In this paper, we prove the existence of analytic relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold doubly-connected V-states bifurcating from any annulus of arbitrary size.
In this work, we are concerned with the quasi-geostrophic shallow-water equations with a parameter , which is a two dimensional active scalar equation taking the form
The involved quantities are the divergence-free velocity field v and the potential vorticity which is a scalar function. The parameter λ stands for the inverse Rossby radius defined in the literature by
where g is the gravity constant, H is the mean active layer depth and is the Coriolis frequency, assumed to be constant. Notice that the case corresponds to the velocity-vorticity formulation of Euler equations. The system (1.1) is commonly used to track the dynamics of the atmospheric and oceanic circulation at large scale motion. For a general review about the asymptotic derivation of the these equations from the rotating shallow water equations we refer for instance to [36, p. 220].
The main purpose of this paper is to explore the emergence of time periodic solutions in the patch form close to the annulus of radii 1 and b for the system with fixed and . Recall that a vortex patch means a solution of (1.1) with initial condition being the characteristic function of a bounded domain , that is . Actually, this structure is conserved in time due to the transport structure of (1.1), and one gets
with is the flow map associated to v, defined through the ODE
In this framework, of bounded datum with compact support, existence and uniqueness follow in a standard way from Yudovich approach implemented for Euler equations and which can be adapted here in a similar way. These structures can be considered as a toy model to simulate hurricanes motion in the context of geophysical flows. In the smooth boundary case, their dynamics is completely described by the evolution of the interfaces surrounding the patch according to the contour dynamics equation given by
where is a parametrization of the boundary of the patch and is an outward normal vector to the boundary. We may refer to [25,26] for a detailed derivation of this equation for active scalar models. We are particularly interested in the existence of ordered structures moving without shape deformation, called V-states. More precisely, we shall focus on the existence of uniformly rotating vortex patches about their center of mass, that can be fixed at the origin, and with a constant angular velocity , namely
In the present work we explore the case of doubly-connected V-states with m-fold symmetry. To fix the terminology, a bounded open domain is said doubly-connected if
where and are two bounded open simply-connected domains with . This means that the boundary of is given by two interfaces, one of them is contained in the open region delimited by the second one. According to the structure of , every radial initial domain generates a trivial stationary solution, and therefore a V-state rotating with any angular velocity. Basic examples are given by the discs in the simply-connected case or the annuli in the doubly-connected case. The first non-trivial examples of uniformly rotating solutions for Euler equations are Kirchhoff ellipses which rotate with the angular velocity where a and b are the semi-axes of the ellipse (see [31] and [4, p. 304]). In [9] Deem and Zabusky established numerically the existence of simply-connected rotating patches with m-fold symmetry for . An analytical proof based on bifurcation theory and complex analysis tools was performed by Burbea in [5] showing the existence of m-fold (for any ) symmetric V-states bifurcating from Rankine vortices with angular velocity . In the spirit of Burbea’s work, a lot of results on m-fold V-states have been obtained both for simply and doubly-connected cases for Euler, and equations in the past decade. We may refer to [6,8,10,16,17,19,22,35]. From this long list, we shall make some comments on two contributions from [10,22] related to the current work. In [22, Thm. B], the authors proved for Euler equations that under the condition
one can find two branches of m-fold doubly-connected V-states bifurcating from the normalized annulus , defined by
at the following angular velocities
Burbea’s result has been extended for in [10, Thm. 5.1], where it is shown the existence of branches of m-fold symmetric V-states () bifurcating from Rankine vortex , with being the unit disc, at the angular velocitity
where and are the modified Bessel functions of first and second kind, respectively. We may refer to the Appendix A for the definitions and some basic properties of these functions. We also notice that more analytical and numerical experiments were carefully explored in [10,11] dealing in particular with the imperfect bifurcation and the response of the bifurcation diagram with respect to the parameter λ.
We emphasize that different studies around this subject have been recently investigated by several authors, we refer for instance to [12,13,15,18,21,24] and the references therein.
The main contribution of this paper is to establish for the existence of branches of bifurcation in the doubly-connected case, generalizing the result of [22]. More precisely, we prove the following result.
