极值曲面方程Lipschitz解的适定性和长时间行为
Well-posedness and long-time behavior of Lipschitz solutions to extremal surface equations
摘要：
证明了在一维空间中极值曲面方程的Lipschitz解等价于非严格双曲守恒律方程组的$L^\infty(\R)$中的熵解.我们得到了系统Cauchy问题的一个显式表示公式和熵解的唯一性。利用这个公式，我们还得到了显式行波熵解在$t \rightarrow +\infty$范数下的收敛性和收敛速度。此外，当初始数据在有限空间间隔之外为常数时，熵解在有限时间后变成显式行波。最后证明了熵解的$L^1$稳定性。
英文摘要：
We show that in one space dimension Lipschitz solutions of extremal surface equations are equivalent to entropy solutions in $L^\infty(\R)$ of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation formula and the uniqueness of the entropy solutions to the Cauchy problem of the system. By using this formula, we also obtain the convergence and convergence rates as $t \rightarrow +\infty$ of the entropy solutions to explicit traveling waves in the $L^1(\R)$ norm. Moreover, when initial data are constants outside of a finite space interval, the entropy solutions become the explicit traveling waves after a finite time. Finally, we prove $L^1$ stabilities of the entropy solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
