Waves of maximal height for a class of nonlocal equations with homogeneous symbols
Abstract.
We discuss the existence and regularity of periodic travelingwave solutions of a class of nonlocal equations with homogeneous symbol of order , where . Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic travelingwave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol . Thereby we recover its unique highest periodic, peaked travelingwave solution, having the property of being exactly Lipschitz at the crest.
Key words and phrases:
Highest wave; singular solution; nonlocal equation with homogeneous symbol; fractional KdV equation2010 Mathematics Subject Classification:
35B10, 35B32, 35B65, 35S30, 45M151. Introduction
The present study is concerned with the existence and regularity of a highest, periodic travelingwave solution of the nonlocal equation
(1.1) 
where denotes the Fourier multiplier operator with symbol , . Equation (1.1) is also known as the fractional Korteweg–de Vries equation. We are looking for periodic travelingwave solutions , where denotes the speed of the rightpropagating wave. In this context equation (1.1) reduces after integration to
(1.2) 
where is an integration constant. Since the symbol of is homogeneous, any bounded solution of the above equation has necessarily zero mean; in turn this implies that the integration constant is uniquely determined to be
The question about singular, highest waves was already raised by Stokes. In 1880 Stokes conjectured that the Euler equations admit a highest, periodic travelingwave having a corner singularity at each crest with an interior angle of exactly . About 100 years later (in 1982) Stokes’ conjecture was answered in the affirmative by Amick, Fraenkel, and Toland [AFT]. Subject of a recent investigation by Ehrnström and Wahlén [EW] is the existence and precise regularity of a highest, periodic travelingwave solution for the Whitham equation; thereby proving Whitham’s conjecture on the existence of such a singular solution. The (unidirectional) Whitham equation is a genuinely nonlocal equation, which can be recovered from the well known Korteweg–de Vries equation by replacing its dispersion relation by one branch of the full Euler dispersion relation. The resulting equation takes (up to a scaling factor) the form of (1.1), where the symbol of the Fourier multiplier is given by . In order to prove their result, Ehrnström and Wahlén developed a general approach based on the regularity and monotonicity properties of the convolution kernel induced by the Fourier multiplier. The highest, periodic travelingwave solution for the Whitham equation is exactly Hölder continuous at its crests; thus exhibiting exactly half the regularity of the highest wave for the Euler equations. In a subsequent paper, Ehrnström, Johnson, and Claasen [EJC] studied the existence and regularity of a highest wave for the bidirectional Whitham equation incorporating the full Euler dispersion relation leading to a nonlocal equation with cubic nonlinearity and a Fourier multiplier with symbol . The question addressed in [EJC] is whether this equation gives rise to a highest, periodic, traveling wave, which is peaked (that is, whether it has a corner at each crest), such as the corresponding solution to the Euler equations? Overcoming the additional challenge of the cubic nonlinearity, the authors in [EJC] follow a similar approach as implemented for the Whitham equation in [EW] and prove that the highest wave has a singularity at its crest of the form ; thereby still being a cusped wave. Concerning a different model equation arising in the context of shallowwater equations, Arnesen [A] investigated the existence and regularity of a highest, periodic, travelingwave solution for the Degasperis–Procesi equation. The Degasperis–Procesi equation is a local equation, but it can also be written in a nonlocal form with quadratic nonlinearity and a Fourier multiplier with symbol , which is acting itself –in contrast to the previously mentioned equations– on a quadratic nonlinearity. For the Degasperis–Procesi and indeed for all equations in the socalled bfamily (the famous Camassa–Holm equation being also such a member), explicit peaked, periodic, travelingwave solutions are known [CH, DHK]. Using the nonlocal approach introduced originally for the Whitham equation in [EW], the author of [A] adapts the method to the nonlocal form of the Degasperis–Procesi equation and recovers not only the existence of a highest, peaked, periodic traveling wave, but also proves that any even, periodic, highest wave of the Degasperis–Procesi equation is exactly Lipschitz continuous at each crest; thereby excluding the existence of even, periodic, cusped travelingwave solutions.
Of our concern is the existence and regularity of highest, traveling waves for the fractional Korteweg–de Vries equation (1.1), where . In the case when , (1.1) can be viewed as the nonlocal form of the reduced Ostrovsky equation
For the reduced Ostrovsky equation, a highest, periodic, peaked travelingwave solution is known explicitly [Ostrovsky1978] and its regularity at each crest is exactly Lipschitz continuous. Recently, the existence and stability of smooth, periodic travelingwave solutions for the reduced Ostrovsky equation, was investigated in [GP, HSS]. In [GP2], the authors prove that the (unique) highest, periodic travelingwave solutions of the reduced Ostrovsky equation is linearly and nonlinearly unstable. We are going to investigate the existence and precise regularity of highest, periodic travelingwave solutions of the entire family of equations for Fourier multipliers , where . Based on the nonlocal approach introduced for the Whitham equation [EW], we adapt the method in a way which is convenient to treat homogeneous symbols, and prove the existence and precise Lipschitz regularity of highest, periodic, travelingwave solutions of (1.1) corresponding to the symbol , where . The advantage of this nonlocal approach relies not only in the fact that it can be applied to various equations of local and nonlocal type, but in particular, that it is suitable to study entire families of equations simultaneously; thereby providing an insight into the interplay between a certain nonlinearity and varying order of linearity. The main novelty in our work relies upon implementing the approach used in [EW, EJC, A] for equations exhibiting homogeneous symbols. For a homogeneous symbol, the associated convolution kernel can not be identified with a positive, decaying function on the real line. Instead we have to work with a periodic convolution kernel. The lack of positivity of the kernel can be compensated by working within the class of zero mean function, though. Moreover, we affirm that starting with a linear operator of order strictly smaller than in equation (1.1) a further decrease of order does not affect the regularity of the corresponding highest, periodic travelingwave.
1.1. Main result and outline of the paper