Letand. There existssuch that for every, with, there exist two curves ofm-fold doubly-connected V-states bifurcating from the annulusdefined in (
1.5
), at the angular velocitieswhereis defined in (
1.7
) andwithandbeing the modified Bessel functions of first and second kind. In addition, the boundary of each V-state is analytic.
Before sketching the proof some remarks are in order.
The spectrum is continuous with respect to λ and b. In particular, when we shrink we find the spectrum of Euler equations detailed in (1.6). However, when we shrink we obtain in part the simply connected spectrum (1.7). In other words,
These asymptotics are obtained for sufficiently large values of m. For more details see Lemma 3.2.
Now, we intend to discuss the key steps of the proof of Theorem 1.1. The following notation will be used throughout the paper.
We denote by the unit disc. Its boundary, the unit circle, is denoted by .
Let be a continuous function. We shall use the following notation throughout the paper
where stands for the complex integration.
First, in Section 2, we reformulate the vortex patch equation by using conformal maps. Indeed, consider an initial doubly-connected domain , with and are two simply-connected domains close to the discs of radii 1 and b respectively. We introduce for the conformal mappings taking the form
Thus, from the contour dynamics equation, rotating doubly-connected V-states amounts to finding non-trivial zeros of the non-linear functional , defined for and by
with
For this aim, we shall implement Crandall–Rabinowitz’s Theorem, starting from the elementary observation that the annulus defined by (1.5) generates a trivial line of solutions for any , which will play the role of the bifurcation parameter. In the same section together with the Appendix B, we also study the regularity of G and prove that it is of class with respect to the functional spaces introduced in Section 2.2. Then, in Section 3, we compute the linearized operator at the equilibrium state and prove that it is a Fourier matrix multiplier. More precisely, for
we have
where
We refer to Proposition 3.1 for more details and point out that some difficulties appear there when computing some integrals related to Bessel functions. Then, the kernel for the linearized operator is non trivial for , as defined in Theorem 1.1, with m large enough. The restriction to higher symmetry is needed first to ensure the condition
required in the transversality condition of Crandall–Rabinowitz’s Theorem and second to get the monotonicity of the sequences (to get a one-dimensional kernel), obtained from tricky asymptotic analysis on the modified Bessel functions. For more details, we refer to Proposition 4.1. We point out that the degenerate case corresponding to where the transversality is no longer true was studied in [23] for Euler equations (). It requires to expand the functional at higher order in order to understand the local structure of the bifurcation diagram. In our case, the dependence of with respect to the parameter b is more involved and similar approach may be implemented with a high computational cost. The previous bifurcation occurs a priori in regularity, but using an elliptic regularity argument, we prove in Lemma 4.1 the analyticity of the boundary for these V-states.
Functional settings
In this section, we shall reformulate the problem of finding V-states looking at the zeros of a nonlinear functional G. We also introduce the function spaces used in the analysis and study some regularity aspects for the functional G with respect to these functions spaces.
Boundary equations
In this subsection we shall obtain the system governing the patch motion. The starting point is the vortex patch equation (1.3), which writes using the complex notation
where is a parametrization of the boundary of . Assuming that the patch is uniformly rotating with an angular velocity Ω, we can choose a parametrization γ in the form
One readily has
Now, to study the second term in the equation (2.1), one needs an explicit formulation of the velocity field v. It has been proved in [10,27] that the velocity field associated to equations writes in the context of vortex patches as an integral on the boundary, namely
where the domain is oriented with the convention “matter on the left” due to Stokes’ Theorem and where is the modified Bessel function of second kind. We shall refer to Appendix A for the definitions and properties of modified Bessel functions. By using (2.2), we obtain
Consequently using again (2.2), we get
Putting together (2.3) and (2.5), the equation (2.1) can be rewritten
Let us assume that our starting domain is doubly-connected, that is
where and are simply-connected bounded open domains of . Then combining (2.4) and (2.6), one should obtain for all ,
where denotes a tangent vector to the boundary at the point z. The minus sign in front of the integral on is here because of the orientation convention for the application of Stokes’ Theorem. Following the works initiated by Burbea, see for instance [5,10,25,26], we should give the equation(s) to solve by using conformal mappings. For this purpose, we shall recall Riemann mapping Theorem.
(Riemann Mapping).
