Singularity formation for Burgers' equation with transverse viscosity
Singularity formation for Burgers' equation with transverse viscosity
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DOI：
10.24033/asens.2513
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发表时间：
2018-03
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影响因子：
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通讯作者：
Charles Collot;T. Ghoul;N. Masmoudi
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文献类型：
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作者：
Charles Collot;T. Ghoul;N. Masmoudi
本文考虑了具有横向粘性的Burgers方程$$\partial_tu+u\partial_xu-\partial_{yy}u=0，\ \（x，y）\in \mathbb R^2，\ \ u：[0，T）\times \mathbb R^2\rightarrow \mathbb R.$$我们构造并精确描述了一个家庭的解决方案，在有限时间内成为奇异的梯度成为无界。对于首阶，该解由Burgers方程沿着$x$变量的向后自相似解给出，其标度参数根据抛物方程沿着$y$变量发展，其中之一是二次半线性热方程。我们发展了一个新的框架，适用于这种混合双曲/抛物爆破问题，重新构建平坦的半线性热方程的爆破剖面，和Burgers方程的冲击自相似性。
We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \ u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.