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首页> 外文期刊>Differential and integral equations >NONUNIFORM DEPENDENCE AND WELL POSEDNESSFOR THE PERIODIC HUNTER-SAXTON EQUATION
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NONUNIFORM DEPENDENCE AND WELL POSEDNESSFOR THE PERIODIC HUNTER-SAXTON EQUATION

机译：周期猎人-萨克森方程的非均匀依赖性和适当性

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摘要

It is proved that the flow map for the Hunter-Saxton (HS) equation from the homogeneous Sobolev space H~s(T) into the space C([0, T], H~s (T) is continuous but not uniformly continuous on bounded subsets. To demonstrate this sharpness of continuity, two sequences of bounded solutions to the HS equation are constructed whose distance at the initial time converges to zero and whose distance at any later time is bounded from below by a positive constant. To achieve this result, appropriate approximate solutions are chosen and then the actual solutions are found by solving the Cauchy problem with initial data taken to be the value of approximate solutions at time zero. Then, using well-posedness estimates, it is shown that the difference between solutions and approximate solutions is negligible.
机译：证明了从均匀Sobolev空间H〜s（T）到空间C（[0，T]，H〜s（T）的Hunter-Saxton（HS）方程的流图是连续的，但不是一致连续的为了证明这种连续性的尖锐性，构建了HS方程的两个有界解序列，它们在初始时间处的距离收敛为零，而在以后任何时间处的距离都由正常数从下方限定。结果，选择合适的近似解，然后通过将初始数据取为零时的近似解的值来解决柯西问题，从而找到实际解，然后使用适定性估计表明，解之间的差异近似解可以忽略不计。

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