The Korteweg-de Vries equation on an interval

Abstract

The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas' unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega.

Boundary behavior of the solution to the linear Korteweg‐De Vries equation on the half line

“…47 the novel unified‐transform methodology by applying it to an array of classical mathematical‐physics problems of paramount significance, whereas Colliander and Kenig, 48 Bona, Sun, and Zhang, 49 and Holmer 50 utilized modern analysis techniques to study local as well as global well‐posedness of IBVPs for the KdV in the low‐regularity setting, thereby arriving at impressive findings. In addition, fascinating advances have been reported by Himonas, Mantzavinos, and Yan 51 with regard to the KdV posed on a finite interval with data in Sobolev spaces. The interested reader is referred to the above outlined works, and references therein, for further details.…”

The Time-fractional Airy Equation on the Metric Graph

“…The Airy equation on an interval was studied with Fokas unified transform method [9] and [10]. The potential theory for solutions of this equation was developed [11] and [12].…”

An elementary proof of the lack of null controllability for the heat equation on the half line

“…However, in later works it was proven to produce novel results for a general class of linear PDEs; see [2,3]. Recently researchers utilized the UTM to produce rigorous wellposedness results in Sobolev and Bourgain spaces for dispersive PDEs; see for instance [4] and [8] for the local and global wellposedness analysis of nonlinear Schrödinger type PDEs and [5] for a similar analysis on the Korteweg-de Vries equation.…”