Benjamin-Ono Equation on a Half-Line
We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
In this paper we study the large time asymptotic behavior of solutions to the initial-boundary value problem for the Benjamin-Ono equation on a half-line:
where is the Hilbert transformation, and means the principal value of the singular integral. We note that in the case of the whole line we have the relations since the operator can be written as follows: where is the usual Fourier transform, and denotes the inverse Fourier transform. This equation is of great interest in many areas of Physics (see [1, 2]). The Cauchy problem (1.1) was studied by many authors. The existence of solutions in the usual Sobolev spaces was proved in [3–9] and the smoothing properties of solutions were studied in [10–14]. In paper  it was proved that for small initial data in solutions decay as in norm at the same rate as for the case of the linear Benjamin-Ono equation, where
The initial-boundary value problem (1.1) plays an important role in the contemporary mathematical physics. For the general theory of nonlinear equations on a half-line we refer to the book , where it was developed systematically a general theory of the initial-boundary value problems for nonlinear evolution equations with pseudodifferential operators on a half-line, where pseudodifferential operator on a half-line was introduced by virtue of the inverse Laplace transformation of the product of the symbol which is analytic in the right complex half-plane, and the Laplace transform of the derivative . Thus, for example, in the case of we get the following definition of the fractional derivative
Here and below is the main branch of the complex analytic function in the complex half-plane , so that (we make a cut along the negative real axis Note that due to the analyticity of for all the inverse Laplace transform gives us the function which is equal to for all In spite of the importance and actuality there are few results about the initial-boundary value problem for pseudodifferential equations with nonanalytic symbols. For example, in paper  there was considered the case of rational symbol which have some poles in the right complex half-plane. There was proposed a new method for constructing the Green operator based on the introduction of some necessary condition at the singularity points of the symbol . In the paper  one of the authors considered the initial-boundary value problem for a pseudodifferential equation with symbol and nonlinearity
As far as we know the case of nonanalytic conservative symbols was not studied previously. In the present paper we fill this gap, considering as example the Benjamin-Ono equation (1.1) with a symbol . There are many natural open questions which we need to study. First we consider the following question: how many boundary data we should pose on problem (1.1) for its correct solvability? Also we study traditionally important problems of a theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. We adopt here the approach of book  based on the estimates of the Green function. The main difficulty for nonlocal equation (1.1) on a half-line is that the symbol is non analytic in the complex plane. Therefore we cannot apply the Laplace theory directly. To construct Green operator we proposed a new method based on the integral representation for sectionally analytic function and theory of singular integrodifferential equations with Hilbert kernel and the discontinues coefficients (see [18, 19]).
To state precisely the results of the present paper we give some notations. We denote . Direct Laplace transformation is
and the inverse Laplace transformation is defined by
Weighted Lebesgue space is where
for , and
Sobolev space is
We define a linear functional :
Now we state the main results.
Theorem 1.1. Suppose that the initial data with are such that the norm is sufficiently small. Then there exists a unique global solution to the initial-boundary value problem (1.1). Moreover the following asymptotic is valid in for , where is defined below by the formula (2.191), and the constant
Remark 1.2. Note that the time decay rate of the solution is faster comparing with the case of the corresponding Cauchy problem. So the nonlinearity in (1.1) is not the super critical case for our problem.
Remark 1.3. In the case of the negative half line we expect that the solutions have an oscillation character, and the time decay rate of the solution is the same as the case of the corresponding Cauchy problem. so the nonlinearity in (1.1) will be the super critical case.
In subsequent consideration we will have frequently to use certain theorems of the theory of functions of complex variable, the statements of which we now quote. The proofs can be found in .
Theorem 2.1. Let be a complex function, which obeys the Hölder condition for all finite and tends to a definite limit as such that for large the following inequality holds: Then Cauchy type integral constitutes a function analytic in the left and right semiplanes. Here and below these functions will be denoted and , respectively. These functions have the limiting values and at all points of imaginary axis , on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formula:
Subtracting and adding the formula (2.3) we obtain the following two equivalent formulas:
which will be frequently employed hereafter.
