisfy √ πj , j = 1, 2, .... (2.10) b = 2d The assumption (2.10) guarantees that the problem (2.8) with the vorticity ω of the form (2.9) possesses a unique solution u with u′ (d) = 0, provided all δj are small enough. To prove that (2.8) has a unique solution with ω given by (2.9), we write u = u0 + v and consider the problem v ′′ + bv + N  δj ωj (u0 + v) = 0, v(0) = v(d) = 0. j=1 We can rewrite this problem in an operator form as follows: Lv + Q(v, δ) = 0, where the operators L, Q : {u ∈ C 2,α ([0, d]) : u(0) = u(d) = 0} → C α ([0, d]) 856 V. Kozlov and E. Lokharu JMFM N are defined by Lv = v ′′ + bv and Q(v, δ) = j=1 δj ωj (u0 + v). The assumption (2.10) guarantees that operator L has a trivial kernel, which ensures it’s invertibility. This allows us to apply implicit function theorem in order to find a family of stream solutions u(z; δ) in a neighbourhood of δ = 0 which depends analytically on δ (see [14] for a good overview). Note that (2.10) also implies that uy (d; δ) = 0 for all small δ, which ensures that there is no stagnation on the surface. 2.2. Linear Approximation of the Water-Wave Problem Let us consider a linear approximation of the problem. For this purpose we formally linearize equations (2.5)–(2.7) near some stream solution (u, d). Thus, we put Φ̂ = u + ǫΦ + O(ǫ2 ), η = d + ǫζ + O(ǫ2 ). Using this ansatz in (2.5)–(2.7), we find   2ζuzz zζx uz + ω ′ (u)Φ = O(ǫ), + Φzz − Φx − d d x Φ(x, 0) = Φ(x, d) = 0,  ′  u (d) 1 Φz |z=d − − ′ ζ = O(ǫ). d u (d) We can simplify equations by letting zζuz . d The latter transformation was used in [9] and [10]. The formula above implies ζ = −Ψ|z=d /u′ (d) and then the linear part of the above problem transforms into Ψ=Φ− Ψxx + Ψzz + ω ′ (u)Ψ = 0 Ψ|z=0 = 0 (2.11) Ψz |z=d − κΨ|x=d = 0, where κ = 1/[u′ (d)]2 − ω(1)/u′ (d). Separation of variables in the system (2.11) leads to the following Sturm–Liouville problem: − φzz − ω ′ (u)φ = μφ on (0, d), φ(0) = 0, φz (d) = κφ(d). (2.12) The spectrum of (2.12) depends only on the triple (ω, d, u) and consists of a countable set of simple real eigenvalues {μj }∞ j=1 ordered so that μj < μl for all j < l. Furthermore, only a finite number of eigenvalues may be negative. The normalized eigenfunction corresponding to an eigenvalue μj will be denoted by φj . 2 Thus, the set of all eigenfunctions {φj }∞ j=1 forms an orthonormal basis in L (0, d). Solving the linear problem (2.11), we find that the space of bounded and even in the x-variable solutions is finite-dimensional and is spanned by the functions Ψ(x, z) = cos( |μj |x)φj (z), (2.13) where μj ≤ 0. Let us turn to the problem (2.12) for the linear vorticity ω0 (p) = bp. Proposition 2.1. For any N ≥ 1 and d > 0 there exists b > 0 satisfying (2.10) such that the Sturm– Liouville problem (2.12) for the triple (ω0 , d, u0 ) has exactly N negative eigenvalues while all other eigenvalues are positive. In the formulation above u0 stands for the unique solution of (2.8) for the pair (ω0 , d). An analog of Proposition 2.1 was proved in [1]. Vol. 20 (2018) N-Modal Steady Water Waves 857 Proof of Proposition 2.1. For a given d > 0, we put  2 πN b= −ǫ d for some ǫ = 0, which can be positive or negative. Note that (2.10) is satisfied automatically for small ǫ. Now we can calculate √ √ ǫd 1 sin2 bd √ sin bd √ = √ + O(ǫ2 ), ǫ → 0. − b κ= b cos2 bd 2πN cos bd For any j > 1 the eigenfunction corresponding to μj is given by C sin relation at z = d in (2.12) gives b + μj cos b + μj d = κ sin b + μj z and then the boundary b + μj d, which implies that eigenvalues μj are uniformly separated from zero for all j > 1, provided ǫ is small. More precisely, there is a positive constant β > 0 such that |μj | > β for all j > 1 and all |ǫ| < β. It is known that eigenvalues μj alternates with eigenvalues λD j of the Dirichlet problem: −φzz − bφ = λD φ on (0, d), φ(0) = φ(d) = 0. λD 1 D More precisely, we have μ1 < and for any two neighbour eigenvalues λD i < λi+1 there is exactly one D D eigenvalue μj ∈ (λD i , λi+1 ). Finally, the choice of the constant b implies λN = ǫ. Thus, there are exactly D N − 1 to the left of λN −1 and μN < ǫ. Furthermore, we necessary have |μN | < β, which implies that μN < −β and μN +1 > β, provided |ǫ| < β. Thus all eigenvalues μ1 , ..., μN are negative, while all others are positive. This finishes the proof.  Note that it follows from the proof of the proposition that the constant κ in (2.12) has the same sign as ǫ and then can be both positive or negative. The sign of κ plays no role in our proof and one can show that multi-modal waves appear for both positive and negative values of κ. 3. Inverse Spectral Problem For a given N > 1 and d > 0, we consider the constant b provided by Proposition 2.1 and let (u0 , d) be the stream solution (2.8), which corresponds to the vorticity function ω0 (p) = bp and the depth d. According to the Proposition 2.1, the Sturm–Liouville problem (2.12) for the triple (ω0 , d, u0 ) has exactly N negative eigenvalues λ1 < · · · < λN < 0 and all other eigenvalues are positive. Now, let the functions ωk , k = 1, . . . , N , in (2.9) be fixed. Then for small δ = (δ1 , . . . , δN ) there exists a unique solution uδ to the problem (2.8) with ω = ωδ given by (2.9) and we can consider eigenvalue problem (2.12) for the triple (ωδ , d, uδ ). Let μ1 < · · · < μN be all negative eigenvalues of that Sturm–Liouville problem, while φj , j > N are eigenfunctions corresponding to negative eigenvalues respectively. Note that both eigenvalues μj ∈ IR and the corresponding eigenfunctions φj ∈ L2 (0, d) depend analytically on δ (for details see Chapter VII in [12] and examples 1.15 and 2.12 therein). Let us define the map T : δ = (δ1 , . . . , δN ) → μ = (μ1 , . . . , μN ). (3.1) which is analytic in a neighborhood of δ = 0. The next theorem provides the invertibility of T and by this we solve the spectral inverse problem locally: for a given set of eigenvalues located near λ1 , ..., λN , we find a corresponding potential through the vorticity distribution ωδ and the stream solution (uδ , d). Theorem 3.1. There exist functions ωk ∈ C0∞ (0, 1), k = 1, . . . , N , such that the Jacobian matrix of the map (3.1) is invertible at δ = 0. 