On periodic geophysical water flows with discontinuous vorticity in the equatorial f-plane approximation

Abstract

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f-plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ₁ adjacent to the surface situated above another layer of constant non-zero vorticity γ₂≠γ₁ adjacent to the bed. For certain vorticities γ₁,γ₂, we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows.

This article is part of the theme issue ‘Nonlinear water waves’.

1. Introduction

This work is concerned with the study of wave–current interactions occurring in geophysical water flows governed by the equatorial f-plane approximation. The currents incorporated in the flows that we consider here present sudden changes captured by a discontinuous vorticity function. The vorticity is a measure of the local spin in the fluid and its relevance stems from various phenomena associated with wave–current interactions; cf. [1,2]. A relevant physical scenario of sheared superposed currents is the case of a layer of high vorticity–adjacent to wind-generated waves [3,4]—situated above a layer hosting currents that have resulted from sediment transport [5]. Numerical simulations for flows exhibiting discontinuous vorticity [6,7] have indicated the appearance of flow characteristics not present in flows exhibiting a continuous vorticity [8]. These aspects were the driving motivation for the development of a rigorous theory for the existence of wave trains interacting with currents presenting a discontinuous vorticity. The previous task was accomplished by Constantin & Strauss in [9], where they established the existence of periodic symmetric two-dimensional waves propagating at constant speed at the surface of a layer of water in a flow allowing for a (bounded) discontinuous vorticity. Extending in a natural and skillful way the method they used in [10], they proved the existence of waves of small and large amplitude. Thereafter, there have appeared works proving the existence of water waves allowing for additional complications, like the presence of stagnation points in flows of constant vorticity (see [11] for small-amplitude waves and [12] for waves of large amplitude), the incorporation of surface tension (as an additional restoring force) for flows with constant vorticity [13,14] and with discontinuous (piecewise constant) vorticity [15], of surface tension and unbounded vorticity [16], and of even piecewise constant vorticity and stagnation points [17]. The scenario of capillary–gravity water flows presenting a bounded vorticity was analysed in [18], where the existence of waves of small amplitude was proved.

An approach, seemingly more suitable for numerical simulations, was developed by Henry and consists in fixing the mean depth (as the bifurcation parameter) rather than the mass flux. This novel reformulation was introduced in [19,20], where the existence of small- and large-amplitude steady periodic water waves which propagate over a flow with a flat bed and exhibiting a continuous vorticity distribution was proved.

The other major aspect we consider here is that of the geophysical effects. The latter are a characteristic of the water flows whose motion is influenced by the Earth's rotation reflected through the manifestation of the Coriolis force. The Coriolis effects enhance the already complex dynamics of flows near the equatorial Pacific. These flows present some peculiarities like a strong stratification (evidentiated by the presence of an interface, called thermocline, separating two layers of constant density), the presence of strong depth-dependent underlying currents (cf. [21,22]) with flow reversal at a depth of approximately 100–200 m and a wide range of other wave propagation phenomena. It is possible to mitigate these intricacies through the employment of suitable approximations of the geophysical governing equations; one of these is the f-plane approximation which is appropriate for oceanic flows within a restricted meridional range of about 2° latitude from the Equator [21,23–27]. Another way of escaping these enormous difficulties is the utilization of variational formulations; cf. [28,29].

The rigorous mathematical study of geophysical water flows was initiated by Constantin in [30], where the modelling of wave–current interactions in the f-plane approximation for underlying currents of constant vorticity was presented. By a suitable and substantial modification of the Gerstner wave solution to the case of three-dimensional flows Constantin also obtained in [31] an exact solution (in the Lagrangian framework) of the nonlinear geophysical water waves in the β-plane near the Equator. These studies were followed by papers presenting explicit exact solutions describing flows accommodating flow reversal and allowing for underlying currents in the β-plane setting; cf. [32–34]. Moreover, a fully nonlinear and three-dimensional system of model equations involving the β-plane approximation was analysed recently by Constantin & Johnson [35]. The additional aspect of centrifugal forces in the β-plane approximation was addressed recently by Constantin & Johnson [22] by means of an approach using spherical and cylindrical coordinates (see also [36]) and by Henry in [37], where an exact and explicit solution describing equatorially trapped waves was obtained.