Letdenote the unit open ball andbe a simply connected bounded domain. Then there exists a unique bi-holomorphic map called also conformal map,taking the formwithand.
Notice that in the previous theorem, the domain is only assumed to be simply-connected and bounded. In particular, the existence of the conformal mapping does not depend on the regularity of the boundary. However, information on the regularity of the conformal mapping implies some regularity of the boundary. This is given by the following result which can be found in [37] or in [34, Thm. 3.6].
(Kellogg–Warschawski).
We keep the notations of Riemann mapping Theorem. If the conformal maphas a continuous extension towhich is of classwithand, then the boundaryis a Jordan curve of class.
Assuming that and are respectively small deformations of the discs of radii 1 and b, so that the shape of is close to the annulus defined in (1.5), we shall consider the parametrizations by the conformal mapping satisfying
and
We shall now rewrite the equations by using the conformal parametrizations and . First remark that for , a tangent vector on the boundary at the point is given by
Inserting this into (2.7) and using the change of variables gives
where
with
Then, finding a non trivial uniformly rotating vortex patch for (1.1) reduces to finding zeros of the non-linear functional
As stated in the introduction, these non trivial solutions may be obtained by bifurcation techniques from trivial solutions which are annuli. Let us recover with this formalism that indeed the annuli rotate for any angular velocity. This is given by the following result.
Let. Then the annulusdefined in (
1.5
) is a rotating patch for (
1.1
) for any angular velocity.
Taking by in (2.8), we get
Using the changes of variables and the fact that , we have
Indeed for , we have by (A.3) and the change of variables
Similarly, we find
This proves Lemma 2.1. □
Function spaces and regularity of the functional
We introduce here the function spaces used along this work. Throughout the paper it is more convenient to think of -periodic function as a function of the complex variable . To be more precise, let , be a continuous function, then it can be assimilated to a -periodic function via the relation
Hence, when f is smooth enough, we get
Since and differ only by a smooth factor with modulus one, we shall in the sequel work with instead of which appears more suitable in the computations. In addition, if f is of class and has real Fourier coefficients, then we can easily check that
We shall now recall the definition of Hölder spaces on the unit circle.
Let .
We denote by the space of continuous functions f such that
We denote by the space of functions with α-Hölder continuous derivative
For , we set
and
where
We denote
We can encode the m-fold structure in the functional spaces by setting
and
The spaces and (resp. and ) are equipped with the strong product topology of (resp. ). We also denote
Observe that in the previous function spaces, we imposed the Fourier coefficients to be real. This corresponds to considering 1-fold domains symmetric with respect to the real axis. Due to the rotation invariance of the problem, we can always assume that the axis of symmetry is indeed the real axis. Since we shall look for m-fold solutions, this choice of function spaces is not really restrictive. Nevertheless, a deeper reason is related to the one dimensional kernel condition to be checked in order to apply the Crandall–Rabinowitz Theorem (see Proposition 4.1). If the coefficients were allowed to be complex, this would imply a real dimension of the kernel strictly bigger than 1, which has to be avoided.
We shall now investigate the regularity of the nonlinear functional G defined by (2.8). Indeed, Crandall–Rabinowitz’s Theorem C.1 requires some regularity assumptions to apply and this is what we check here. The ingredients of the proof are classical and they are postponed to the Appendix B.
Let,,and. There existssuch that
is well-defined and of class.
The restrictionis well-defined.
The partial derivativeexists and is continuous.
Spectral study
In this section, we study the linearized operator at the equilibrium state and look for the degeneracy conditions for its kernel.
Linearized operator
In this subsection, we compute the differential and show that it acts as a Fourier multiplier. More precisely, we prove the following proposition.
Let,and. Then for alland for all, if we writewe have for allwhere for all, the matrixis defined bywithandRecall that the modified Bessel functionsandare defined in Appendix
A
.