858 V. Kozlov and E. Lokharu JMFM Proof. Let us calculate the Jacobian matrix of the map T . Since uδ depends analytically on δ (as an operator function with values in C 2,α ([0, d])) as it was noted in Subsection 2.1, we have uδ (z) = u0 (z) + δ1 u1 (z) + · · · + δN uN (z) + O(|δ|2 ), |δ| → 0, where u1 (0) = u1 (d) = · · · = uN (0) = uN (d) = 0. Using this ansatz, we find that every uj is subject to u′′j + buj = −ωj (u0 ), uj (0) = uj (d) = 0 for all 1 ≤ j ≤ N . Solving these equations, we find √ z √ 1 sin bz −√ ωj (u0 (p)) sin b(z − p)dp, uj (z) = cj √ b b 0 where d √ 1 √ cj = ωj (u0 (p)) sin b(d − p)dp. sin( bd) 0 (3.2) Let us turn to the dispersion equation for the triple (ωδ , d, uδ ) which is given by the following eigenvalue problem −φ′′ − ω ′ (uδ )φ = μφ on (0, d);   1 ωδ (1) φ′ (d) = − φ(d), φ(0) = 0. kδ2 kδ Here k = u′δ (d). The corresponding eigenfunctions φl and eigenvalues μl , l = 1, ..., N depend smoothly on δ, so that μl = λl + δ1 μl1 + · · · δN μlN + O(|δ|2 ), |δ| → 0, l = 1, ..., N. In order to find the coefficients μlj , we write the eigenvalue problem above in operator form: Aφ = μφ, where the self-adjoint operator A is defined by   1 ωδ (1) − ψ(d)φ(d) + Aφ, ψ = − kδ2 kδ d d 0 φ′ ψ ′ dz − 0 ωδ′ (u)φψdz, where φ, ψ ∈ W01,2 (0, d) and the last space consists of functions from W 1,2 (0, d) vanishing at 0. Here ·, · stands for the standard inner product in L2 (0, d). Thus, if A = A0 + δ1 A1 + · · · + δN AN + O(|δ̄|2 ), |δ̄| → 0, where operators Aj are defined by u′j (d) Aj φ, ψ = k02  for φ, ψ ∈ W01,2 (0, d). Using (3.2), we find u′j (d) 1 √ =− sin( bd) d 0  2 − b φ(d)ψ(d) − k0 √ ωj (u0 ) sin bzdz = − d ωj′ (u0 )φψdz 0 d 0 √ cos2 bz ωj′ (u0 ) 2 √ dz sin bd (3.3) √ √ after integration by parts and observation that u0 = sin bz/ sin bd. Then, after some algebra one obtains μlj = Aj φl , φl , where φl stands for the normalized eigenfunction corresponding to the eigenvalue λl of the Sturm–Liouville problem for δ = 0. Therefore, using (3.3), we conclude d √ 2 φ2 (d)  b− . (3.4) ωj′ (u0 )(Al cos2 bz − φ2l )dz, Al = 2 l 2 √ μlj = k0 k0 sin bd 0 Vol. 20 (2018) N-Modal Steady Water Waves 859 √ √ Let us transform the right-hand side in (3.4). Since u0 = sin bz/ sin bd, we can reduce integration over [0, d] to a smaller interval of where the function u0 is monotone. We put 2π Λ= √ b which is the period of the function u0 . √ Consider first the case sin bd > 0. Let d∗ be the smallest positive root of the equation u0 (d∗ ) = 1. Then the function u0 is monotone on the interval [0, d∗ ]. Let z ∈ (0, d∗ ). To describe all roots of u0 (ẑ) = u0 (z) on the interval (0, d), which are different from z, we introduce the integer M as the largest integer satisfying  3 < d. (3.5) Λ M+ 4 Then the roots of u0 (ẑ) = u0 (z) are given by   Λ 3 Λ± z+ , k = 1, . . . , M. (3.6) zk± = k + 4 4 Now we can write (3.4) as  M d∗  √ ′ 2 2 + 2 − 2 μlj = ωj (u0 ) (2M + 1)Al cos bz − φl (z) − (φl (zk ) + φl (zk )) dz. (3.7) 0 k=1 Note that the function φl solves the problem −φ′′l − bφl = λl φl , φl (0) = 0, φ′l (d) = κ0 φl (d). √ √ Hence, depending on the sign of b + λl , it is either Cl sin b + λl z or C2 sinh b + λl z. Let us consider the case when b + λ1 > 0 and hence φl (z) = Cl sin b + λl z for all l = 1, . . . , N . Then φ2l (z) + = where Cl2  M  (φ2l (zk+ ) + φ2l (zk− )) k=1 (M + 1/2) − cos(2 Bl = M  k=1 Therefore b + λl z) − Bl cos(2   cos 2 b + λl (k + 3/4)Λ . d∗ μlj = 0 where Since the functions  b + λl (z + Λ/4)) , ωj′ (u0 )fl (z)dz,  √  fl (z) = (M + 1/2) Al − Cl2 + Al cos(2 bz)   +Cl2 cos(2 b + λl z) + Bl cos(2 b + λl (z + Λ/4) . √ 1, cos(2 bz), cos(2 b + λl z), cos(2 (3.8) (3.9) b + λl (z + Λ/4)), l = 1, . . . , N, are linear independent, the set of functions √ cos( bz), fl (z), l = 1, . . . , N, (3.10) 860 V. Kozlov and E. Lokharu JMFM √ √ is also linear independent. Indeed, the only case when they can be dependent is b = 2 b + λl for certain l, but then either Al or Al − Cl2 is non-zero in the representation (3.9) for fl which is sufficient for linear independence of the system (3.10). In the case b + λ1 ≤ 0 we have b + λ2 > 0. Otherwise we will have two eigenfunctions corresponding to different eigenvalues √ √ and which are no orthogonal to each other. Now we must replace the function sin b + λ1 by sinh b + λ1 and, repeating then the above argument, we obtain the linear independence of the system (3.10). Using linear independence of the functions (3.10), we can find functions αj = αj (p) in C0∞ (0, 1) such that d∗ d∗ √ αj (u0 (z)) cos( bz)dz = 0, αj (u0 (z))fl (z)dz = δlj and (3.11) 0 0 where δlj is the Kronecker delta. If we put Z p ωj (p) = αj (s)ds = 0 0 αj (u0 (z))u′0 (z)dz, p = u0 (Z), then we see that ωj ∈ C0∞ (0, 1) due to the second equality in (3.11) and ωj′ (p) = αj (p). Therefore the √ matrix (3.8) is invertible √ by the first equality in (3.11). This completes the proof in the case sin bd > 0. Assume now that sin bd < 0. Then the function u0 is negative on (0, Λ/2) and it is monotonically increasing from 0 to 1 on the interval [Λ/2, d∗ + Λ/2]. Reasoning as above we obtain the representation (3.8) but on the interval (Λ/2, d∗ + Λ/2) with 3/4 replaced by 1/4 in formulas like (3.5) and (3.6). The remaining part of the proof in this case is the same as above.  A similar inverse problem was considered in [8], where the authors used inverse spectrum Sturm– Liouville theory (see [7] for details). Unfortunately we could not implement a similar argument based on the inverse Sturm–Liouville theory and so we used another approach based on a perturbation argument. 4. Existence of N -Modal Waves For a given N > 1 and d > 0 let b be the constant from Proposition 2.1 so that the Sturm–Liouville problem (2.12) has N negative eigenvalues λ1 , . . . λN and all remaining eigenvalues are positive. By Theorem 3.1 we can choose real-valued functions ω1 , . . . , ωN ∈ C0∞ (0, 1) such that the map (3.1) is invertible in a certain neighborhood Γ ⊂ RN of the point (λ1 , . . . , λN ). Furthermore, we assume that for every (μ1 , ..., μN ) ∈ Γ we have μj < 0, j = 1, ..., N. In what follows we will use μ = (μ1 , . . . , μN ) ∈ Γ as a parameter of our water wave problem and denote by ω(p; μ), u(z; μ) the perturbed vorticity ωδ(µ) and stream solution uδ(µ) . We choose also μ∗ = (μ∗1 , . . . , μ∗N ) ∈ Γ so that μ∗j /μ∗l are rational numbers for all 1 ≤ j, l ≤ N . In this case all bounded solutions (2.13) to the linear problem (2.11) for the triple (ω(p; μ∗ ), d, u(z; μ∗ )) have a common period which we denote by Λ∗ . Since the set Γ is open, we can always find some μ∗ with this property. The aim of the present paper is to give an affirmative answer to the question raised in [9]: does higher-order bifurcation occur so that N -modal waves exist for all N ≥ 1? Theorem 4.1. For any integer N ≥ 1 and d > 0 there exist constants β, ǫ > 0, a linear vorticity distribution ω0 (p) = bp with b > 0 and a smooth function μ : [−β, β]N → Γ with the following property: for any t := (t1 , ..., tN ) ∈ IRN such that |t| < β and |t|2 /|tj | ≤ ǫ the nonlinear water wave problem (2.5)–(2.7) with the vorticity ω(p; μ(t)) possesses a unique Λ∗ -periodic and even solution (Ψ, η) corresponding to the Bernoulli constant r(t) = [uz (d; μ(t))2 + 2d]/3 such that η(x) = d + t1 cos( |μ∗1 |x) + · · · + tN cos( The constants β and δ depend only on N, d, μ∗ and Λ∗ . |μ∗N |x) + O(|t|2 ). Vol. 20 (2018) N-Modal Steady Water Waves 861 The waves constructed in Theorem 4.1 are symmetric with respect to the vertical line x = 0 and this is essential for the proof of the theorem. The only existence result for non-symmetric two-dimensional waves in the absence of surface tension is [15], while there are several numerical results: [20–22]. There are also some results on two-dimensional non-symmetric gravity-capillary steady water waves (see [4]). 4.1. Functional-Analytic Setup In order to write equations (2.5)–(2.7) in an operator form, we define nonlinear operators     zηx zηx zηx Φ̂x − Φ̂z − Φ̂z F̂1 (Φ̂, η; μ) = Φ̂x − η η η x z  2 d Φ̂zz + ω(Φ̂; μ), + η   η 2 3r(μ) − 2η F̂2 (Φ̂, η; μ) = Φ̂2z |z=d − 2 , d 1 + ηx2 where r(μ) = [u′ (d; μ)]2 − 2d. The definitions above imply that F̂1 (u(·; μ), d; μ) = F̂2 (u(·; μ), d; μ) = 0 for all μ ∈ Γ. Next, we put zuz (z; μ)(η(x) − d) . d The purposes of this change of variables are twofold. First, it gives a formal linearization near the stream solution u(z; μ). Second, it allows to eliminate the profile η from the equations. Indeed, the definition above implies that Φ| (4.1) η(x) = d − ′z=d . u (d) Thus, we naturally define   Φ| zuz (η − d) Fj (Φ; μ) = F̂j Φ + u + , d − z=d ; μ , j = 1, 2. d uz (d) Φ(x, z) = Φ̂(x, z) − u(z; μ) − Therefore, the problem (2.5)–(2.7) with the vorticity ω(·; μ) and Bernoulli constant r(μ) reads as F(Φ; μ) = 0, (4.2) where F = (F1 , F2 ) : X × Γ → Y := Y1 × Y2 and the spaces are defined by 2,α X = {Φ ∈ Cper (S̄) : Φ(x, 0) = 0, Φ(x, z) = Φ(−x, z) for all (x, y) ∈ S} and 0,α 1,α Y1 = Cper (S̄), Y2 = Cper (IR), Y = Y1 × Y2 . Here and elsewhere the subscript per denotes Λ∗ -periodicity and evenness in the horizontal x-variable. As the norms in these spaces we will use · C 0,α ([−Λ∗ /2,Λ∗ /2]×[0,d]) and · C 1,α ([−Λ∗ /2,Λ∗ /2]) respectively. Proposition 4.2. The Fréchet derivative DΦ F at Φ = 0 is given by [DΦ F1 (0, μ)](Φ) = Φxx + Φzz + ω ′ (u)Φ [DΦ F2 (0, μ)](Φ) = Φz |z=d − κΦ|z=d , (4.3) (4.4) 862 V. Kozlov and E. Lokharu JMFM where 1 ω(1; μ) . − [uz (d; μ)]2 uz (d, μ) κ= Proof. The statement follows from a direct calculation.  4.2. Spectral Decomposition According to our notations μ = (μ1 , ..., μN ) stands for the first N eigenvalues of the Sturm–Liouville problem (2.12) for the triple (ω(·, μ), d, u(·, μ)). Note that by construction these eigenvalues are negative, while all others are positive. Let φ1 (z; μ), ..., φN (z; μ) be the corresponding eigenfunctions. We define projectors PΦ = N  Φj φj , j=1 P = id − P. Here Φj = Φ, φj and ·, · stands for the standard scalar product in L2 (0, d). The projectors P and P are orthogonal projectors in L2 (0, d) and are well defined operators on X. Note that the complete set of eigenfunctions {φj }j∈N is a basis