| Publication type: |
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Article in peer-reviewed journal |
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| Subject: |
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Mathematics/Analysis of PDEs
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| Title: |
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Asymptotic properties of entropy solutions to fractal Burgers equation |
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| Author(s): |
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Nathaël Alibaud 1, Cyril Imbert ( ) 2, Grzegorz Karch 3 |
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| Laboratory: |
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| Abstract: |
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We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty , respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case alpha \in (0,1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions. |
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| Fulltext language: |
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English |
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| Journal: |
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SIAM Journal on Mathematical Analysis / SIAM Journal of Mathematical Analysis |
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| Audience: |
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international |
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| Publication date: |
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2010-03-10 |
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| Volume: |
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42 |
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| Issue: |
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1 |
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| Page, identifiant, ...: |
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354-376 |
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| Keyword(s): |
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fractal Burgers equation – asymptotic behavior of solutions – self-similar solutions – entropy solutions |
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| Classification: |
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MSC 35K05, 35K15 |
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| Comment: |
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23 pages. This version contains details that are skipped in the published version. |
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| ANR Project: |
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| Project Id |
partially supported by ANR project "EVOL" |
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