Since , then for given , we have
But, with the notation introduced in Appendix B, we can write
We write
It has already been proved in [10, Prop. 5.8] that for all ,
where
By a similar calculus, we get
In view of (B.8), we can write
with
By using the change of variables and the fact that , we deduce
Moreover, from (2.10), we know that
So using that
we obtain from (A.3),
Now, by (A.6) and (A.3), one obtains for all ,
Notice that the inversion of symbols of summation and integration is possible due to the geometric decay at infinity given by (A.11). Then, we deduce by (2.11) that
By using the change of variables and the fact that , we infer
But
and
Moreover, by writing the line integral with the parametrization and making the change of variables , we get as in (2.10)
Since , we obtain
An integration by parts together with (3.5) and (3.6) gives
Therefore,
Finally,
Similar computations taking into account the modification with b, change of signs and the fact that yield
According to (B.9), we can write
with
The change of variables implies
But by symmetry and (3.6)
Hence,
By using the change of variables and the fact that , we have
which also writes
We denote
Since , we have
Integrating by parts with (3.5) and using (3.6) yield
Therefore,
Similar computations taking into account the modification with b, change of signs and the fact that imply
Gathering (3.1), (3.2), (3.7), (3.10), (3.3), (3.9), (3.8) and (3.4), we get the desired result. The proof of Proposition 3.1 is now complete. □
Asymptotic monotonicity of the eigenvalues
This subsection is devoted to the proof of Proposition 3.2 concerning the asymptotic monotonicity of the eigenvalues needed to ensure the one dimensional kernel assumption of Crandall–Rabinowitz’s Theorem. But first, we have to prove their existence and this is the purpose of the following lemma.
Letand. There existssuch that for all integer, there exist two angular velocitiesfor which the matrixis singular.
The determinant of is
where
It is a polynomial of degree two in Ω which has at most two roots. Let us compute its discriminant. After straightforward computations, we find
Using the asymptotic expansion of large order (A.10), we infer
As a consequence,
where
We can rewrite as
According to (A.7) and (A.3), we find on , which implies in turn the strict decay property of on . Therefore, since , we get
Now since , we obtain from (A.2),
Finally,
Thus
Therefore, for there exist two angular velocities and for which the matrix is singular. These angular velocities are defined by
This ends the proof of Lemma 3.1. □
We shall now study the monotonicity of the eigenvalues obtained in Lemma 3.1. This is a crucial point to obtain later the one dimensional condition for the kernel of the linearized operator given by Proposition 3.1.
Letand. There existswithwhereis defined in Lemma
3.1
such that
The sequenceis strictly increasing and converges to.
The sequenceis strictly decreasing and converges to.
Then, we have for allwith,
The convergence is an immediate consequence of (3.11), (3.15), (3.16) and (3.14). Then, we turn to the asymptotic monotonicity. For that purpose, we study the sign of the difference
for n large enough.
▶ We first study the difference term before the square roots. We can write
By vitue of (A.11), we deduce
Therefore,
We conclude that
▶ The next task is to look at the asymptotic sign of the difference . We can write
with
By using (A.11), we have
Hence, the following asymptotic expansion holds
As a consequence,
In addition,
and
where is defined in (3.16). From (3.15), (3.16), (3.19), (3.20) and (3.21), we obtain
▶ Combining (3.18) and (3.22), we get
We conclude that there exists such that
i.e. the sequence (resp. ) is strictly increasing (resp. decreasing). This achieves the proof of Proposition 3.2. □
We shall now study both important asymptotic behaviours
The first one corresponds to the Euler case and the second one corresponds to the simply-connected case. We remark that we formally recover (at least partially) [22, Thm. B.] and [10, Thm. 5.1.] looking at these limits. More precisely, we have the following result.
The spectrum is continuous in the following sense.
Let. There existssuch thatwhereis defined in (
1.6
).
Let. There existssuch thatwhereis defined in (
1.7
).
