Analytic semigroup generated by the linearization of a Riemann-Dafermos solution
The Riemann problem for the conservation law \(u_T + f(u)_X=0 \) is investigated. A Riemann-Dafermos solution is considered, i.e. a solution of Dafermos regularization \(u_T + f(u)_X=\varepsilon T u_{XX}\) that is close to a Riemann solution of the initial problem. In a self-similar coordinate \(x= \frac{X}{T}\) the Riemann-Dafermos solution becomes stationary. In the Banach space of bounded continuous functions with the appropriate weighted norm the linear variational system around such stationary solution is written in the form \(U_{\tau}=AU.\) The linear operator \(A\) is proved to be sectorial and hence \(A\) generates an analytic semigroup. Some important consequences of this result are discussed.
scientific article; zbMATH DE number 2186759