Zero diffusion-dispersion limit for the Benjamin–Ono–Burgers equation

We prove that the solution of the viscous Benjamin–Ono equation converges, as diffusion and dispersion parameters tend to zero (under a suitable balance condition), to the unique entropy solution of the inviscid Burgers equation. The key tool in our proof is Schonbek’s Lp-compensated compactness method. Specifically, we prove a uniform L4-estimate using a modification of a con- served quantity for the inviscid Benjamin–Ono equation and a suitable differ- ential inequality argument.