in the spaces H0n (0, d) = {φ ∈ H n (0, d) : φ(0) = 0} for n = 0 and n = 1. In general the eigenfunction are orthogonal with respect to the following bilinear form d B(φ, ψ) = 0 [φ′ ψ ′ − ω ′ (u)φψ]dz − κφ(d)ψ(d), which is well defined on H01 (0, d) × H01 (0, d). Integrating by parts, we find that B(φ, φj ) = μj φ, φj , j ∈ N, φ ∈ C02,α ([0, d]). In what follows we will often use the following identity d 0 (−φzz − ω ′ (u(z)))φ)φj dz = −[φz (d) − κφ(d)]φj (d) + B(φ, φj ) (4.5) = −[φz (d) − κφ(d)]φj (d) + μj φ, φj , which is valid for any function φ ∈ C02,α ([0, d]) and j ≥ 1. Let us write Eq. (4.2) in the following form, where linear and nonlinear parts of the operator are separated: Φxx + Φzz + ω ′ (u)Φ = N1 (Φ; μ) (4.6) Φz |z=d − κΦ|z=d = N2 (Φ; μ), (4.7) where N1 and N2 are nonlinear parts of the operators F1 and F2 respectively. Thus, multiplying (4.6) by φj , j = 1, ..., N and using (4.5), we obtain equations for projections: and Φ′′j − μj Φj = N1 , φj − N2 φj (d),  xx + Φ  zz + ω ′ (u)Φ  = P(N1 ) + N2 Φ  z (x, d) − κΦ(x,  d) = N2 Φ j = 1, ..., N ; N  φj (d)φj (z) (4.8) (4.9) j=1 (4.10) Vol. 20 (2018) N-Modal Steady Water Waves 863  = PΦ. It is clear that if some functions Φj and Φ  ∈ ran(id − P ) solve Eqs. where Φj = Φ, φj and Φ (4.8)–(4.10), then the function Φ(x, z) = N  j=1  z) Φj (x)φj (z) + Φ(x, solves (4.2).  can be resolved from Eqs. (4.9)–(4.10) as an operator of We will show below that the function Φ (Φ1 , ..., ΦN ) and μ in a neighbourhood of the origin. We note that an asymptotic analysis of solutions to ordinary differential equations with operator coefficients based on spectral decomposition was developed by Kozlov and Maz’ya in [18] for linear problems and in [19] for non-linear ones. 4.3. The Reduction to a Finite-Dimensional System First, we consider the linear part of (4.9)–(4.10) which is given by  xx + Φ  zz + ω ′ (u)Φ  =f Φ  z (x, d) − κΦ(x,  d) = g Φ  0) = 0. Φ(x, (4.11) (4.12) (4.13)  be the subspace of X which consists of functions Φ satisfying Let X P[Φ(x, ·)] = Φ(x, ·) for all x ∈ IR or equivalently Φ(x, ·), φj (·) = 0, j = 1, . . . , N. (4.14) f (x, ·), φj (·) = g(x)φj (d), j = 1, . . . , N, (4.15) Furthermore, we define the range space Y to be a subspace of Y which consists of all (f, g) ∈ Y satisfying for all x ∈ IR. These compatibility conditions for the problem (4.11)–(4.13) are obtained by multiplication  → Y of (4.11) by φj and application of (4.5) and (4.14). Let us define a linear operator L := (L1 , L2 ) : X by =Φ  xx + Φ  zz + ω ′ (u)Φ  L1 Φ =Φ  z (x, d) − κΦ(x,  d). L2 Φ Now we are ready to prove  → Y is a linear isomorphism and the norm of it’s inverse is estimated Lemma 4.3. The mapping L : X by a constant depending only on b, d, μN +1 and Λ∗ .  ⊂ Y and its kernel is contained in the set of Proof. First, we note that operator L is bounded and L(X) linear combinations of the functions (2.13). To construct a solution we consider first the problem (4.11)– (4.13) in the spaces X and Y . Since this problem is elliptic and its kernel belongs to the set spanned by the functions (2.13) for every right-hand side (f, g) ∈ Y subject the compatibility conditions (4.15) this problem has a unique solution Φ ∈ X orthogonal to the kernel of the problem. Let us show that this  