Our aim here is to prove the existence of gravity water waves of small amplitude arising as the free surface of geophysical flows allowing for discontinuous (merely bounded) vorticity. The flows we consider are governed by the f-plane approximation. We show that the methods used in [9] also bear relevance in our setting. This is particularly useful when proving that the equations of motion together with their boundary conditions can be expressed in a weak form by means of a formulation for the height function. After proving in §2 the equivalence of the weak formulations, we construct in §3, via local bifurcation results of Crandall–Rabinowitz type, a local curve of solutions to the water wave problem. We conclude in §4 by a detailed investigation of the local bifurcation for a layer of constant non-zero vorticity γ₁ adjacent to the free surface above another layer of constant non-zero vorticity γ₂≠γ₁. For this scenario, we derive the dispersion relation for small-amplitude waves. This relation indicates how the relative speed of the bifurcating laminar flow at the free surface varies with respect to certain parameters like the wavelength, the mean depth of the flow, and—in the case of a piecewise constant vorticity—the position of the vorticity jumps. We like to mention that, it was shown in [38,39] that the dispersion relations corresponding to the fixed mean depth approach coincide with those in [9,40] for the fixed mass flux approach. The derivation of dispersion relations for water flows exhibiting a (continuous) non-constant vorticity was performed by Karageorgis [41].

2. Preliminaries

It was proved in [42] that gravity wave trains can propagate at the free surface of a rotational water flow of constant vorticity and governed by the equatorial f-plane approximation only if the flow has a two-dimensional character. Accordingly, we will consider here two-dimensional water flows bounded below by an impermeable flat bed and above by a free surface, which in a rotating framework with the origin at a point on Earth's surface, with the x-axis pointing horizontally due east, the z-axis pointing upwards has the equation z=η(x). Denoting with (u,w) the velocity field, with P the pressure in the fluid, with ω∼0.73×10⁻⁴ the Earth's speed of rotation, with g the gravitational acceleration, and considering travelling waves, the motion of the flow is governed (in the frame of reference (x,z) moving at constant wave speed c) by the conservation of momentum equations

2.1

and the equation of mass conservation

2.2

which are supplemented by appropriate boundary conditions, as follows. They are the kinematic boundary conditions

2.3

and

2.4

On the free surface, the condition that decouples the motion of the air above the free surface from the motion of water is

2.5

where P_atm is the constant value of the atmospheric pressure. Moreover, the local spin of the water is captured by the vorticity function

2.6

We shall work under the assumption of no stagnation points, which is written as

2.7

throughout the fluid domain

2.8

We will recall now the several equivalent formulations of the water wave problem for the case of a regular vorticity. We refer the reader to [1] for details. The analysis for the case of discontinuous merely bounded vorticity function will be performed subsequently.

It was proved in [43] that the employment of the stream function, defined, up to a constant, by

2.9

allows us to reformulate the set of equations (2.1)–(2.6) as

2.10

with p₀ being the relative mass flux,¹ given by

The no-stagnation condition (2.7) induces [10] (in the case of a regular vorticity) the existence of a function such that the vorticity function Ω(x,z) satisfies

2.11

Using the latter property of the vorticity function, we readily obtain from the first two equations in (2.10) that the expression

2.12

is constant throughout the fluid domain. The latter relation constitutes the f-plane version of Bernoulli's Law. It allows us to reformulate the original water wave problem (2.1)–(2.6) as

2.13

Moreover, also due to (2.7), we can introduce new variables by means of the Dubreil–Jacotin transformation

2.14

defined by

2.15

An equivalent formulation for the original water wave problem can be obtained by means of the height function defined through

We readily see that

2.16

which implies that the water wave problem is equivalent to

2.17

where the no-stagnation condition (2.7) is written as

2.18

We remark now that the system (2.17) can be put in the form

2.19

where

2.20

Let Γ_min be the minimum of the function . We then observe that Γ_min≤0.

