Research Article | Open Access
Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
We study the mixed initial-boundary value problem for the capillary wave equation: , where . We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
This paper is concerned with the initial-boundary value problem (IBV problem) for the nonlinear capillary wave equation with mixed (Robin) boundary conditions posed on a half-unbounded domain: Here is a fractional derivative defined by Mixed boundary value problems arise in a variety of applied mathematics, engineering, and physics, such as gas dynamics, nuclear physics, chemical reaction, studies of atomic structures, and atomic calculations. Therefore, the mixed problems have attracted much attention and have been studied by many authors. For detailed description of the mixed boundary conditions, see [1–3] and the references cited therein. This paper is the first attempt to investigate the inhomogeneous mixed initial-boundary value problem for the dispersive fractional nonlinear equation, considering as an example the famous capillary water wave equation (1). Fractional differential equations appear in many applications of the applied sciences, such as the fractional diffusion and wave equations , subdiffusion and superdiffusion equations , electrical systems , viscoelasticity theory , control systems , bioengineering , and finance . Many articles have appeared in the literature, where fractional derivatives are used for a better description of certain material properties. Thus, for example, the fractional NLS model (1) comes from the study of the long-time behavior of solutions to the water waves equations . The operator corresponds to the dispersive relation of the linearized gravity water wave equations for one-dimensional interfaces with surface tension. Furthermore, thanks to the absence of resonances at the quadratic level, one expects the nonlinear dynamics of water waves to be governed by nonlinearity of cubic type like those appearing in (1). Papers [9–11] addressed some other applications of fractional Schrödinger equations. Works concerning the Cauchy problem for fractional type Schrödinger equations, which address the existence of small solutions, and in particular the question of modified scattering, include [12–14]. In paper , it was shown that (1) with dispersive fractional derivative operator of order admits global solutions whose long-time behavior is not linear. Global existence results and asymptotic behavior of small solutions of Cauchy problem for capillary water wave equation were obtained in . The initial-boundary problems have been much less studied than the Cauchy problems in spite of their importance. The inhomogeneous problems are often called forced ones, when an external force is applied to a system. Frequently the forcing is putted as the inhomogeneous boundary condition. In the case of the initial-boundary value problems there appear new difficulties comparing with the Cauchy problems due to the boundary. For example, in the case of the initial-boundary value problems it is not clear how many of the boundary conditions are required for the well-posedness of the initial-boundary value problem. The answer to this question relies on the construction of the Green function for the linear capillary water wave equation that is interesting on its own. Also it is necessary to take into account the boundary effects which affect the behavior of the solutions. Also usually we ask as less as possible regularity on the initial and boundary data, since the regularity of the solution implies the compatibility conditions on the initial and boundary data. Observe that for the Cauchy problem there is no such complication, in general, and we can ask more regularity on the initial data.
It is well-known that boundary value problems with homogeneous boundary conditions are easier than the corresponding homogeneous problems. However, we present in this paper remarkable results, such as global in-time existence of solutions and its large time asymptotic behavior. In book , it was proved that in the case of mixed problem for dissipative equations the solutions obtain more rapid time decay comparing with the case of the Cauchy problem. This phenomenon was also observed for some dispersive equations, such as intermediate-long wave and Benjamin-Ono equations, posed on the positive half-line [17, 18]. However, there are several important examples of equations whose small solutions do not behave like linear ones, as it is the case for the Schrödinger equation . As we will show below, the same happens for the case of the capillary wave equation (1) with mixed boundary conditions: the cubic nonlinearity is also critical with respect to the large time decay. Theorem 1 below shows that (1) admits global solutions whose long-time behavior is not linear. In particular, a correction of logarithmic type (see (9)) is needed in order to obtain the decay and the scattering of solutions. Our approach is based on the estimates of the integral equation in the Sobolev spaces. To construct the Green operator on a half-line we adopt the analytic continuation method proposed in the paper , based on the Riemann theory. To get smooth solutions we use a method of the decomposition of the critical cubic nonlinearity. A major complication of IBV problem for nonlocal equation (1) on a half-line is that its symbol is nonanalytic; therefore, we can not apply the Laplace theory directly. The construction of the Green operator involves the solution of the inhomogeneous Riemann problem with discontinuous coefficients. Another difficulty is that the symbol is dispersive. To get the asymptotics of solution, we need to solve the nonlinear singular integrodifferential equation with Hilbert kernel. We believe that the method developed in this paper could be applied to a wide class of dispersive nonlinear nonlocal equations.
