## Advances in Differential Equations

- Adv. Differential Equations
- Volume 13, Number 1-2 (2008), 1-26.

### On the Cauchy problem for a coupled system of KdV equations: critical case

M. Panthee and M. Scialom

#### Abstract

We investigate some well-posedness issues for the initial value problem associated to the system \begin{equation*} \begin{cases} u_{t}+\partial_x^3u+\partial_x(u^2v^3) =0,\\ v_{t}+\partial_x^3v+\partial_x(u^3v^2)=0, \end{cases} \end{equation*} for given data in low order Sobolev spaces $H^s(\mathbb R)\times H^s(\mathbb R)$. We prove local and global well-posedness results utilizing the sharp smoothing estimates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever $s\geq 0$ by using global smoothing estimates. In particular, for data satisfying $\delta < \|(u_0, v_0)\|_{L^2\times L^2} < \|(S, S)\|_{L^2\times L^2}$, where $S$ is solitary wave solution, we get global solution whenever $s>3/4$. To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplified by Fonseca, Linares and Ponce [6, 7].

#### Article information

**Source**

Adv. Differential Equations, Volume 13, Number 1-2 (2008), 1-26.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867358

**Mathematical Reviews number (MathSciNet)**

MR2482535

**Zentralblatt MATH identifier**

1157.35472

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

#### Citation

Panthee, M.; Scialom, M. On the Cauchy problem for a coupled system of KdV equations: critical case. Adv. Differential Equations 13 (2008), no. 1-2, 1--26. https://projecteuclid.org/euclid.ade/1355867358