Remark 2.1 —

As the vorticity function that we are considering here is discontinuous (such as a step function) we cannot expect that solutions to the water wave problem to be smooth. This is the reason why we need alternative formulations of the water wave problem. The reformulation (2.19) serves precisely this purpose.

Theorem 2.2 —

Given α∈(0,1) and r:=2/(1−α), the following formulations are equivalent

• (i) the velocity formulation (2.1)–(2.2) with the boundary conditions (2.3)–(2.5) and (2.7) for P,u,v∈W^1,r_per(D_η)⊂C^α(D_η);

• (ii) the stream function formulation (2.13) with ψ_z>0 for , γ∈L^r[p₀,0] and ;

• (iii) the height formulation (2.19) with the condition (2.18) for and Γ∈W^1,r[p₀,0].

Proof. —

We prove that (i) implies (ii). First, arguing as in [9], we infer that . Equation (2.2) allows us to define (uniquely up to a constant) by means of (2.9). The kinematic boundary conditions (2.3) and (2.4) allow us to define ψ=0 on z=η(x) which implies that ψ=p₀ on z=−d. To complete the proof of (i)⇒(ii), we need to prove the validity of the first two equations in (2.13). By the change of variables (2.14) and using (2.16), we obtain that

2.21

and

2.22

Therefore, for we have

Thus

where the last equality follows by applying the curl operator to the vector field defined by the two equations in (2.1). Consequently, Ω=γ(p) for some function γ∈L^r[p₀,0]. The last equality implies immediately that Δψ=γ(ψ), that is the first equality in the formulation (2.13) is proved.

As γ∈L^r[p₀,0], we have that Γ defined by (2.20) belongs to W^1,r[p₀,0]. We will prove next that and that the chain rule holds. We note first that

imply that . We prove now that

2.23

To this end, we choose a test function and compute the action of ∂_x(Γ(ψ)) on ϕ as

2.24

2.25

2.26

2.27

2.28

which proves the claim made in (2.23). It is even simpler to prove that

It now follows that the gradient of the expression

2.29

vanishes throughout D_η. Evaluating E on the surface z=η(x), and taking into account that ψ=0 on z=η(x), the definition of Γ and the dynamic boundary condition, we see that the top boundary condition in (2.13) holds true.

To prove that (ii)⇒(i), we define u,w∈W^1,r_per(D_η) by (2.9) and Γ by means of (2.20), and set

It is now immediate that the differentiation of P defined above gives that u and w satisfy

2.30

Moreover, P(x,η(x))=P_atm and the kinematic boundary conditions (2.3) and (2.4) are satisfied because of the third and fourth equations in (2.13).

For checking the implication (ii)⇒(iii), we define the variable q and p by means of (2.15). This leads to the relations

2.31

and

2.32

which allow to rewrite the formula Ω=u_z−w_x in terms of the variables q and p, a process that gives the partial differential equation in (2.17). Writing the second equation in (2.13) by means of the formulae (2.31) and (2.32), we obtain the second equation in (2.17), if we also use that the free surface y=η(x) can be written as p=0. Clearly, the third condition in (2.13) is equivalent to the third condition in (2.17).

To prove that (iii)⇒(ii), we argue along the lines of [9] and use that the mapping is a global C^1+α diffeomorphism from (−L,L)×(p₀,0) to {(x,z)∈D_η:x∈(−L,L)}. Defining ψ(x,y) to be the second component of the inverse of the previous bijection, one can show using a similar route as in [9] that ψ satisfies the system (2.13). ▪

3. Existence of non-trivial solutions

(a). Laminar solutions

A special type of solutions to (2.19) (presenting no q dependence), having thus purely horizontal streamlines, is given by

3.1

where the parameter λ satisfies λ=(c−u)² at the flat free surface and is related to Q by

3.2

(b). The linearization of the water wave problem

The goal of this section is to find non-trivial solutions to the water wave problem by resorting to the Crandall–Rabinowitz theorem on bifurcation from a simple eigenvalue. To this end, we will use the weak reformulation of the governing equations (2.19).

