Q-measure-valued solution of a hyperbolic partial differential equation

Abstract

In this article, we introduced a new measure termed as Q-measure to represent a weak* limit of the barycenter of a sequence of Borel measurable function. The Q-measure is defined by replacing the sequence of measurable functions via the barycenter of the sequence of Borel measurable functions in the Young measure. We introduced the weak stability of Q-measure. Using this new measure, we discussed the measure-valued solution of hyperbolic conservation law. The main goal of this paper is to numerically approximate an entropy measure solution using a modified algorithm. Finally, numerical approximations of entropy measure-valued valued solutions and two-dimension of measure-valued solutions for the Euler system of equations are shown.

Keywords

• Young measure
• Weak* limit
• Hyperbolic conservation law