## Differential and Integral Equations

- Differential Integral Equations
- Volume 32, Number 11/12 (2019), 679-704.

### On the Cauchy problem for the periodic fifth-order KP-I equation

#### Abstract

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation $$ { \partial_t} u - { \partial_x}^5 u -{ \partial_x}^{-1} { \partial_y}^2u + u{ \partial_x} u = 0, ~(t,x,y)\in\mathbb R\times\mathbb T^2 . $$ We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbf E = \{u\in L^2,~{ \partial_x}^2 u \in L^2, ~{ \partial_x}^{-1} { \partial_y} u \in L^2\}$ which is the natural energy space associated with this equation.

#### Article information

**Source**

Differential Integral Equations, Volume 32, Number 11/12 (2019), 679-704.

**Dates**

First available in Project Euclid: 22 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1571731515

**Mathematical Reviews number (MathSciNet)**

MR4021259

**Zentralblatt MATH identifier**

07144909

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

#### Citation

Robert, Tristan. On the Cauchy problem for the periodic fifth-order KP-I equation. Differential Integral Equations 32 (2019), no. 11/12, 679--704. https://projecteuclid.org/euclid.die/1571731515