We describe now the functional analytic setting that is able to make our problem amenable to the usage of the Crandall–Rabinowitz theorem on bifurcation from simple eigenvalues. Setting

we will work with the Banach spaces

3.3

for some α∈(0,1), and Y =Y ₁×Y ₂ for

3.4

endowed with the norm

3.5

with being the distributions in which are 2π periodic in the q-variable.

Motivated by the reformulation (2.19), we define the operator by

3.6

and

3.7

which rewrites the water wave problem as . Note now that if we set with

we have that

that is, the first condition of the Crandall–Rabinowitz theorem is satisfied. We proceed now to check the second condition of the latter mentioned theorem and find out that the linearization of the operator around u≡0 is

3.8

(c). The kernel of

We start now to identify values of λ for which is one-dimensional. To reach this goal, we adapt an approach used in [9].

We note first that the membership of a function to is equivalent to m being solution to the system

3.9

with m even and L-periodic in the q-variable and

Keeping in mind that the even function m has the Fourier series representation

it follows that m solves (3.9) if and only if each M_k solves the Sturm–Liouville problem

3.10

As we are interested in solutions of period 2π, we will examine (3.10) for k=1. We relate now (3.10) to the minimization problem

3.11

Arguing as in [10] yields the existence of a function M that realizes the minimum of the right-hand side in (3.11) and satisfies the Sturm–Liouville problem

3.12

We now see that, for the existence of solutions for (3.10), it is necessary that μ(λ)=−1 for some . We first see that, for λ>g−2ωc−2Γ_min, we have for all p∈[p₀,0]. This implies that

3.13

for all ϕ∈H¹(p₀,0) with ϕ(p₀)=0. The latter inequality shows, using also the definition (3.11), that μ(λ)≥−1 for all λ>g−2ωc−2Γ_min. This shows that the existence of some λ>−2Γ_min with μ(λ)=−1 is secured if we show that μ(λ)≤−1 for some λ>−2Γ_min. We give in the next lemma a condition that guarantees the existence of such a λ.

Lemma 3.1 —

Let . Assume that

3.14

where . Then there exists λ>−2Γ_min with μ(λ)<−1.

Proof. —

Consider the expression

3.15

where . As the limit of E(k) as equals the right-hand side of (3.14), we can find a such that

3.16

We set now

and define, for , the function

3.17

We note that

Moreover,

which implies that

Using that p₀≤p_n≤p₁≤0, we compute

3.18

where

3.19

and

3.20

Using (3.16) and as p_n→p₁ as we can find an ϵ∈(0,g−2ωc) such that there is an with the property that

For the above value of ϵ we can find now an n≥N large enough such that B_n<ϵ. From (3.19) and (3.20) we infer now the existence of a φ_n∈H¹(p₀,0) with φ_n(p₀)=0, φ_n≢0, that satisfies

an inequality which implies that μ(−2Γ_min)<−1. Using now the continuity of the function wherever μ(λ)<0, we infer the existence of some λ>−2Γ_min with μ(λ)<−1. ▪

Remark 3.2 —

As a and μ are still C¹-functions of λ, one can show as in [10] that μ(λ) is a strictly increasing function of λ provided μ(λ)<0. This shows that, under condition (3.14), there is a unique λ*>−2Γ_min such that μ(λ*)=−1.

We are now ready for the assertion concerning the dimension of the kernel of .

Proposition 3.3 —

If λ*>−2Γ_min is the unique solution of μ(λ)=−1, then the kernel of is one-dimensional.

Proof. —

From the previous consideration in this subsection we know that contains the one-dimensional space generated by , with M being the eigenfunction corresponding to k=1 in (3.10) and satisfying M(0)=1.

