On a limit of perturbed conservation laws with saturating diffusion and non-positive dispersion

We consider a conservation law with convex flux, perturbed by a saturating diffusion and non-positive dispersion of the form ut+f(u)x=ε(ux1+ux2)x-δ(|uxx|n)x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t + f(u)_x = \varepsilon \Big ( {u_{x}\over \sqrt{1 + u_x^2}} \Big )_x - \delta (|u_{xx}|^n)_x$$\end{document}. We prove the convergence of the solutions {uε,δ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u^{\varepsilon , \delta }\}$$\end{document} to the entropy weak solution of the hyperbolic conservation law, ut+f(u)x=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t + f(u)_x = 0,$$\end{document} for all real number 1≤n≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le n \le 2$$\end{document} provided δ=o(εn(n+1)2;εn+1n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta = o(\varepsilon ^{n(n+1)\over 2}; \varepsilon ^{n+1\over n })$$\end{document}.