General boundary value problems of the Korteweg-de Vries equation on a bounded domain

Abstract

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite intervalsubject to the nonhomogeneous boundary conditions,aij∂ j x u(0, t) + bij∂ j x u(L, t) , i = 1, 2, 3, and aij, bij (j, i = 0, 1, 2, 3) are real constants. Under some general assumptions imposed on the coefficients aij, bij, j, i = 0, 1, 2, 3, the IBVPs (0.1)-(0.2) is shown to be locally well-posed in the space H s (0, L) for any s ≥ 0 with φ ∈ H s (0, L) and boundary values hj, j = 1, 2, 3 belonging to some appropriate spaces with optimal regularity.

Initial boundary value problem for Korteweg–de Vries equation: a review and open problems

“…The main point here is to demonstrate the smoothing properties for solutions of the IBVP (3.12). In order to overcome this difficulty, Capistrano-Filho et al in [18] needed to study the following IBVP…”

“…Theorem G can also be extended to the case of −1 < s ≤ 0 using the same approach developed in [8]. Finally, still concerning with well-posedness problem, while the approach developed recently in [18] studies the nonhomogeneous boundary value problems of the KdV equation on (0, L) with quite general boundary conditions, there are still some boundary value problems of the KdV equation that the approach do not work, for example…”

“…Finally, in a recently work, Capistrano-Filho et al in [18] studied the well-posedness of IBVP (2.2)-(2.3). The authors proposed the following hypotheses on those coefficients a i j , b i j , j, i = 0, 1, 2, 3: For s ≥ 0, consider the set…”

“…Theorem H [18] Let s ≥ 0 with s = 2 j−1 2 , j = 1, 2, 3..., and T > 0 be given. If one of the assumptions below is satisfied, (i) (A1), (B1) and (C) hold, (ii) (A1), (B2) and (C) hold, (iii) (A2), (B1) and (C) hold, (iv) (A2), (B2) and (C) hold, then, for any r > 0, there exists a T * ∈ (0, T ] such that for any…”

Formulation of Problems for Stationary Dispersive Equations of Higher Orders on Bounded Intervals with General Boundary Conditions

“…Usually, simple boundary conditions at x = 0 and x = L such as D i u(0) = D i u(L) = D l u(L) = 0, i = 0,..., l − 1 for (1) were imposed, see [1,21,22]. Different kind of boundary conditions for KdV and Kawahara equations was considered in [16,[23][24][25]. We must mention [26] where general mixed problems forlinear multidimensional (2b + 1)-hyperbolic equations were studied by means of functional analisys methods.…”