(i) In view of (A.9), we deduce
In what follows, we fix . By virtue of (3.23), the matrices defined in Proposition 3.1, satisfy the following convergence
After straightforward computations, we find
This polynomial of degree two in Ω has the discriminant
Thus, provided , i.e. for
we have two roots
Then, we recover the result found in [22, Thm. B.]. Now, observe that the sequence is decreasing. Then there exists and such that
We use the integral representation (A.8), allowing to write
Now using the integral representation (A.1), we find
The classical inequalities
provide the following estimate for
We conclude that
On the other hand, we set for ,
Remark that (A.5) implies
By the dominated convergence theorem, one has
Now one obtains from (3.6)
Putting together the last two equality with (3.23) yields
Added to (3.6), we have
Then, making appeal to the power series decompositions (A.5) and (A.2), we get
Combining (3.13), (3.25), (3.26) and (3.23) one obtains
Hence, there exists such that
Therefore, we deduce from (3.11) and (3.23) that,
(ii) In what follows, we fix . By using the asymptotic (A.9), we find
Using the power series decomposition (A.2), the decay property of and the asymptotic (3.23), we get
Thus, we obtain from (3.13), (3.25) and (3.23)
Notice that
Consider the function φ defined by , . From (A.4), we get
Hence φ is strictly decreasing on . Moreover, in view of the asymptotic (A.9), we infer
Thus, using also (A.3), we obtain
Therefore, we deduce that there exists such that
In addition, using (A.10) and up to increase the value of one gets
Coming back to (3.27), we infer the existence of such that
Thus, we get from (3.11)
Then, we partially recover the result found in [10, Thm. 5.1.]. We also obtain, up to increase the value of ,
Unfortunately, we cannot prove bifurcation from these eigenvalues. □
Bifurcation from simple eigenvalues
We prove here the following result which implies the main Theorem 1.1 by a direct application of Crandall–Rabinowitz’s Theorem C.1.
Let,,andsuch that. Then the following assertions hold true.
There existssuch thatis well-defined and of class.
(ii) Let . We write
Proposition 3.1 gives
For , we have
Thus, the kernel of is non trivial and it is one dimensional if and only if
The previous condition is satisfied in view of Proposition 3.2. Hence, we have the equivalence
Therefore, we can select as generator of the following pair of functions
(iii) We consider the set defined by
Clearly, is a closed sub-vector space of codimension one in . It remains to prove that it coincides with the range of . Obviously, we have the inclusion
We are left to prove the converse inclusion. Let . We shall prove that the equation
admits a solution in the form (4.1). According to (4.2), the previous equation is equivalent to the following countable set of equations
For , the existence follows from the definition of . Thanks to (4.3), the sequences and are uniquely determined by
or equivalently,
It remains to prove the regularity, that is . For that purpose, we show
We may focus on the first component, the second one being analogous. We set
and
If we denote , then we can write
where
The convolution must be understood in the usual sense, that is
We shall use the classical convolution law
By using the decay property of the product and the asymptotic (A.9), we have
We also have
Hence
We now prove that and are with regularity .
▶ Regularity of:
First observe that by Cauchy–Schwarz inequality and the embedding , we have
We now have to prove that . We show that it coincides, up to slight modifications, with which is of regularity . For that purpose, we show that we can differentiate term by term.
We denote (resp. ) the sequence of the partial sums (resp. the sequence of the remainders) of the series of functions . One has
Using Cauchy–Schwarz inequality, we obtain similarly to (4.7)
Hence
One has
We set
By continuity of the Szegö projection defined by
from into itself (see [17] for more details) added to the fact that , we deduce that . Applying Bernstein Theorem of Fourier series gives that is the uniform limit of its Fourier series, namely
Gathering (4.8) and (4.9), we conclude that we can differentiate term by term and get
As a consequence,
▶ Regularity of:
By using (3.12) and (A.11), we have the asymptotic expansion
with, using Proposition 3.2,
and, using (3.16),
We denote
We can write
Thus we can write
Now since , we have
By using the link regularity/decay of Fourier coefficients, we deduce that
Similarly to (4.10), we can obtain
By the same method, we can also differentiate term by term and obtain
Notice that from (4.12), we can write
where
Using again the continuity of the Szegö projection, we have
Using (4.15), (4.16) and (4.5), we deduce that
Thus
Gathering (4.14), (4.17) and (4.15), we conclude that
Putting together (4.4), (4.18), (4.10), (4.6) and (4.5), we finally conclude
(iv) is a simple eigenvalue since . From (B.1) and (B.2), we deduce
Thus,
Notice that the previous expression belongs to the range of if and only if the vector
is a scalar multiple of one column of the matrix . This occurs if and only if
Putting (4.19) together with implies
Now remark that the above equation is equivalent to
Since and , then in view of (4.19), the first equation can’t be solved. Then, necessary, the second equation must be satisfied. But we notice that it corresponds to a multiple eigenvalue (), which is excluded here. Therefore, we conclude that
This ends the proof of Proposition 4.1. □
The previous proposition allows to construct, for any fixed , , and two branches of m-fold doubly-connected V-states with regularity bifurcating from the annulus at the angular velocities for the equations. Actually, we have the following better result for the regularity of the boundary.
