Analysis of the initial value problem for two specific non-strictly hyperbolic systems

Abstract: In this talk, we are going to discuss the following problems:

1. At first we study the initial value problem for the zero-pressure gas dynamics system in non-conservative form. The techniques of adhesion approximation and modified adhesion approximation are used in the construction of weak asymptotic solution. Explicit formula for the weak asymptotic solution and generalized solution have been studied under plane-wave type initial data.

2. Secondly, we further investigate the modified adhesion model for the zero-pressure gas dynamics system when the initial data contains δ-measures. Interaction of δ-waves and the interaction of δ-waves with classical shock/rarefaction waves are discussed. In our analysis we have used the vanishing viscosity method.

3. Lastly, we study the initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in {−1, 0, 1}. We give a construction of the explicit formula for a weak solution. The Lax formula has been used for the derivation of the formula for the velocity component. We have also studied the initial-boundary value problem with a weak formulation of the boundary condition.