Theorem 2.2. An arbitrary function given on the contour , satisfying the Hölder condition, can be uniquely represented in the form where are the boundary values of the analytic functions and the condition holds. These functions are determined by formula
Theorem 2.3. An arbitrary function given on the contour , satisfying the Hölder condition, and having zero index, is uniquely representable as the ratio of the functions and , constituting the boundary values of functions, and analytic in the left and right complex semiplane and having in these domains no zero. These functions are determined to within an arbitrary constant factor and given by formula
We consider the following linear initial-boundary value problem on half-line
where for , we define
where the function is given by formula
for Here and below
and are a left and right limiting values of sectionally analytic function given by
All the integrals are understood in the sense of the principal values.
Proposition 2.4. Let the initial data be . Then there exists a unique solution of the initial-boundary value problem (2.9), which has integral representation
Proof. To derive an integral representation for the solutions of the problem (2.9) we suppose that there exists a solution of problem (2.9), which is continued by zero outside of :
Let be a function of the complex variable which obeys the Hölder condition for all finite and tends to as We define the operator
Since the operator is defined by a Cauchy type integral, it is readily observed that constitutes a function analytic in the entire complex plane, except for points of the contour of integration . Also by Sokhotzki-Plemelj formula we have for Here and are limits of as tends to from the left and right semi-plane, respectively.
We have for the Laplace transform Since is analytic for all we have Therefore applying the Laplace transform with respect to to problem (2.9) we obtain for where We rewrite (2.22) in the form with some function such that for all and for Applying the Laplace transformation with respect to time variable to problem (2.24) we find for Here the functions and are the Laplace transforms for and with respect to time, respectively. We will find the function using the analytic properties of function in the right-half complex planes Re and We have for In view of Sokhotzki-Plemelj formula via (2.27) the condition (2.28) can be written as where the sectionally analytic functions and are given by Cauchy type integrals: To perform the condition (2.29) in the form of nonhomogeneous Riemann problem we introduce the sectionally analytic function: where Taking into account the assumed condition (2.25) and making use of Sokhotzki-Plemelj formula (2.3) we get for limiting values of the functions and Also observe that from (2.30) and (2.32) by formula (2.4) Substituting (2.29) and (2.34) into this equation we obtain nonhomogeneous Riemann problem
It is required to find two functions for some fixed point , : , analytic in and , analytic in , which satisfy on the contour the relation (2.36). Here, for some fixed point , , the functions are called the coefficient and the free term of the Riemann problem, respectively.
Note that bearing in mind formula (2.33) we can find unknown function which involved in the formula (2.27) by the relation The method for solving the Riemann problem is based on the Theorems 2.2 and 2.3.
In the formulations of Theorems 2.2 and 2.3 the coefficient and the free term of the Riemann problem are required to satisfy the Hölder condition on the contour . This restriction is essential. On the other hand, it is easy to observe that both functions and do not have limiting value as . The principal task now is to get an expression equivalent to the boundary value problem (2.36), such that the conditions of theorems are satisfied. First, let us introduce some notation and let us establish certain auxiliary relationships. Setting we introduce the function where for some fixed point () We make a cut in the plane from point to point through Owing to the manner of performing the cut the functions , are analytic for and the function is analytic for
We observe that the function given on the contour satisfies the Hölder condition and under the assumption does not vanish for any . Also we have Therefore in accordance with Theorem 2.3 the function can be represented in the form of the ratio where Now we return to the nonhomogeneous Riemann problem (2.36). Multiplying and dividing the expression by and making use of the formula (2.43) we get where Replacing in (2.36) the coefficient of the Riemann problem by (2.45) we reduce the nonhomogeneous Riemann problem (2.36) to the form Now we perform the function given by formula (2.31) as where Firstly we calculate the left limiting value Since there exists only one root of equation such that for all therefore, taking limit from the left-hand side of complex plane, by Cauchy theorem we get The last relation implies that can be expressed by the function which is analytic in where By Sokhotzki-Plemelj formula (2.4) we express the left limiting value in the term of the right limiting value as where Bearing in mind the representation (2.48) and making use of (2.52), (2.55), and (2.45) after simple transformations we get where Replacing in (2.47) by (2.56), we reduce the nonhomogeneous Riemann problem (2.47) in the form where In subsequent consideration we will have to use the following property of the limiting values of a Cauchy type integral, the statement of which we now quote. The proofs may be found in .Lemma 2.5. If is a smooth closed contour and a function that satisfies the Hölder condition on , then the limiting values of the Cauchy type integral also satisfy this condition.
