An error estimate for viscous approximate solutions to degenerate anisotropic convection-diffusion equations

We consider a viscous approximation for a nonlinear degenerate convection-diffusion equations in two space dimensions, and prove an $L^1$ error estimate. Precisely, we show that the $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ difference between the approximate solution and the unique entropy solution converges at a rate $\mathcal{O}({\text{eps}}^{1\mathrm{/}2})$, where $\text{eps}$ is the viscous parameter.