论文标题
汉堡 - 希尔伯特方程的冲击形成
Shock formation for the Burgers-Hilbert equation
论文作者
论文摘要
我们证明了Burgers-Hilbert方程的有限时间。我们使用有限的$ h^5 $ -NORM构建流畅的初始数据，以使解决方案的空间衍生物的$ l^\ infty $ norm爆炸。爆炸是一个单点的渐近自相似冲击，具有明确的计算爆炸曲线。爆炸配置文件是Hölder$ 1/3 $连续性的风口浪尖。爆炸时间和位置是用明确的ODE来描述的。我们的证明使用转换来调制自相似变量，稳定的自相似于Inviscid Burgers方程的自相似解决方案的定量属性，在自相似变量的$ l^2 $ estimate中，以及希尔伯特变换的点估计和传输方程。
We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar shock at one single point with an explicitly computable blowup profile. The blowup profile is a cusp with Hölder $1/3$ continuity. The blowup time and location are described in terms of explicit ODEs. Our proof uses a transformation to modulated self-similar variables, the quantitative properties of the stable self-similar solution to the inviscid Burgers equation, an $L^2$-estimate in self-similar variables, and pointwise estimates for Hilbert transform and for transport equations.