Since satisfies on the Hölder condition, on basis of this Lemma the function also satisfies this condition. Therefore in accordance with Theorem 2.2 it can be uniquely represented in the form of the difference of the functions and , constituting the boundary values of the analytic function , given by formula Therefore the problem (2.58) takes the form The last relation indicates that the function analytic in and the function analytic in constitute the analytic continuation of each other through the contour Consequently, they are branches of unique analytic function in the entire plane. According to generalize Liouville theorem this function is some arbitrary constant . Thus, bearing in mind the representations (2.59) and (2.52) we get Since there exists only one root of equation such that for all therefore, in the expression for the function the factor has a pole in the point . Also the function has a pole in the point Thus in general case the problem (2.36) is insolvable. It is soluble only when the functions and satisfy additional conditions. For analyticity of in points it is necessary that We reduce (2.64) to the form Multiplying the last relation by and taking limit we get that This implies that for solubility of the nonhomogeneous problem (2.36) it is necessary and sufficient that the following condition is satisfied:
Therefore, we need to put in the problem (2.9) one boundary data and the rest of boundary data can be found from (2.66). Thus, for example, if we put from (2.66) we obtain for the Laplace transform of where Now we prove that the coefficient of does not vanish for all . We represent the function in the form where Since, for making use of analytic properties of the function by Cauchy Theorem we have where is some fixed point, To calculate the function we will use the identity (2.43). Observe that the function is analytic for all . Therefore, setting the relation (2.43) into definition of and making use of Cauchy Theorem we find Thus, from (2.72) and (2.73) we obtain the following relation for the function Substituting this formula into (2.67) we get Now we return to problem (2.36). From (2.63) under the conditions and (2.75) the limiting values of solution of (2.36) are given by From (2.76) with the help of the integral representations (2.61) and (2.50), for sectionally analytic functions and , making use of Sokhotzki-Plemelj formula (2.3) and relation (2.45) we can express the difference limiting values of the function in the form We now proceed to find the unknown function involved in the formula (2.27) for the solution of the problem (2.9). Replacing the difference in the relation (2.38) by formula (2.77) we get It is easy to observe that is boundary value of the function analytic in the left complex semi-plane and therefore satisfies our basic assumption for all Having determined the function bearing in mind formula (2.27) and conditions we determine required function where the function is given by formula (2.55): Now we prove that, in accordance with last relation, the function constitutes the limiting value of an analytic function in
With the help of the integral representations (2.61), (2.31), and (2.50) for sectionally analytic functions and , and making use of Sokhotzki-Plemelj formula (2.3) we have Substituting these relations into (2.80) we express the function in the following form: If it is taken into account that by virtue of the relation (2.45), the last expression agrees with formula Expressing the function in the last equation in terms of we arrive at the following relation: where by virtue of (2.49) and (2.66), Thus the function is the limiting value of an analytic function in Note the fundamental importance of the proven fact that the solution constitutes an analytic function in and, as a consequence, its inverse Laplace transform vanishes for all We now return to solution of the problem (2.9).
Under assumption the integral representation (2.61) takes form where is defined by (2.75). Substituting this relation into (2.86) and taking inverse Laplace transform with respect to time and inverse Fourier transform with respect to space variables we obtain where the function was defined by formula (2.12). Proposition 2.4 is proved.
Now we collect some preliminary estimates of the Green operator . Let the contours be defined as
where can be chosen such that all functions under integration are analytic and for
Proof. We rewrite formula (2.12) in the form
Firstly we consider the sectionally analytic function given by Cauchy type integral (2.98).
On basis of the definition (2.98) its limiting value can be represent in the form where Making use of analytic properties of the functions for and for by Cauchy theorem we have where is some fixed point,
To calculate the function we will use the following identity: Observe that the function is analytic for all Therefore, setting the relation (2.103) into definition of and making use of Cauchy theorem we find Thus from (2.102) and (2.104) we obtain the following relation: where In the same way (see also proof of relation (2.74)) we can prove that Also we observe that Inserting into definition (2.96) the expression (2.105) for we obtain the function in the form Replacing in formula (2.97) the functions and by (2.108) and (2.107), respectively, we reduce the function in the form Therefore inserting into definition (2.95) expressions (2.109) and (2.110), for and , respectively, we obtain the function in the form where Also, note that since we obtain So, where Using relation we rewrite last formula in the following form: On the basis of definitions (2.116) and in accordance with the Sohkotzki-Plemelj formula (2.3) we have So we get Now we consider for Note that is analytic in domain , and and So we get where Changing the contour of integration with respect to by Cauchy theorem we get where