2. Main Result and Notation
To state the results of the present paper, we give some notations. The usual direct and inverse Laplace transformations we denote by and . The Fourier transform and the inverse Fourier transform are defined as The usual Fourier sine transform and the Fourier cosine transform are given by Define the “distorted” Fourier sine transform and the inverse “distorted” Fourier sine transform as follows:where
For a detailed study of the properties of and , see below Lemma 3.
Also we introduce the Green operator on a half-line as Let be a complex function, defined in , which obeys the Hölder condition for all finite and tends to a definite limit as . Then, Cauchy type integral constitutes a function which is analytic in the left and right complex semiplanes. Here and below these functions will be denoted by 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 (see  for these definitions and a discussion of related issues): The usual Lebesgue space , where the norm if and if . Weighted Lebesgue space is , where the norm .
Weighted Sobolev space is , where . The usual Sobolev space is , so the index we omit if it does not cause a confusion. Also , and .
Let , where is small. Different positive constants we denote by the same letter . We denote ,.
Our main result is as follows.
Theorem 1. Let , and and the norm . Then, there exists such that for all the initial-boundary value problem (1) has a unique global solution . Furthermore, there exists a unique final state such that the large time asymptotics is true, where , and
3. Sketch of Proof
Firstly, we construct the Green operator defined by the linear IBV problem posed on a half-line:To derive an integral representation for the solution of problem (11), we adopt the analytic continuation method proposed in the paper . The Laplace transform of the fractional operator with respect to space and time variables is where , , and are Laplace transforms of ,, and , respectively.
Thus, since applying the Laplace transforms to (11) with respect to space and time variables we obtain Let be some complex function analytic in the left-half complex plane and , , such that Thus, from (14), we obtain Note that there are three unknown types of data, , , and in (16). To find some of these data we use analytic properties of , which is analytic in by the construction. Also, as we see in the left-hand side of (16), is not analytic. We define the “analyticity switching” functions . Denote . Note that for and the equality has only one root such that in . We make a cut along the negative imaginary axis. Let We prove that for Therefore, via Index Zero Theorem, we have where Via Sokhotski-Plemelj formula, we have where Applying (19) and (21) into (16), we obtain For the analyticity of the function in the right-half complex plane, we need to put the following conditions: Thus, we must put in the problem (11) only one type of boundary data. Another type of unknown boundary data can be find from relation (24). For example, if we consider Robin boundary condition, , boundary data and are completely determined by where is Laplace transform of . We will prove that is analytic in under condition . Finally substituting (25) and (24) into (23) and applying inverse Laplace transform, we will prove that the solution (11) is represented by , where This fact is exploited in Proposition 2 below. Therefore, the IBV problem (1) can be rewritten as The estimates of the integral equation (27) in the Sobolev spaces yield an a priori estimate of . To get the -estimate of , we will prove the asymptotics
We introduce the new function as Then, we obtain from (1) and (27) In Lemma 7 below, we will prove that the nonlinearity is decomposed into the resonant term and the remainder such that where and . Also we prove Then, the estimates of and follow. Therefore, along with (28) and (29) we prove the -estimate of . Also we obtain asymptotics of the solution. We construct the Green operator for the initial-boundary value problem in the next section. Section 5 is devoted to several lemmas involved in the proof of the main result. In Section 6, we prove our main result.
4. Green Operator
We consider linearized version of problem (1). We prove the following proposition.
Proof. Substituting (25) and (24) into (23) and applying inverse Laplace transform we prove that the solution (11) is represented: whereDenote where was introduced by (20). Taking the Laplace transform of (34) with respect to and variables, we get where with Now we consider given by (39). Using analytic properties of integrand functions for , we rewrite the function in the following form: wherewhere Since , we get which implies . We make the changes of the variables and in the first and second terms of (43) correspondingly. Using that, by the definitions and , we obtain Now we consider and given by (22). Via (19) and Cauchy Theorem taking residue in the point , we get and by the same way As a consequence of (47) and (48), from (35) it follows thatSubstituting (49) into (46), we attain where By Jordan Lemma taking residue in the point , we rewrite in the form Here Since integrand is even function with respect to -variable, we get for the last summand of (52) Therefore, takes the form where Via Lemma 11 . Therefore, we make the change of variables , to get Also since via Lemma 3 , Thus, finally we obtain where operators and are defined by (6). By the same way as in the proof of (59), we getApplying this relation into , we obtain Therefore, via (61) and (59), Proposition 2 follows.
Firstly, we prove the main properties of the operators , and defined in (6).
Lemma 3. The following estimates are valid:
Proof. Applying Sokhotski-Plemelj formula, we get