The Inviscid Limit of Viscous Burgers through Shock Formation and Development

We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order ${\nu }^{1\mathrm{/}4}$ up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to $t=0$, we prove a pointwise error bound on a fixed interval $0\le t\le \bar{\epsilon }$, where $\bar{\epsilon }>0$ is independent of the viscosity $\nu$. The bound consists of a term of order ${\nu }^{1\mathrm{/}4}$ and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.