高阶非线性色散偏微分方程中激波的形成：熵的非唯一性和不存在性
Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy
摘要：
证明了三阶一维非线性色散方程存在由爆破型相似解描述的单点梯度突变。爆破后的解具有激波型自相似扩张。由于这种扩张不是唯一的，这意味着激波型解的原则不唯一性，也不存在任何熵型描述的真正的唯一的解决方案。一个困难的自由边界设置，与冲击指定的额外条件，是必要的，以恢复在这样的问题的唯一性。
英文摘要：
It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such extensions are not unique, this implies the principle nonuniqueness of shock-type solutions and also nonexistence of any entropy-type description of proper unique solutions. A difficult free-boundary setting, with extra conditions specified on shocks, are necessary to restore uniqueness in such problems.
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