WELL-POSEDNESS OF BOUNDARY-VALUE PROBLEMS FOR THE LINEAR BENJAMIN-BONA-MAHONY EQUATION

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary-value problem is found to be ill posed.

1. Introduction.Recently, the work of Fokas and others [1] has demonstrated that some ideas arising in the analysis of nonlinear so-called integrable partial differential equations (PDEs) may be applied to solve boundary-value problems (BVPs) for linear PDEs.The method, referred herein as the method of Fokas (MoF), has been applied with great success to solve problems previously beyond the reach of conventional analytic techniques [1,2].The MoF provides explicit solutions to boundary-value problems for linear evolution equations with variable coefficients [3,4], systems of linear constant coefficient evolution equations [5] and movingboundary-value problems [6].The MoF has been extended to incorporate twodimensional problems in non-separable domains [7] as well as elliptic PDEs [8].
In this paper, we apply the MoF to the linear Benjamin-Bona-Mahony (BBM) equation [9]: (1) The BBM equation is a well-known model for long waves in shallow water.Due to the presence of the mixed derivative, the linear BBM equation has a rational dispersion relation which tends to zero for large wave numbers.This non-polynomial nature of the dispersion relation means that a modification of the MoF is required in order to use the method.Although we do not consider examples other than the linear BBM equation, the procedure presented applies to problems of the form where M (k) and L(k) are polynomials.Using the MoF, it is possible to find explicit solutions for various boundary-value problems associated with such PDEs.In fact, such equations may be analyzed with the same efficiency as those without mixed derivatives.Although, mixed derivative PDEs do not often appear in the classroom, they are important in many applications [10].Indeed, inspired by problems in water-waves, Fokas and Pelloni discuss boundary-value problems for Boussinesq type systems [11].However, those mixed-partial derivative equations do not demonstrate the behavior we report.The only investigation of boundary-value problems for linear BBM we are aware of, considers the Dirichlet problem on the half-line [12].
Our work includes explicit solutions for Dirichlet, Neumann and Robin boundary conditions for the half-line and for the finite interval.
As already stated, the non-polynomial nature of the dispersion relation for the linear BBM equation has a significant impact on the application of the MoF.It is the cause of the essential singularities that arise in the analysis which obstruct the deformation of contour integrals.Further, the so-called global relation (see below) ceases to be valid at certain points in the finite complex plane.This results in the PDE being ill-posed for specific cases of the Robin boundary condition.Specifically, for these cases of the Robin boundary condition, initial and boundary conditions may not be arbitrarily specified.Instead, there exists a discontinuous relation between the initial and boundary conditions.The following two theorems specify the situation for the problem on the half-line and finite interval respectively.
Theorem 1.1.The BVP has a unique solution when α = β.Further, the existence of a smooth solution requires u 0 (0) = g(0) = 0.If α = β the problem is ill-posed in the sense that the initial and boundary conditions may not be prescribed arbitrarily.If homogeneous boundary conditions are imposed, the initial condition must be identically zero for a smooth solution to exist.By uniqueness, this implies the only solution with homogeneous boundary conditions is the trivial one.If α = β and non-homogeneous boundary conditions are prescribed, then the constraint between initial and boundary conditions û0 (−i) + g (0) = 0, must hold for a solution to exist.
Theorem 1.2.The BVP has a unique solution when (α, γ) = ±(β, δ).The existence of a smooth solution requires u 0 (0) = g(0) = 0 and u 0 (L) = h(0) = 0.If (α, γ) = ±(β, δ) the problem is ill-posed in the sense that the initial and boundary conditions may not be prescribed arbitrarily.If homogeneous boundary conditions are imposed, the initial condition must be identically zero for a smooth solution to exist.By uniqueness, this implies the only solution with homogeneous boundary conditions is the trivial one.If (α, γ) = ±(β, δ) and non-homogeneous boundary conditions are prescribed, then the constraint between initial and boundary conditions must hold for a solution to exist.
The methods discussed in this paper extend in a straightforward manner to equations with mixed partial derivatives other than BBM.We believe, part of the value of the current work lies in the extension of the MoF itself and not merely the above two theorems.As a final remark, we point out that the results presented extend to the forced equation For small time, we may regard the nonlinear term in the full BBM equation as a forcing.Since BBM is a semi-linear equation, we conjecture that the ill-posedness results apply to the nonlinear problem as well.
