论文标题
在非线性声学中应用的准电波方程的抛物线近似
Parabolic approximation of quasilinear wave equations with applications in nonlinear acoustics
论文作者
论文摘要
这项工作涉及抛物线扰动与平滑界域上的误线性波方程的收敛分析。特别是，我们考虑具有二次类型非线性的波动方程，涵盖了非线性声学的两个经典模型，即Westervelt和Kuznetsov方程。通过采用高阶能量分析，我们可以通过线性速率在标准能量范围内获得其解决方案与相应的Inviscid方程解决方案的收敛性，假设较小的数据和足够短的时间。但是，可以在低阶规范中施加的初始数据的小，而不是能量分析中所需的规范。它仅来自确保研究波方程的非分类。此外，我们以非负，可能消失的声音扩散率在有限的域上解决了两个经典模型的本地良好问题。
This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analysis, we obtain convergence of their solutions to the corresponding inviscid equations' solutions in the standard energy norm with a linear rate, assuming small data and a sufficiently short time. The smallness of initial data can, however, be imposed in a lower-order norm than the one needed in the energy analysis. It arises only from ensuring the non-degeneracy of the studied wave equations. In addition, we address the open questions of local well-posedness of the two classical models on bounded domains with a non-negative, possibly vanishing sound diffusivity.