Let,and. Consider am-fold doubly-connected V-state close toforequations, rotating with an angular velocity Ω and associated with an initial domain, whereandare simply-connected domains satisfyingand parametrized by the following conformal mappingsIfis small enough, then the boundariesandare analytic.
The proof is done in the spirit of [20, Section 5.4] by applying [30, Thm. 3.1’]. We highlight that the positive number r quantifies the smallness of and in the topology. We mention that (2.6) can also be written as follows
where Ψ is the velocity potential given by
Therefore, integrating the relation (4.20), there exists for each a constant such that
Fix . By compactness of , there exist , and (small) such that we can write
Fix and denote
Solving the Helmoltz problem (4.21) as in [27], the stream function writes
where denotes the planar Lebesgue measure. From (A.5) and (A.2), we can write
where , are bounded at 0 and is integrable at the origin. Notice that corresponds to the classical Euler velocity potential. Since is of regularity then one can classically prove that
For instance, the regularity is obtained by using [14, Exercise 4.8 (a)]. As for the regularity, one may use in particular the “Main Lemma” in [33] applied to the Calderón–Zygmund type operator . The term being less singular, we get
and then
One can easily find from (4.21) that
Observe that the functions and are analytic. Thus it remains to prove that
where is a normal unitary vector to . We can write
The normal unitary vector can be expressed as follows in terms of the conformal mapping
On one hand, denoting and , we have for ,
On the other hand,
where
We shall now prove that the terms , and are small. Let us start with . Recalling the notation (B.3), one has
From (B.6), we get
Now fix and denote
We mention that the triangle inequality and the mean value theorem imply that is bi-Lipschitz, namely
Recall that behaves like a logarithm at 0 and using (A.5) we can write
Therefore, for any , we have
Thus, applying [17, Lem. 1], we infer
Putting together (4.26), (4.27) and (4.30), we deduce
From the previous computations, one also obtains
As for , we may use Taylor formula to write
The triangular inequality and the mean value theorem imply
Hence using (4.29), (4.28), (B.4) and (B.5), we deduce
Moreover, according to the computations carried out in Proposition 3.1 (see also [27, Lem. 3.2]), we have
Therefore, in view of (4.25), (4.31), (4.32), (4.33), (4.34) and Proposition 3.2, we infer
and
Combining (4.23), (4.24), (4.35) and (4.36), we deduce by triangular inequality
and
The Crandall–Rabinowitz Theorem implies that Ω is close to . Hence, according to Proposition 3.2, we can say
Thus, up to take r sufficiently small, we get (4.22). Consequently, is analytic from which we deduce by reconstruction that is also analytic. □
Footnotes
Formulae on modified Bessel functions
We shall collect some useful information on modified Bessel functions. For more details we refer to [1,38]. We define first the Bessel functions of order by
Notice that when we have the following integral representation, see [32, p. 115].
We define the Bessel functions of imaginary argument by
and
For , we define . We give now useful properties of modified Bessel functions.
Proof of Proposition 2.1
In this appendix, we prove the regularity result stated in Proposition 2.1. The techniques involved are now classical and the following proof follows closely the lines of the proof of [8, Prop. 4.1].
(i) The proof proceeds in three steps. The first step is to show the well-posedness of the function from to for some r small enough. Then, in the second step, we shall prove the existence and give the computation of the Gâteaux derivative of . Finally, in the third step, we shall prove that these Gâteaux derivatives are continuous. This will show the regularity of .
▶ Step 1: Show thatis well-defined:
For this purpose, we split into two terms, the self-induced term and the interaction term ,
where
➢ We refer to [10, Prop. 5.7] for the study of . Only the differs, but has no consequence. We recall here the results. There exists such that for all , we have
is of class .
The restriction is well-defined.
Moreover, we have
where
Actually, this is the most difficult part of this proof since in this case, the integrals appearing have singular kernel and the proof uses some results about singular kernels. As we shall see in the remaining of the proof, the terms concerning are not singular.
