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Seminar第1912期 具有时间阻尼的一维可压缩欧拉方程

创建日期 7/11/2019 福平   浏览次数  154 返回    
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报告主题:具有时间阻尼的一维可压缩欧拉方程
报告人:梅茗 教授 (加拿大Champlain College & McGill University)
报告时间:2019年7月11日(周四)15:00
报告地点:校本部G507
邀请人:朱佩成

报告摘要:This talk deals with the Cauchy problem for the 1D compressible Euler equations with time-dependent damping, where the time-vanishing damping in the like form of $\frac{\mu}{(1+t)^\lambda}$ makes the variety of the dynamic system. For $0<\lambda<1$ and $\mu>0$, or $\lambda=1$ but $\mu>2$, where $\lambda=1$ and $\mu=2$ is the critical case, when the derivatives of the initial data are small, but the initial data themselves are allowed to be arbitrarily large, the solutions are proved to exist globally in time; for these global solutions, we further technically determine what will be their corresponding asymptotic profiles, particularly in the critical case with \lambda=1, and then show the convergence with optimal rates; while, when the derivatives of the initial data are large at some points, then the solutions are still bounded, but their derivatives will blow up at finite time. For $\lambda=1$ and $0<\mu<1$, the derivatives of solutions will blow up for all initial data, including the small initial data. In order to prove the global existence of the solutions with large initial data, we introduce a new energy functional, which crucially helps to build up the maximum principle for the corresponding Riemann invariants, and the uniform boundedness for the local solutions, these keys finally guarantee the global existence of the solutions. The results presented here essentially improve and develop the existing studies. Finally, some numerical simulations in different cases are carried out, which further confirm our theoretical results.


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