Assume now that m∈X belongs to the kernel of . Then, for all p∈[p₀,0] it holds that

with coefficients

that belong to C^1+α[p₀,0]. It is now easy to see that m_k satisfies (3.10). Our task is now to show that all m_k=0 for all . We show first that m₀=0. Indeed, from the differential equation and from the boundary condition at p=p₀ from (3.10) we obtain that

From the boundary condition at p=0 we obtain that

3.21

We assume for the sake of contradiction that C₀≠0. Then, we have from above that λ* coincides with the unique point λ₀ where the strictly convex function

attains its minimum on . But this is a contradiction, because μ(λ*)=−1, while μ(λ₀)=0 as we will see shortly. Indeed, for any φ∈H¹(p₀,0) with φ(p₀)=0 we have

3.22

which implies that μ(λ₀)≥0. Equation (3.21) now forces C₀=0, which leads to m₀=0.

Assuming now that m_k≠0 for some k≥2, we obtain that

3.23

which yields that μ(λ*)<−1, which is a contradiction. Therefore, m_k=0 for all . Moreover, there exists a constant such that m₁=aM, which concludes the proof. ▪

(d). The characterization of the range of

Proposition 3.4 —

A pair , with belongs to the range of if and only if

3.24

where φ* is a generator of .

Proof. —

If belongs to the range, then there exists v∈X such that

3.25

a system that yields (3.24) after multiplying the first equation with φ* and integrating over D. We now prove the converse and assume that (3.24) holds true. Introducing the notation

we note that the equation is solved for some v∈X if and only if

3.26

and

3.27

for

where

Arguing as in [9], it suffices to solve (3.25) in the space X₀ for . To this end, we will prove that, given ϵ∈(0,1) and , the problem

3.28

has a unique solution , which satisfies

3.29

for some constant C that depends only on . We note now that a weak solution of (3.28) verifies the equality

3.30

for all test functions with ϕ|_B=0. It is easy to see that the right-hand side of the above equation defines a bounded linear functional on the Hilbert space

Moreover, the left-hand side of the above equation defines a bounded bilinear coercive form on . To prove the coercivity, we write, for ,

with ϕ_k∈C³[p₀,0] given by

By the Parseval equality we have, for all k≥1,

Adding the previous equalities for all k≥1, we obtain

3.31

a relation that clearly implies the coercivity. By the Lax–Milgram theorem (cf. [44]) we infer the existence and uniqueness of a weak solution of (3.28).

We consider now, for k≥1, the sequence of approximate problems

3.32

where , for , and satisfying

3.33

Applying elliptic regularity results (cf. [45]), we find that the unique solution of v^(ϵ,k) to the problem (3.32) belongs to . Moreover, using Schauder estimates (A 2), we find that

3.34

for some constant C that depends only on .

We will show in the sequel that is a bounded sequence for each fixed ϵ∈(0,1). To this end, we assume for the sake of contradiction that there is some ϵ∈(0,1) and a subsequence v^(ϵ,k_n) for which

But the latter equality together with (3.33) and (3.34) imply that . Dividing in (3.34) by we obtain that the functions are bounded in . By the compactness of the embedding we may extract a subsequence u_nk such that

Furthermore, it is also immediate that and u is a weak solution of

3.35

Using u as a test function in the weak formulation above, we find that

relation which combined with (3.31) gives u≡0. The latter is a contradiction with . Therefore, is a bounded sequence. Again by the compactness of the embedding we can find a subsequence that converges to some which solves (3.28). In addition, by Schauder estimates (A 2) we infer that and that v^(ϵ) satisfies (3.29).

