Nonincreasing number of alternate changes of solutions for a one dimensional scalar conservation law

Abstract

In this paper, solutions of one dimensional conservation law $\It{u_{t}} + \It{f(u)_{x}} = 0 $ are treated. This PDE governs various one-dimensional phenomena involving fluid dynamics and models the foundation and propagation of shock waves. In general, the conservation law has no classical solutions due to discontinuities. i.e. shock waves. So, vanishing viscosity method is used for observing the behavior of solutions rather than handling the original equation. By adding a second order derivative with small coefficient, we can construct a second-order semilinear parabolic equation which has smooth solutions. With some theorems about regularity of solutions, zero sets and maximum-principle, nonincreasing property of intersection points w.r.t time t of the approximated equation can be proved. And as the viscosity coefficient tends to zero, we obtain the similar property of solutions as limits of solutions of parabolic equation. That is, the number of alternate changes of two solutions is nonincreasing in t. Here the alternate changes means the sign changes of difference of two solutions.