Let us formulate our main theorem, which provides the existence of a global bifurcation branch of nontrivial, smooth, periodic and even travelingwave solutions of equation (1.1), which reaches a limiting peaked, precisely Lipschitz continuous, solution at the end of the bifurcation curve.
Theorem 1.1 (Main theorem).
For each integer there exists a wave speed and a global bifurcation branch
of nontrivial, periodic, smooth, even solutions to the steady equation (1.2) for , emerging from the bifurcation point . Moreover, given any unbounded sequence of positive numbers , there exists a subsequence of , which converges uniformly to a limiting travelingwave solution that solves (1.2) and satisfies
The limiting wave is strictly increasing on and exactly Lipschitz at .
It is worth to notify that the regularity of peaked travelingwave solutions is Lipschitz for all . The reason mainly relies in the smoothing properties of the Fourier multiplier, which is of order strictly bigger than , see Theorem 4.6.
The outline of the paper is as follows: In Section 2 we introduce the functionalanalytic setting, notations, and some general conventions. Properties of general Fourier multipliers with homogeneous symbol and a representation formula for the corresponding convolution kernel are discussed in Section 3. Section 4 is the heart of the present work, where we use the regularity and monotonicity properties of the convolution kernel to study a priori properties of bounded, traveling wave solutions of (1.1). In particular, we prove that an even, periodic travelingwave solution , which is monotone on a half period and whose maximum equals the wave speed, is precisely Lipschitz continuous. Eventually, in Section 5 we investigate the global bifurcation result. By excluding certain alternatives for the bifurcation curve, we conclude the main theorem. In Section 6 we apply our result to the reduced Ostrovsky equation, which can be reformulated as a nonlocal equation of the form (1.2) with Fourier symbol . We recover the well known explicit, even, peaked, periodic travelingwave given by
on and extended periodically. Moreover, we prove that any periodic travelingwave is at least Lipschitz continuous at its crests; thereby excluding the possibility of periodic, travelingwaves exhibiting a cusp at its crests. Let us mention that the Fourier multiplier for the reduced Ostrovsky equation can be written as a convolution operator, whose kernel can be computed explicitly, see Remark 3.7. Furthermore, relying on a priori bounds on the wave speed coming from a dynamical system approach for the reduced Ostrovsky equation in [GP], we are able to obtain a better understanding of the behavior of the global bifurcation branch.
2. Functionalanalytic setting and general conventions
Let us introduce the relevant function spaces for our analysis and fix some notation. We are seeking for periodic solutions of the steady equation (1.2). Let us set , where we identify with . In view of the nonlocal approach via Fourier multipliers, the Besov spaces on torus form a natural scale of spaces to work in. We recall the definition and some basic properties of periodic Besov spaces.
Denote by the space of test functions on , whose dual space, the space of distributions on , is . If is the space of rapidly decaying functions from to and denotes its dual space, let be the Fourier transformation on the torus defined by duality on via
Let be a family of smooth, compactly supported functions satisfying
and for any , there exists a constant such that
For and , the periodic Besov spaces are defined by
with the common modification when ^{1}^{1}1One can show that the above definition is independent of the particular choice of . If and , then
Moreover, for , the Besov space consisting of functions satisfying
is called periodic Zygmund space of order and we write
Eventually, for , we denote by the space of Hölder continuous functions on . If and , then denotes the space of times continuously differentiable functions whose th derivative is Hölder continuous on . To lighten the notation we write for .
As a consequence of Littlewood–Paley theory, we have the relation for any with ; that is, the Hölder spaces on the torus are completely characterized by Fourier series. If , then is a proper subset of and
Here, denotes the space of Lipschitz continuous functions on . For more details we refer to [T3, Chapter 13].
We are looking for solutions in the class of periodic, bounded functions with zero mean, the class being denoted by
In the sequel we continue to use the subscript to denote the restriction of a respective space to its subset of functions with zero mean.
If and are elements in an ordered Banach space, we write () if there exists a constant such that (). Moreover, the notation is used whenever and . We denote by the nonnegative real half axis and by the set of natural numbers including zero. The space denotes the set of all bounded linear operators from to .
3. Fourier multipliers with homogeneous symbol
The following result is an analogous statement to the classical Fourier multiplier theorems for nonhomogeneous symbols on Besov spaces (e.g. [BCD, Proposition 2.78]):
Proposition 3.1.
Let and be a function, which is smooth outside the origin and satisfies
Then, the Fourier multiplier defined by
belongs to the space .
Proof.
In view of the zero mean property of , the proof can be carried out in a similar form as in [AB, Theorem 2.3 (v)], where it is show that a function belongs to if and only if
∎
The above proposition yields in particular that
defines a bounded operator form to for any ; thereby it is a smoothing operator of order .
We are interested in the existence and regularity properties of solutions of
(3.1) 
The operator is defined as the inverse Fourier representation
where for and . In view of the convolution theorem, we define the integral kernel
(3.2) 
so that the action of is described by the convolution
One can then express equation (3.1) as
In what follows we examine the kernel . We start by recalling some general theory on completely monotonic sequences taken from [Guo, Widder].
Definition 3.2.
A sequence of real numbers is called completely monotonic if its elements are nonnegative and
where and .
Definition 3.3.
A function is called completely monotone if it is continuous on , smooth on the open set , and satisfies
For completely monotonic sequences we have the following theorem, which can be considered as the discrete analog of Bernstein’s theorem on completely monotonic functions.
Theorem 3.4 ([Widder], Theorem 4a).
A sequence of real numbers is completely monotonic if and only if
where is nondecreasing and bounded for .
There exists a close relationship between completely monotonic sequences and completely monotonic functions.
Lemma 3.5 ([Guo], Theorem 5).
Suppose that is completely monotone, then for any the sequence is completely monotonic.
We are going to use the theory on completely monotonic sequences to prove the following theorem, which summarizes some properties of the kernel .
Theorem 3.6 (Properties of ).
Let . The kernel defined in (3.2) has the following properties:

is even, continuous, and has zero mean.

is smooth on and decreasing on .

for any . In particular, is integrable and is Hölder continuous with if , and continuously differentiable if .
Proof.
Claim a) follows directly form the definition of and . Now we want to prove part b). Set
Clearly is completely monotone on . Thus, Lemma 3.5 guarantees that is a completely monotonic sequence. By Theorem 3.4, there exists a nondecreasing and bounded function such that
In particular
The coefficients can be written as
where
for some bounded function . Thereby,
In particular, we deduce that
Notice that we can compute explicitly as
Thus, for , we have that
Consequently, on the interval , the kernel is represented by
From here it is easy to deduce that is smooth on and decreasing on , which completes the proof of b). Regarding the regularity of claimed in c), let be arbitrary. On the subset of zero mean functions of an equivalent norm is given by
Thereby, is in if and only if the function
is integrable over . Now, this follows by a classical theorem on the integrability of trigonometric transformations (cf. [Boas, Theorem 2] ), and we deduce the claimed regularity and integrability of . The continuity properties are a direct consequence of Sobolev embedding theorems, see [Demengel, Theorem 4.57]. ∎
Remark 3.7.
The proof of Theorem 3.6 includes a general approach on the relation between the symbol and the monotonicity property of the corresponding Fourier multiplier. However, there exists even a more explicit expression of in terms of the Gamma function (cf. [PBM, Section 5.4.3]) given by
Moreover, we would like to point out that if , , we have that
where is the th Bernoulli polynomial. If (which corresponds to the case of the reduced Ostrovsky equation), or , then , , and
Lemma 3.8.
Let . The operator is parity preserving on . Moreover, if are odd functions satisfying on , then either
or on .
Proof.
The fact that is parity preserving is an immediate consequence of the evenness of the convolution kernel. In order to prove the second assertion, assume that are odd, satisfying on and that there exists such that . Using the zero mean property of and , we obtain that
where denotes the minimum of on . In view of being nonconstant and for all , we conclude that
which is a contradiction unless on . ∎
4. A priori properties of periodic travelingwave solutions
In the sequel, let be fixed. We consider periodic solutions of
(4.1) 
The existence of solutions is subject of Section 5, where we use analytic bifurcation theory to first construct small amplitude solutions and then extend this bifurcation curve to a global continuum terminating in a highest, traveling wave. Aim of this section is to provide a priori properties of travelingwave solutions . In particular, we show that any nontrivial, even solution , which is nondecreasing on the half period and attaining its maximum at is precisely Lipschitz continuous. This holds true for any , see Theorem 4.7.
We would like to point out that the subsequent analysis can be carried out in the very same manner for periodic solutions, where is the length of a finite half period.
Let us start with a short observation.
Lemma 4.1.
Proof.
In what follows it is going to be convenient to write (4.1) as
(4.2) 
In the next two lemmata we establish a priori properties of periodic solutions of (4.2) requiring solely boundedness.
Lemma 4.2.
Let be a solution of (4.2), then and
Proof.
Lemma 4.3.
Let be a solution of (4.2), then
Proof.
If , there is nothing to prove. Therefore it is enough to assume that is a nontrivial solution. From Lemma 4.2 we know that is a continuously differentiable function. In view of being a function of zero mean and being continuous, we deduce the existence of such that
By the mean value theorem, we obtain that
for some and
where we used that has zero mean. Again by Lemma 4.2 we can estimate the term above generously by
Using that solves (4.1), we obtain
Dividing by yields the statement.
∎
From now on we restrict our considerations on periodic solutions of (4.2), which are even and nondecreasing on the half period .
Lemma 4.4.
Proof.
Assuming that we can take the derivative of (4.2) and obtain that
Due to the assumption that on it is sufficient to show that
(4.3) 
to prove the statement. In view of being odd with on , the desired inequality (4.3) follows from Lemma 3.8. In order to prove the second statement, let us assume that . Differentiating (4.2) twice yields
In particular, we have that
We are going to show that , which then (together with the first part) proves the statement. Using the evenness of and , we compute
Notice that the first integral on the right hand side tends to zero if , so does the second term in view of being differentiable and continuous on . Concerning the last integral, we observe that
since and are negative on . ∎
We continue by showing that any bounded solution of (4.2) that satisfies is smooth.
Theorem 4.5.
Let be a bounded solution of (4.2). Then:

If uniformly on , then .

Considering as a periodic function on it is smooth on any open set where .
Proof.
Let uniformly on . Recalling Proposition 3.1, we know that the operator maps into for any . Moreover, if then the Nemytskii operator
maps into itself for . From (4.2) we see that for any solution we have
Thus,
(4.4) 
for all . Eventually, (4.2) gives rise to
Hence, an iteration argument in guarantees that . In order to prove the statement on the real line, recall that any Fourier multiplier commutes with the translation operator. Thus, if is a periodic solution of (4.2), then so is for any . The previous argument implies that for any , which proves statement (i). In order to prove part (ii) let be an open subset of on which . Then, we can find an open cover , where for any we have that is connected and satisfies . Due to the translation invariance of (4.2) and part (i), we obtain that is smooth on for any . Since is the union of open sets, the assertion follows. ∎
Theorem 4.6.
Let be an even solution of (4.2), which is nondecreasing on . If attains its maximum at , then cannot belong to the class .
Proof.
Assuming that , the same argument as in Lemma 4.2 implies that the function is twice continuously differentiable and its Taylor expansion in a neighborhood of is given by
(4.5) 
for some where . Since attains a local maximum at , its first derivative above vanishes at the origin whereas the second derivative is given by
We aim to show that in a small neighborhood of zero the right hand side is strictly bounded away from zero. Set . Using that and are even functions with and being negative on , we find that
for some constant . Since is even (cf. Lemma 3.8) and continuous, there exists and a constant such that
Thus, considering the Taylor series (4.5) in a neighborhood of zero, we have that
which in particular implies that
Passing to the limit , we obtain a contradiction to . ∎
We are now investigating the precise regularity of a solution , which attains its maximum at .
Theorem 4.7.
Let be an even solution of (4.2), which is nondecreasing on . If attains its maximum at , then the following holds:

and is strictly increasing on .

, that is is Lipschitz continuous.

is precisely Lipschitz continuous at , that is
(4.6)
Proof.

Assume that is a solution which is even and nondecreasing on . Let and . Notice that by periodicity and evenness of and the kernel , we have that
The integrand is nonnegative, since for and for and by assumption that is even and nondecreasing on . Since is a nontrivial solution and is not constant, we deduce that
(4.7) for any . Moreover, we have that
for any . Hence if and only if . In view of (4.7), we obtain that
Thereby, is strictly increasing on . In view of Theorem 4.5, is smooth on .

In order to prove the Lipschitz regularity at the crest, we make use of a simple bootstrap argument. We would like to emphasize that the following argument strongly relies on the fact that we are dealing with a smoothing operator of order , where . Let us assume that is not Lipschitz continuous and prove a contradiction. If is merely a bounded function, the regularization property of implies immediately the is a priori Hölder continuous. To see this, recall that
Using