Forcing operators on star graphs applied for the cubic fourth order Schr\"odinger equation

TL;DR

This paper investigates the well-posedness of the cubic fourth order Schrödinger equation on star graphs using boundary forcing operators, achieving results in low regularity Sobolev spaces and demonstrating the approach's efficiency over traditional methods.

Contribution

It introduces a boundary forcing operator approach to study the well-posedness of the cubic fourth order Schrödinger equation on star graphs in low regularity spaces, simplifying analysis with harmonic analysis tools.

Findings

Proved well-posedness in low regularity Sobolev spaces.

Demonstrated the effectiveness of boundary forcing operators.

Potential applicability to other nonlinear dispersive equations.

Abstract

In a recent article \textit{"Lower regularity solutions of the biharmonic Schr\"odinger equation in a quarter plane", to appear on Pacific Journal of Mathematics [15]}, the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [15] concerning the study of the cubic fourth order Schr\"odinger equation in a star graph structure $\mathcal{G}$. Precisely, consider $\mathcal{G}$ composed by $N$ edges parameterized by half-lines $(0,+\infty)$ attached with a common vertex $\nu$. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the \textit{boundary forcing operator approach}. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.