Convergence of a family of perturbed conservation laws with diffusion and non-positive dispersion (Q1985827)

The authors study the following perturbed conservation law \[ u_t+f(u)_x=\varepsilon u_{xx}-\delta (|u_{xx}|^n)_x, \quad u(0,x)=u_0(x), \] where \(1\le n\le 2\) and \(\varepsilon,\delta\) are small positive parameter. The flux \(f(u)\) is supposed to be convex. It is shown that in the limit as \(\varepsilon,\delta\to 0\), where \(\delta=o(\varepsilon^{\frac{3n-1}{2}};\varepsilon^{\frac{5n-1}{4n-2}})\), the approximate sequence \(u_{\varepsilon,\delta}\) converges to Kruzhkov entropy solution of the original problem \(u_t+f(u)_x=0\), \(u(0,x)=u_0(x)\).