2. The local relation and Lax pairs.In this section, an algorithmic procedure for deriving one-parameter divergence forms associated with linear PDEs is presented.Such a divergence form, referred to as the local relation is the starting point for the MoF.Further, this section highlights the connection of the method with techniques for nonlinear integrable PDEs through the existence of Lax pairs for linear PDEs.
The local relation may be obtained from the divergence form for the PDE.To obtain the divergence form, consider the formal adjoint of (1) Multiplying equation (1) by v, (5) by u and adding the resulting equations, we obtain after some rearrangement A one-parameter family of solutions to (5) is given by separation of variables as Using this v we obtain which is the required one-parameter family of divergence forms.An alternate means to obtain the local relation highlights the connection of the MoF to integrable PDEs.
In particular, one can rewrite linear PDEs as the compatibility condition of two linear ordinary differential equations.In other words, a linear PDE possesses a Lax pair.In the following we shall derive Lax pairs for linear PDEs with and without mixed partial derivatives.Consider the linear constant-coefficient differential equation This PDE has a Lax pair of the form where X is a differential operator acting on u(x, t) with coefficients depending on the spectral parameter k.Imposing the compatibility of these two equations and assuming that u(x, t) solves (8) one obtains As an example, for the heat equation with "dispersion relation"1 ω(k) = k 2 , we obtain the Lax pair The local relation follows immediately.Indeed the above equations may be written as Equation (8) represents the evolution of u(x, t) with dispersion relation ω(k).Similarly, one may consider (2) to represent the evolution of a quantity u(x, t) with dispersion relation L(k)/M (k).Consequently, we seek a Lax pair of the form Further, it is assumed that M (k) > 0 for k ∈ R and L(k), M (k) do not share any roots.As before, imposing compatibility of the two ordinary differential equations for µ and imposing that u(x, t) solves (2), we obtain that the appropriate differential operator is given by For the specific case of the linear BBM equation with L(k) = ik and M (k) = 1 + k 2 , we obtain 3. Solutions of the linear BBM equation on the half-line.The BBM equation models long waves in shallow water.It is natural to consider the boundary-value problem for the equation posed on the semi-infinite line: In what follows, we start with formal calculations assuming a smooth solution exists which has sufficiently rapid decay at infinity.These calculations lead us to a solution expression u(x, t) for most values of α and β.Specifically we have the following proposition.
Proposition 3.1.For the above boundary-value problem with α = β and g(0 and C is an arbitrarily small closed contour around k = i.
Assuming sufficient regularity of the initial and boundary condition functions, we claim the solution expression is a classical solution to the above boundary-value problem.For the particular case when α = β, the arguments below lead us to conclude the problem is ill posed in the sense that the initial and boundary condition functions may not be chosen arbitrarily.Instead there exists a discontinuous relation between them in order for the problem to be solvable.However we obtain yet a solution expression.Indeed we have the following proposition.Proposition 3.2.For the boundary-value problem (10a-10c) with α = β = 1 and g(0 is a solution.Further, for a smooth solution g(t) and u 0 (x) satisfy the following condition Integrating the local relation ( 9) over the region {(x, s) : x ≥ 0, s ∈ (0, t)}, applying Green's Theorem and integrating by parts, we obtain the global relation with where û0 (k) and û(k, t) represent the Fourier transform of the initial condition and of the solution at time t respectively.They are defined by where u 0 (x) and u(x, t) are defined to be zero for x < 0. Similarly, g0 (ω, t) and g1 (ω, t) are the finite-time transforms of the boundary data given by g0 (ω, t) = t 0 e ωs u(0, s)ds, g1 (ω, t) = t 0 e ωs u x (0, s)ds.
Solving the global relation (11) for û(k, t) and applying the inverse Fourier transform, we obtain the following integral expression for the solution where the integral on the real line has been deformed to a closed contour C around k = i as shown in Figure 1.This is possible since the respective integrands are analytic functions which decay in the upper-half plane for large k.
The integral expression (12) depends on unprescribed boundary data.As is usual in the MoF, this is resolved by using the invariances of ω(k).Note that ω(k) is invariant under the transform k → 1/k.Using this transformation, the global relation becomes The time transform of the boundary condition (10c) is given by e ωs g(s)ds.Solving the above equations for g0 (ω, t) and g1 (ω, t), and substituting the result in (12) we obtain where The right-hand side of (13) depends on u(x, t) itself, through the presence of û(1/k, t) in the second term.As such, (13) does not represent an explicit solution formula, and more work is required.We consider two possible cases.
