A higher dispersion KdV equation on the half-line

The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces of the same optimal regularity as in the case of the initial value problem for this equation on the whole line

The Korteweg-de Vries equation on an interval

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 051507 (2019) and may be found at https://doi.org/10.1063/1.5080366.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

Global Existence of Some Infinite Energy Solutions for a Perfect Incompressible Fluid

This paper provides results on local and global existence for a class of solutions to the Euler equations for an incompressible, inviscid fluid. By considering a class of solutions which exhibits a characteristic growth at infinity we obtain an initial value problem for a nonlocal equation. We establish local well-posedness in all dimensions and persistence in time of these solutions for three and higher dimensions. We also examine a weaker class of global solutions