Indeed, multiplying (4.11) by φj , j = 1, . . . , N , and integrating over (0, d), we get solution belongs to X. d 0 ′′ (Φ (x, z)φj (z) − μj Φ(x, z)φj (z))dz = 0, 864 V. Kozlov and E. Lokharu JMFM where we have applied (4.5). Hence d Φ(x, z)φj (z)dz = 0 N  cj cos( |μj | x), j = 1, . . . , N. j=1  This prove Since the solution is periodic and is orthogonal to the kernel cj = 0, which implies Φ ∈ X. the existence and uniqueness. It remains to estimate the norm of L−1 . Applying the Schauder estimate (see Theorem 7.3 [3]) to the system (4.11)–(4.13) we find  Φ ≤ C[ f C 2,α (RΛ∗ ) Y1 + g  + Φ Y2 L2 (RΛ∗ ) ], (4.16) where RΛ∗ = [−Λ∗ /2, Λ∗ /2] × [0, d] and the constant C depends on b, d and Λ∗ . Let us show that  Φ L2 (RΛ∗ ) ≤ C ∗[ f Y1 + g Y2 ] (4.17)  and integrate over z ∈ [0, d], with C ∗ depending on b, d, μN +1 and Λ∗ . We multiply equation (4.11) by Φ which gives d 0  xx Φdz  + Φ d 0  zz + ω ′ (u)Φ]  Φdz  = [Φ d 0  f Φdz. (4.18) To calculate the second integral on the left-hand side, we use (4.5) to show that d − 0 Because the series  zz + ω ′ (u)Φ]φ  j dz = −φj (d)g(x) + μj [Φ = Φ +∞  j=N +1 d 0  j dz. Φφ  φj φj Φ, converges uniformly on [0, d] (see Theorem 2.27, [2]), we find d − 0  zz + ω ′ (u)Φ]  Φdzs  [Φ = ∞  j=N +1 ≥ μN +1  φj ]2 − Φ(x,  d)g μk [ Φ, d 0  2 dz − Φ(x,  d)g. Φ Now we integrate (4.18) over RΛ∗ . After integration by parts, we get  2 ′     Φx dxdz + − [Φzz + ω (u)Φ]Φdxdz ≤ RΛ∗ RΛ∗ Combining this inequality with (4.19), we arrive at RΛ∗ 2 Φ dxdz ≤ μ−1 N +1 RΛ∗ For an arbitrary β > 0, we write RΛ∗ and, similarly, [−Λ∗ /2,Λ∗ /2]  |f Φ|dxdz ≤β  d)g|dx ≤ |Φ(x,  |f Φ|dxdz + [−Λ∗ /2,Λ∗ /2] RΛ∗  2 dxdz + β −1 Φ RΛ∗  z (x, z)g|dxdz |Φ  ≤ βΛ∗ d Φ (4.19) 2 C 2,α (RΛ∗ ) RΛ∗  |f Φ|dxdz.   d)g|dx . |Φ(x, f 2 dxdz RΛ∗ + β −1 g 2 L2 ([−Λ∗ /2,Λ∗ /2]) . Vol. 20 (2018) N-Modal Steady Water Waves Composing the last three inequalities, we obtain Λ∗ d 2βΛ∗ d  2  22 Φ Φ X+ f L (RΛ∗ ) ≤ μN +1 βμN +1 2 Y1 + 865 Λ∗ d g βμN +1 2 Y2 . Now this inequality and (4.16) with β= μN +1 , C4Λ∗ d where C is the constant from (4.16), leads to (4.17). This delivers the required estimate for the inverse.  Before formulating the next theorem, let us define the space 2,α Xred = [Cper (IR)]N , 2,α (IR) consists of all even and Λ∗ -periodical functions from C 2,α (IR). where the space Cper  and a Theorem 4.4. For any m ≥ 1 there exist open neighborhoods of the origin U ⊂ Xred , V ⊂ X m  smooth operator h ∈ C (U ; V ) such that the function Φ := h(Φ1 , ..., ΦN ) is the unique solution of the boundary problem (4.9)–(4.10), provided (Φ1 , ..., ΦN ) ∈ U . Proof. The statement follows directly from Lemma 4.3 and the implicit function theorem applied to the system (4.9)-(4.10).  Theorem 4.4 allows to reduce the problem for small-amplitude waves to a finite dimensional system of the form Φ′′j − μj Φj = Nj (Φ̄; μ), (4.20) 2,α where Φ̄ = (Φ1 , ..., ΦN ) and Φj ∈ Cper (IR), j = 1, ..., N . The nonlinear operators Nj are smooth and DΦ̄ Nj (0; μ) = 0. 4.4. The Lyapunov–Schmidt Reduction and the Existence of N -Modal Waves The kernel of the linear operator on the left-hand ⎛ ⎛ ⎞ cos( |μ∗1 |x) ⎜ ⎜cos( ⎟ 0 ⎜ ⎜ ⎟ ξ1 = ⎜ , ξ = ⎜ ⎟ 2 .. ⎝ ⎝ ⎠ . 