We will prove next that the sequence is bounded for all sequences ϵ_n→0. Assuming that for ϵ_n→0, we get from (3.29) that and the functions are bounded in . As before, by the compactness of the embedding we find a subsequence (w_nk) that converges in to a function w with . From the boundary value problem for v^(ϵ_nk) we have (by letting ) that is a weak solution of the problem

3.36

Additionally, , because the Schauder-type estimates imply that . Therefore, w=βφ* for some . In the weak formulation (3.28), we select the test function ϕ=φ* and obtain the equation

3.37

which becomes

by using the affiliation of φ* to the set of weak solutions to (3.36). As we infer that

which implies that β=0, and, consequently, w=0. The latter conclusion contradicts the fact that . Therefore, the sequence is bounded for all sequences ϵ_n→0. From the compactness of the embedding we infer the existence of a subsequence (v^(ϵ_nk))_k≥1 that converges in to some which is a solution of (3.25). Using Schauder estimates (A 2), we have the , which completes the proof. ▪

(e). The transversality condition

We consider now the last condition from the Crandall–Rabinowitz theorem; that is, we need to check that does not belong to the range of .

A short calculation shows that

3.38

Assuming that would belong to the range of , then by the characterization (3.24) we would have that

3.39

But, as φ* satisfies (3.9), the left-hand side of the above expression equals

which is strictly negative.

The previous findings about the kernel and the image of as well as the application of the Crandall–Rabinowitz theorem allow us to summarize our discussion in the following result concerning the existence of a local curve of solutions emerging from the laminar flow solutions (3.1).

Theorem 3.5 —

Assume that the condition (3.14) is satisfied. Then there is an ε>0 and a C¹-curve

of solutions to the problem (2.19) satisfying h_p>0. The curve contains precisely one solution that is independent of q, namely H(p,λ*). Moreover, for |s|<ε we have

3.40

where H(p,λ) are the laminar flow solutions (3.1), and M is the solution of (3.10) with k=1 and M(0)=1.

Proof. —

The result stated follows from the consideration in the present section by applying the Crandall–Rabinowitz Theorem on bifurcation from simple eigenvalues; cf. [46]. ▪

4. Analysis of a dispersion relation

We illustrate here the local bifurcation result from the previous section by means of the dispersion relation in the case of a layer of constant non-zero vorticity adjacent to the free surface above a layer of (another) constant non-zero vorticity neighbouring the bed. The reasoning from §3 shows that the necessary and sufficient condition for the existence of waves of small amplitude that are perturbations of the laminar flow solutions (3.1) is that the Sturm–Liouville problem

4.1

has a non-trivial solution M∈C^1,α(p₀,0),M≢0, where, as before, . Choosing p₁∈[p₀,0], the line p=p₁ marks off a rotational layer of vorticity γ₁ adjacent to the free surface corresponding to [p₁,0] from a second rotational layer of vorticity γ₂ adjacent to the flat bed corresponding to [p₀,p₁]. We find first that the function Γ emerges as

4.2

Thus

4.3

We aspire now to find a function M∈C^1,α(p₀,0) that satisfies (4.1). To this end, we set

The functions u and v will have to satisfy

4.4

and

4.5

together with the matching conditions

4.6

and the boundary conditions

4.7

and

4.8

The compatibility conditions (4.6) ensure that M∈C¹([p₀,0]). The asserted C^1,α smoothness follows from Theorem 3 of [9].

We solve first equation (4.4) for U. In doing so, we set

which transforms (4.4) in where s=a(p)/γ₂. Thus, the general solution of (4.4) is

for constants . From the bottom boundary condition (4.7) we find that , which renders U as

4.9

where . Thus,

4.10

Similarly, we find that

4.11

with and c₁,c₂ are some real constants. This gives

4.12

For simplicity, we introduce the notation

Making use of matching conditions (4.6), we obtain the system

4.13

which, after replacing with in the second equation, becomes equivalent to

4.14

We find

4.15

and

4.16

The quantities θ and ρ need to be made explicit in terms of λ, of the average mean depth d and of the depth at which the jump in vorticity occurs. The latter is denoted with d₀>0 and is the average depth corresponding to p₁.