➢ We shall first show that for , we have . According to the algebra structure of , it suffices to show that for , . For that purpose, we consider the operator defined by
But for , we have taking and small functions,
and
Since is continuous on , we have
Moreover, taking , we have by mean value Theorem, since from (A.4) is continuous on , and left triangle inequality
Using that , we conclude that
We deduce that
Applying this with , we find
The last point to check is that the Fourier coefficients of are real. According to the definition of the space , the mapping has real coefficients. We deduce that the Fourier coefficients of are also real. Due to the stability of such property under conjugation and multiplication, we only have to prove that the Fourier coefficients of are real. This is checked by the following computations. By using (A.3) and the change of variables , one has
▶ Step 2: Show the existence and compute the Gâteaux derivatives of:
➢ The Gâteaux derivative of at in the direction is given by
The previous limits are understood in the sense of the strong topology of . As a consequence, we need to prove first the pointwise existence of these limits and then we shall check that these limits exist in the strong topology of . To be able to compute the Gâteaux dérivatives, we have to precise that since the beginning of this study we have identified with . Hence is naturally endowed with the Euclidean scalar product which writes for and
By straightforward computations, we infer
where
and
Since B differs from A only with a conjugation, then, they both satisfy the same estimates in the coming analysis. For all , we have
So
Let . let . Then
But by right and left triangle inequalities, we get
Hence,
Thus,
We conclude that,
which means that .
➢ Concerning the other differentiation, we have
Using the algebra structure of , we obtain
From (B.6), we find
In the same way as for , we infer
Gathering the foregoing computations leads to
that is, .
➢ The last thing to check is that the convergence in (B.7) occurs in the strong topology of . Since there are many terms involved, we shall select the more complicated one and study it. The other terms can be treated in a similar way, up to slight modifications. Let us focus on the first term of the right-hand side of (B.8). We shall prove,
For more convenience, we use the following notation
Consider such that . According to (2.9), we get
Applying mean value Theorem and left triangle inequality, we obtain
Consequently,
This implies that
Let us now consider . In view of the mean value Theorem, one obtains the following estimate
Now remark that we can write
According to (2.11), one obtains
After straightforward computations, we obtain for ,
As a consequence, we infer
Coming back to (B.10) and using the fact that , we conclude
Therefore,
The second step is now achieved.
▶ Step 3: Show that the Gâteaux derivatives ofare continuous:
Now we investigate for the continuity of the Gâteaux derivatives seen as operators from the neighborhood into the Banach space . Using the algebra structure of , we deduce from (B.9) and (B.8) that we only have to study the continuity of the terms , , , , and . As before, we shall focus on the term for and remark that the other terms are similar. We denote
with and . Let us show that
According to (2.9), we get
where
We have directly
Now set
By a new use of the mean value Theorem and left triangle inequality, we obtain
Hence, we deduce
Take . Applying the mean value Theorem yields
By (2.11), we have
where
Notice that it can be written in the following form
with
By the same techniques as already used above, we get
We deduce that
(ii) Looking at Proposition 2.1, it is sufficient to prove the preservation of the m-fold symmetry. Let r be as in Proposition 2.1. Let . Let and be the associated conformal maps
One easily obtains
Hence, by using the change of variables , we have for all and for all ,
By definition (2.8) of , this immediately implies that
So
(iii) Fix . By (B.1) and (B.2), we have for and ,
As a consequence, we deduce that for and ,
This proves the continuity of and achieves the proof of Proposition 2.1. □
Crandall–Rabinowitz’s Theorem
Now, we recall the classical Crandall–Rabinowitz’s Theorem. This result was first proved in [7] and it is one of the most common theorems appearing in the bifurcation theory. A convenient reference in the subject is [29]. We briefly explain the core of local bifurcation theory.
Consider a function with X and Y two Banach spaces. Assume that for all Ω in a non-empty interval I we have . This provides a line of solutions
Now take some with . The implicit function Theorem explains that if is invertible, then the line is the only curve of solutions close to , i.e. for ε small enough. (Local) bifurcation theory is the study of situations where this is not true, that is, close to there exists (at least) another line of solutions. In this case, we say that is a bifurcation point. Crandall–Rabinowitz’s Theorem gives sufficient conditions to construct a bifurcation curve and states as follows.
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