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 Lp setting.
Coclite, G. M., & di Ruvo, L. (2016). Convergence of the Solutions on the Generalized Korteweg–de Vries Equation∗. Mathematical Modelling and Analysis, 21(2), 239-259. https://doi.org/10.3846/13926292.2016.1150358
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