Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

Antonio Olivas Martinez,Georgy A Omel'Yanov

doi:10.1155/2009/101647

Antonio Olivas Martinez, Georgy A Omel'Yanov

Open Access

https://doi.org/10.1155/2009/101647

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Abstract

We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.

Highlights

It is well known that the uniqueness problem for weak solutions of hyperbolic quasilinear systems remains unsolved up to now in the case of arbitrary jump amplitudes
Any traditional asymptotic method does not serve for the problem of nonlinear wave interaction since it leads to the appearance of a chain of partial differential equations, the first of them is nonlinear and, coincides with the original equation
We present the asymptotic ansatz as a natural regularization of 2.3 : uε u u− − u ω1 R − u ω2 u − R ω3, 2.7 where R R x, t, ε ∈ C∞ R1 × R1 × 0, 1 is a function such that

Summary

Introduction

It is well known that the uniqueness problem for weak solutions of hyperbolic quasilinear systems remains unsolved up to now in the case of arbitrary jump amplitudes. The discrepancy in the weak asymptotics method is assumed to be small in the sense of the space of functionals Dx over test functions depending only on the “space” variable x. We consider as the main result the fact that the weak asymptotics method allows to construct the admissible limiting solution without any additional conditions. In what follows we will omit the subindex Δ for u

Asymptotic Ansatz

Preliminary Calculations

Analysis of the Singularity Dynamics

The Shock Wave Propagation over the Centered Rarefaction

Conclusion