Convergence of the Solutions on the Generalized Korteweg–de Vries Equation^*

* The authors are members of the Gruppo Nazionale per l′Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Abstract

We consider the generalized Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L^p setting.

Keywords:

• singular limit
• compensated compactness
• connected compactness
• Korteweg-de Vries equation

AMS Subject Classification:

• 35G25
• 35L65
• 35L05

Notes

* The authors are members of the Gruppo Nazionale per l′Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)