0 Let us write side in (4.20) for μ = μ∗ is ⎛ ⎞ 0 ⎜ |μ∗2 |x)⎟ ⎜ ⎟ , ..., ξ = ⎜ ⎟ N .. ⎝ ⎠ . cos( 0 Φj = tj cos( spanned by ⎞ 0 ⎟ .. ⎟ . ⎟. ⎠ 0 ∗ |μN |x) |μ∗j |x) + ζj , (4.21) where ζj is orthogonal to cos( |μ∗j |x) in L2 (−Λ∗ /2, Λ∗ /2). We note that the vector function (ζ1 , ..., ζN ) is orthogonal to all ξj , j = 1, ..., N , and hence this splitting corresponds to the Lyapunov–Schmidt decomposition under the action of the projection Φ̄ = (Φ1 , ..., ΦN ) to the space XN spanned by ξ1 , ..., ξN . Thus, substituting (4.21) into (4.20) and taking the projection on XN and its compliment, we arrive at the equations ζj′′ − μ∗j ζj = ζj (μj − μ∗j ) + Nj∗ (t, ζ; μ) − Gj (t, ζ; μ) cos( tj (μj − μ∗j ) = Gj (t, ζ; μ). |μ∗j |x), (4.23) The nonlinear parts are defined by Nj∗ (t, ζ; μ) = Nj (Φ̄; μ), Gj (t, ζ; μ) = (4.22) 2 Λ∗ Λ∗ /2 Nj (Φ̄; μ) cos( −Λ∗ /2 |μ∗j |x)dx, 866 V. Kozlov and E. Lokharu JMFM where Φ̄ = (Φ1 , ..., ΦN ) depends on t and ζ via (4.21). The linear operator on the left-hand side in (4.22) defined on the space X∗N , where Λ∗ /2 2,α X∗ = {g ∈ Cper (IR) : g(x) sin −Λ∗ /2 |μ∗j |xdx = 0, j = 1, ..., N } with values in Λ∗ /2 α (IR) : {g ∈ Cper g(x) sin −Λ∗ /2 |μ∗j |xdx = 0, j = 1, ..., N } is invertible, while the right hand side is presented by some terms of order |t|2 plus small perturbations of ζ. Thus, using implicit function theorem, we can resolve ζ = (ζ1 , ..., ζN ) from (4.22) as an operator of t = (t1 , ..., tN ) and μ = (μ1 , ..., μN ), provided |t| and |μ − μ∗ | are small enough. Furthermore, the following estimate is valid: ζj C 2,α (−Λ∗ /2,Λ∗ /2) ≤ C|t|2 . Therefore, system (4.22)–(4.23) is reduced to a system of scalar equations: where the nonlinear term satisfies tj (μj − μ∗j ) = Gj (t, ζ(t, μ)μ),     ∂Gj  (t, μ) ≤ C|t|2 |Gj (t, μ)| ≤ C|t| ,  ∂μ 2 (4.24) (4.25) for all sufficiently small |t| and |μ|, while the constant C is independent of μ. Let us rewrite (4.24) as μj = μ∗j + Gj (t, μ)/tj . Thus, if |t|2 /|tj | ≤ 1/(2C), where C is the constant from (4.25), we can apply fixed point theorem to resolve μ as a function of t so that μj = μ∗j + O(|t|2 /ǫ). Thus, tracking back all changes of variables, we find a solution (Φ, μ) to (4.2) such that Φ(x, z; μ) = N  tj cos( j=1 |μ∗j |x)φj (z) + O(|t|2 ). Now the formula (4.1) recovers the profile as η(x) = d − N  tj cos( ′ (d) u j=1 |μ∗j |x)φj (z) + O(|t|2 ). Using these identities, we can express the stream function via ψ(x, y) = u(yd/η(x))   N  yduy (yd/η(x)) ∗ uy (d))φj (d) + tj cos( |μj |x) φj (yd/η(x)) − η(x) j=1 + O(|t|2 ). This completes the proof of the theorem. Acknowledgements. The authors are thankful to the anonymous referee for the help in improving the article. V. K. acknowledges the support of the Swedish Research Council (VR) Grant EO418401. Compliance with ethical standards Conflict of interest The authors report no disclosure relevant to the manuscript. Prof. Kozlov and Dr. Lokharu report no financial disclosure. The authors have no conflict of interests. Vol. 20 (2018) N-Modal Steady Water Waves 867 Open Access. 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Fluid Mech. 191, 341–372 (1987) Vladimir Kozlov and Evgeniy Lokharu Department of Mathematics Linköping University 581 83 Linköping Sweden e-mail: [email protected] Evgeniy Lokharu e-mail: [email protected] (accepted: October 25, 2017; published online: December 4, 2017)