Acta Mathematica Scientia

, Volume 39, Issue 1, pp 83–93

# Finite Time Emergence of A Shock Wave for Scalar Conservation Laws Via Lax-Oleinik Formula

• Zejun Wang (王泽军)
• Qi Zhang (张琦)
Article

## Abstract

In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law ut + F(u)x = 0. First, we prove a simple but useful property of Lax-Oleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)−qF′(q)+ L(F′(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T*. Meanwhile, we can give the equation of the shock and an explicit value of T*.

## Key words

scalar conservation law Lax-Oleinik formula Riemann problem asymptotic behavior

## 2010 MR Subject Classification

35B40 35L65 76Y05

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