The Majda-Biello system on the half-line

TL;DR

This paper investigates the well-posedness of the Majda-Biello system, modeling Rossby wave interactions, on the half-line with various boundary conditions, establishing conditions based on the parameter lpha and Sobolev space regularity.

Contribution

It provides the first comprehensive analysis of the Majda-Biello system on the half-line, detailing well-posedness conditions for different lpha values and boundary data types using Bourgain space estimates.

Findings

Well-posedness for 0<lpha<1 or 1<lpha<4 in Sobolev spaces H^s, s.

For lpha=1 or lpha>4, well-posedness depends on boundary data type and regularity.

Optimal well-posedness results are established using Fokas solution formula and bilinear estimates.

Abstract

The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter $\alpha$ as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for $0<\alpha<1$ or $1<\alpha<4$ all these problems are well-posed for initial data in Sobolev spaces $H^s$, $s\ge 0$. For $\alpha=1$ or $\alpha>4$ well-posedness holds for Dirichlet data if $s>-3/4$, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For $\alpha=4$ well-posedness of all problems holds for $s\ge 3/4$. The Robin and Neumann boundary data are in $H^{s/3}$ while the Dirichlet boundary data are in $H^{(s+1)/3}$. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained here are optimal.