Viscosity Approximating Solutions to ODE Systems That Admit Shocks, and Their Limits

AbstractWe study nonlinear systems of ordinary differential equations that arise when considering stationary one-dimensional systems of conservation laws with forcing terms defined in a bounded interval. We construct weak entropy solutions of bounded variation which are pointwise and L1 limits of solutions of regularized, i.e., viscous, systems, where the limit is taken in the viscosity parameter. In particular, no oscillations occur either for the viscous solutions or for the inviscid one. We also discuss the possible formation of boundary layers when boundary values are prescribed for the viscous regularized equations. As applications, first we show the existence of transonic solutions of bounded variation with strong shocks for the equation of stationary gas flow in a duct of variable area as a pointwise limit of artificial viscosity solutions. We analyze their properties depending on the kind of duct as well as on the boundary data of the regularized problem. Second we show that the model applies to the hydrodynamic modeling for semiconductor devices for some particular heat conduction terms and added diffusion to the energy equation. In particular, we show that under the assumption of bounds for the state variables, there exists a regular solution for that particular viscous heat conducting model. Also, if the bounds for the state variables are uniform in the vanishing parameters, we obtain the existence of an inviscid weak entropy solution of bounded variation as a pointwise limit of the regular ones