• Case 1. α = β.By deforming the contour C as indicated in Figure 1 before substitution of the transforms of the unknown boundary terms, we assure that the only singularity of the integrals in (13) enclosed by since the integrand is analytic in a neighborhood of k = i.Further, an application of the Residue Theorem for the third term in (13) leads to We conclude that the solution u(x, t) is given by In particular, the Dirichlet (β = 0) and Neumann (α = 0) problems for this PDE have solutions given by the above expression.Since the term proportional to the exponential satisfies the boundary condition, it is necessary that αS(0, t) + βS x (0, t) = 0, which is assured by taking g(0) = u 0 (0) = 0. Thus, the existence of a classical solution requires the compatibility of the boundary and initial conditions at the corner point (x, t) = (0, 0).• Case 2. α = β = 1.The solution to the boundary-value problem given above is not valid when α = β.Without loss of generality, we take α = β = 1.
In this case, both integrals possess singularities at k = i.The residues are calculated to obtain u(x, t) = S(x, t) + e −x g(t This may be interpreted as a linear integral equation for the solution u(x, t) that depends on known initial-boundary data as well as on the unknowns u(0, t) and û(−i, t).If it is possible to express these unknowns in terms of the known initial-and boundary-condition functions then the above expression represents the solution to the problem.However, since by assumption u(x, t) solves the PDE, we may substitute (15) into (10a) to obtain If this relation holds for all time, we obtain the following solution to the problem on the half-line Further, since the terms proportional to e −x satisfy the boundary condition, the expression S(x, t) has the same interpretation as in the case α = β.Consequently we require g(0) = u 0 (0) = 0.However, the initial condition is not satisfied unless as is readily seen by letting t = 0 in (17).It is noted that this expression may be obtained from (16) also.
Given a continuously differentiable function g(t), the constraint (18) requires the initial condition u 0 (x) to be of the form where w(x) is a function orthogonal to e −x using the standard inner product on [0, ∞), and w(0) = 0 with sufficient smoothness and decay for large x to justify the contour deformations in the previous section.Alternatively, given an initial condition u 0 (x) with u 0 (0) = 0 and sufficient decay and smoothness, the above is a constraint on permissible boundary conditions g(t).
The relation between initial and boundary conditions evidently restricts our freedom to arbitrarily choose initial and boundary functions for the case α = Interestingly, if g(t) is identically zero, then u 0 (x) must be identically zero too, as we show below.Indeed, if g(t) ≡ 0, the solution expression is given by Substituting this expression into (16), we find Using the transformation k → 1/k and replacing all contours C to contours C around k = −i, we obtain Since this statement is valid for all t we may expand the exponential in a Taylor series in t, interchange order of summation and integration and equate the coefficient of every power of t to zero.This imposes the following set of conditions on the Fourier transform of the initial condition: However, û0 (k) is an analytic function of k in the lower-half plane and is continuous up to the real line.Consequently, û0 (k) ≡ 0 for k real and the initial condition u 0 (x) is identically zero.It follows that the initial condition is not continuous as a function of the boundary data g(t), at g(t) ≡ 0, since (19) shows that for g(t) ≡ 0, u 0 (x) can be arbitrarily large by choosing w(x) large.
Finally we show that (16) holds for any classical solution to the PDE.Assuming u(x, t) is a solution to the problem (10a-10c), we have Multiplying this equation by e −x and integrating over the domain [0, ∞) yields after a few integrations by part d dt (u x (0, t) + u(0, t)) + ∞ 0 e −y u(y, t)dy = u(0, t).
Noting that the first term on the left-hand side is the time derivative of the boundary condition, we obtain (18).We conclude by noting that the identity (18), valid for any solution to the differential equation ( 16) became a constraint on the initial and boundary conditions.Indeed it may be verified that also when α = β, (16) is readily satisfied by the solution (14) to the problem.Remark 1.Note the modification to the usual MoF that is required due to the mixed partial derivative in equation (1).Firstly, the term u xx appears in the t−derivative of the local relation (7).It is this term that leads to the additional u x (0, 0), u(0, 0) quantities in the global relation (11) and hence resulting in the extra integral terms in equation (13).Secondly, the rational polynomial dispersion relation of a PDE with mixed partial derivatives results in the global relation containing terms with essential singularities in the finite complex plane.As a result, the global relation is no longer valid at particular points in the complex plane and the deformations of the contour integrals must respect this loss of analyticity.Further, the fact that the dispersion relation is bounded and analytic at infinity allows us to deform the integrals to the closed contour C. Finally, and most notably, there exist boundary conditions for PDEs with mixed partial derivatives that force the initial and boundary data to be related to one another for smooth solutions to exist.
