Advances in Differential Equations

On the Cauchy problem for a Boussinesq-type system

Jaime Angulo Pava

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The Cauchy problem for the following Boussinesq system, $$ \begin{cases} u_t + v_x + uu_x = 0\\ v_t - u_{xxx} + u_{x} + (uv)_x = 0 \end{cases} $$ is considered. It is showed that this problem is locally well-posed in $H^s(\mathbb{R}) \times H^{s-1}(\mathbb{R})$ for any $s>3/2$. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined that the special solitary-wave solutions of this system are orbitally stable for the entire range of the wave speed. Combining these facts we can extend globally the local solution for data sufficiently close to the solitary wave.

Article information

Adv. Differential Equations, Volume 4, Number 4 (1999), 457-492.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]


Angulo Pava, Jaime. On the Cauchy problem for a Boussinesq-type system. Adv. Differential Equations 4 (1999), no. 4, 457--492.

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