The horizontal velocity u of the bifurcation inducing laminar flow solutions is only a function of z, which implies that

as it is obvious from (2.31) and (3.1). Therefore,

4.17

and

4.18

For the relative mass flux p₁ we have

4.19

an equation with solutions

4.20

Note that the expression is positive irrespective of the sign of γ. As d₀>0, we have that

4.21

Moreover,

4.22

A similar computation for the total mass flux p₀ gives rise to the equation

4.23

whose discriminant is, by means of (4.19), equal to

4.24

Thus,

from which we derive that

4.25

4.26

where the last equality follows from (4.21). To decide the sign in the above expression, we note that

irrespective of the sign of γ₂. As d>d₀, it follows that

4.27

From (4.27) and (4.21) we conclude that θ=(a(p₁)−a(p₀))/γ₂=d−d₀.

Owing to (4.11) and (4.12), we can write now the surface boundary condition in the form (4.8) as

4.28

Using formulae (4.15) and (4.16), we find that

4.29

and

4.30

Replacing ρ with and θ with d−d₀, we obtain

4.31

Provided it has a unique positive solution, equation (4.31) delivers the (relative) speed at the surface of the bifurcation inducing laminar flow solutions.

Remark 4.1 —

If we set d₀:=d, γ₁:=γ and γ₂:=0 in the formula above, we obtain that satisfies the equation

As , (cf. (4.18)), the latter equation is equivalent to

whose positive solution is

which recovers the dispersion relation (3.12) for constant vorticity flows in the f-plane approximation, from [43]; see also formula (3.27) from [47].

We investigate now in some detail equation (4.31) for specific signs of the two vorticities that give rise to the existence of a unique positive solution of (4.31), which is equivalent to local bifurcation.

Proposition 4.2 —

Assume that γ₁<0 and γ₂<0. Then local bifurcation occurs.

Proof. —

We denote and set p(x) to be the polynomial given by the left-hand side of (4.31). We will prove that p(0)<0. Indeed,

4.32

because the expression is negative; cf. [9]. Therefore, p has at least one positive solution, x₀. We will prove that x₀ is the unique positive root of p. To this end, we assume that p has a second positive root. Then, the third root of p has to be real. As the product of the three roots of p equals −p(0)>0, the third root is necessarily positive. On the other hand, the sum of the roots of p is, by Viète's relations,

4.33

and thus, is negative, which is a contradiction. Consequently, p has precisely one strictly positive root. ▪

While explicit formulae for the solutions of (4.31) are available by means of the Cardano's formula, the expressions of these formulae are too intricate to be of fair insight. We will give in the sequel an estimate on the positive solution of (4.31). To this end, we note first that relation (4.28) is equivalent to

4.34

The latter relation is very useful in proving the following estimate.

Theorem 4.3 —

Assume that γ₂<γ₁. Then

4.35

where λ₊(d) is the positive solution of the equation

and λ₊(d₀) is the positive solution of the equation

Proof. —

We remark that the function

is strictly increasing, because

which is clearly positive. We assume now that . This implies that . The latter inequality, (4.18) and (4.34) yield that , a relation possible only if . As this is in contradiction with the assumption, it follows that

Similarly, one can prove the left-sided inequality in (4.35). ▪

Corollary 4.4 —

Assume that γ₂<γ₁. If u(0)=:u₀ is the speed at the free surface of the bifurcation inducing laminar flows, then we have for the genuine wave speed c the following estimates:

4.36

where

and

Proof. —

The two inequalities in (4.36) are simply a consequence of (4.35) and of . ▪

Appendix A

This section is dedicated to the regularity of solutions to the linearization of the problem (2.19). Given α∈(0,1), let D be a C^1+α-bounded open set. Assume that u satisfies the system

A 1

where the coefficients a_ij satisfy uniform ellipticity and the coefficients c_j satisfy the uniform obliqueness condition

We are now ready to state the following result concerning Schauder-type estimates.

Theorem A.1 —

Assume that α and D are as given at the beginning of this section. Let also . If the problem (A 1) has a solution , then u belongs in fact to with

A 2

where C is a constant that depends only on the bounds of the involved norms of the coefficients, on the domain and on the ellipticity and obliqueness constants.

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Competing interests

I declare I have no competing interests.

Funding

This work was supported by EPSRC grant no. EP/K032208/1.