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*.
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Zejun Wang’s research was supported in part by NSFC (11671193) and the Fundamental Research Funds for the Central Universities (NE2015005). Qi Zhang’s research was supported in part by NSFC (11271182 and 11501290).
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Wang, Z., Zhang, Q. Finite Time Emergence of A Shock Wave for Scalar Conservation Laws Via Lax-Oleinik Formula. Acta Math Sci 39, 83–93 (2019). https://doi.org/10.1007/s10473-019-0107-8
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DOI: https://doi.org/10.1007/s10473-019-0107-8