4. Uniqueness of solutions to linear BBM on the half-line.In the previous section we constructed solutions to the linear BBM equation.The procedure was algorithmic.Depending on the values of the coefficients α and β, we obtained a solution expression (case 1) or we concluded additional constraints were required on the initial and boundary condition functions (case 2).In either case, it is necessary to investigate the uniqueness of the solution of the equation.If we can prove uniqueness, then in case 1, we conclude that ( 14) is the unique solution to the given boundary-value problem.In particular, it follows that the Dirichlet and Neumann problems have unique solutions.Similarly, in case 2, where the initial and boundary conditions are required to satisfy the constraint (18), we conclude that the problem is ill-posed in the sense that we cannot choose initial and boundary conditions arbitrarily.
It is typical to address uniqueness of solutions to evolution PDEs using energy integral arguments.The energy integral for the linear BBM equation is found as shown below.
In the above expression, we think of u(x, t) as the difference of two distinct solutions to the BVP.Since the given initial and boundary conditions are linear, uniqueness follows if the right-hand side of the above expression is non-positive for homogeneous boundary conditions.Clearly this is the case for the Dirichlet problem.However, no direct conclusion is obtained for the Neumann or the Robin problem.We may consider the right-hand side of the last line above as a map from known boundary conditions to unknown values at the boundary.For the Neumann case, if the Neumann-to-Dirichlet map exists and maps zero to zero, we may yet conclude uniqueness.Similarly, for the Robin boundary condition.In what follows we construct explicit Robin-to-Dirichlet and Neumann-to-Dirichlet maps which do indeed map homogeneous data to zero.Consequently, we claim there is a unique solution to the BVP (10a-10c).

Robin-Dirichlet boundary-value map.
The assumption of a classical solution to the BVP (10a-10c) leads to the global relation (11).In this section we show how the global relation may be used to derive a map from known initial-boundary data to unknown boundary data.Let us consider the Robin condition αu(0, t) + βu x (0, t) = g(t), which upon taking the time transform becomes αg 0 (ω, t) + βg 1 (ω, t) = G(ω, t).
Solving for u x (0, t) and g1 (ω, t) (assuming β = 0) in the above equations and substituting in the global relation (11) we obtain Multiplying this expression by e −ωt 1 − k 2 β − αik and integrating over a sufficiently small contour C containing k = −i we obtain the required map.The contribution from g0 (ω, t) vanishes since The first equality is obtained using the change of coordinates l = k/(1 + k 2 ), where C is the image of C under this map.The second equality uses the definition of the time transform.Evidently the integrand is an entire function of l, from which the conclusion follows.If α = β the contribution from û(k, t) also vanishes since due to analyticity of the integrand.An application of the Residue Theorem leads to the following expression for the Dirichlet data This expression is also valid for the Neumann problem (β = 1, α = 0).The expression above simplifies if we impose the corner condition u 0 (0) = g(0) = 0, which was required for existence of smooth solutions in the previous sections.Since the Robin-to-Dirichlet map maps the homogeneous problem to zero Dirichlet data, we obtain uniqueness for the case α = β.
The argument above applies with little modification for the case α = β = 1.We now obtain a contribution from û(k, t) given by where we have used the integral relation ( 16).We obtain the following expression for the Robin-to-Dirichlet map from which the same conclusion is obtained.
5. Linear BBM on the finite interval.Next, we consider the finite-interval BVP.The MoF is even more advantageous in this setting.Classical methods would lead one to consider separation of variables, i.e., we start with u(x, t) = f (x)s(t).Substituting into the differential equation, we obtain Dividing by f (x)s (t) results in a generalized eigenvalue problem in the spatial variable requiring significant analysis in order to establish whether or not the eigenfunctions obtained form a complete set.The MoF not only provides a solution without this additional effort, but it also indicates which boundary-value problems may be ill posed in the same sense we used for the BVP on the half-line.Since the procedure is somewhat similar to what was done for the BVP on the half line, we present less detail than before.By integrating the local relation over the region {(x, s) : 0 ≤ x ≤ L, 0 < s < t} and applying Green's Theorem, we obtain the global relation for the finite interval: where U 0 (k) = (1 + k 2 )û 0 (k) + u x (0, 0) + iku(0, 0) − e −ikL u x (L, 0) − ike −ikL u(L, 0), and h(k, t) is defined similarly to g(k, t) but with u(x, t) evaluated at x = L.The Fourier transform is given by û0 Applying the inverse Fourier transform, we obtain the integral expression The integral of the boundary terms may be deformed off the real line to appropriate contours C 1 and C 2 which are closed curves around k = i and k = −i respectively.
