On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation

TL;DR

This paper establishes local and global well-posedness for the Kadomtsev-Petviashvili-Burgers-I equation in certain Sobolev spaces, advancing understanding of its mathematical properties using Bourgain's spaces.

Contribution

It proves well-posedness for KPBI in $H^{s,0}$ spaces with $s > -1/2$, nearly optimal for smooth flow-map requirements.

Findings

Well-posedness in $H^{s,0}$ for $s > -1/2$

Use of Bourgain's type spaces for analysis

Results are nearly sharp for smooth flow-map

Abstract

We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost sharp if one requires the flow-map to be smooth.