Assume we are given Robin boundary conditions at both x = 0 and x = L. αu(0, t) + βu x (0, t) = g(t), γu(L, t) + δu x (L, t) = h(t) where α, β, γ, and δ are real.Combining these last equations with the transformed version of the global relation as well as with the original global relation, we obtain a system of equations for the unknown boundary terms.For instance, solving (24) for g0 and h0 , we obtain a system of equations for the two remaining boundary terms.To solve this system of equations we are required to invert the matrix For a given set {α, β, γ, δ}, the zeros in the complex k plane of the determinant of this matrix are the singularities which appear in the final expression for the solution.
Since the entries of the matrix are analytic functions of k in C\{0}, the zeros of the determinant are isolated.Hence, by deforming the contours C 1 , C 2 suitably before substitution of the expressions for gi , hi , i = 0, 1, it is possible to ensure the integrals contain at most one singularity, namely at k = i or at k = −i.Consequently, if there exists a set {α, β, γ, δ} ∈ R 4 such that for k = ±i the determinant is zero, then the problem is presumed to be ill posed in a sense similar to that observed in the half-line case.Indeed, setting k = ±i the determinant has a zero only if (α, γ) = ±(β, δ).Hence either one of the following boundary conditions leads to an ill-posed problem: or In all other cases, the solution proceeds in a manner similar to Case 1 (α = β) of the half-line problem.In the following, we present details of the boundary-value problem with boundary conditions (25a-b).In this case, the global relation is where G, H, g0 , h0 are the time transforms of the Robin and Dirichlet data at the left and right boundaries defined analogously as to the half-line case.Looking ahead, we suppress the dependence of these terms on ω and t since the time transforms are invariant under the symmetries of the dispersion relation unlike the functions U and U 0 , which are given by The global relation ( 27) is valid for all k ∈ C\{−i, i}.Using the symmetry k → 1/k we obtain another version of the global relation valid for k ∈ C\{−i, 0, i}, Hence we obtain a system of equations for the unknown Dirichlet data which may be solved to obtain where It may be shown that this constraint is satisfied by any solution to the PDE.Indeed it is the finite interval analogue of (16).We may obtain two additional equations between the three unknowns from (29) by multiplying by e −ωt (1 − k 2 )/(1 + k 2 ) 2 and integrating along a small contour C around k = i.This leads to C e −ωt 1 − k 2 (1 + k 2 ) 2 ∆ −1 (P + R) dk = 0. (32) Evidently, equations (32) and (31) are three equations for three unknowns.This system of equations may be solved to obtain the Robin-to-Dirichlet map as well as a map from known initial and boundary data to the function û(−i, t).Using these definitions for the unknown functions, (30) represents the solution to the boundaryvalue problem on the finite interval.Indeed it is the unique solution since the Robin-to-Dirichlet map for the finite interval maps the homogeneous problem to trivial Dirichlet data, and an energy argument can be used to imply uniqueness.Note that (29), which was obtained from the global relation (27), and (31) are obtained from the assumption of existence of a smooth solution.
A calculation similar to that for the half-line case indicates that the only solution with this particular boundary condition and g(t) = h(t) = 0 is the trivial one.

6.
Conclusions.We have demonstrated the use of the MoF to PDEs with mixed partial derivatives in two independent variables by discussing BVPs for the linear BBM equation in detail.The application of these ideas to other mixed-derivative equations is straightforward.For the linear BBM equation, we have obtained two theorems mentioned in the Introduction.
It is possible to anticipate the non-trivial behavior mentioned in Theorem 1.1 and 1.2.For linear BBM, the differential operator acting on the time derivative has the symbol M (k) = 1 + k 2 , with a null-space spanned by {e x , e −x }.In order to invert this operator, it is necessary for the null-space of the operator to be empty.This is achieved by selecting the boundary condition.To see this consider linear BBM on the finite interval as a second-order forced ordinary differential equation in u t (see [13]).For all boundary conditions except the cases (α, γ) = ±(β, δ), it is possible to construct a Green's function uniquely.Further, the functions e x and e −x are in the null-space of the exceptional boundary conditions.This is the cause of the ill-posedness.This form of ill-posedness should be a generic feature of PDEs with mixed partial derivatives.