位势流方程中的真空问题

Hydrodynamic equations are important research areas for partial differential equations. It is not only rich in the theoretical research of mathematics, but aslo has a deep backgroud in physics. In this project, we mainly study one case about the boundary-value problem of the three dimensional potential equation. We consider the wellposedness of the solution for the supersonic flow past a strong bend or corner. Here the incoming flow is a perturbation of a constant flow. The problem is described in the classical literature 《Supersonic flow and shock waves》(P. 273-278) in hydrodynamics authored by R. Courant and K. O. Friedrichs. If the bend (or corner) is sufficiently strong, then there will appear a complete rarefaction wave (or a complete centered rarefaction wave), i. e. there is a vacuum region between the flow and the bend (or the corner). We will prove the wellposedness and especially the local stability.

流体力学方程是偏微分方程中的重要研究领域，不但富有数学理论研究价值，而且具有非常强的物理背景。本项目主要研究一类三维位势流方程的边值问题，即考虑超音速位势流流经一个大弯曲或者拐角时解的适定性。这里来流为一个常数来流的扰动。问题描述于R. Courant和K. O. Friedrichs流体动力学方面的经典著作《Supersonic flow and shock waves》(P. 273-278)。当弯曲(或者拐角)角度充分大时，当超音速来流流经拐角时，将会产生一个完全疏散的疏散波(或者中心疏散波)，即在流体和弯曲(或者拐角)之间将会产生真空区域，我们将证明解的适定性，特别是局部稳定性。

本项目中，我们主要考虑在临界Lp型Besov空间中可压有粘的MHD方程则的全局适定性和低Mach数极限。更精确地说，我们证明了在等熵情形下，当Mach数趋于0时，可压MHD方程的解在临界Lp框架下会收敛到不可压MHD方程的解。此外，我们还得到了收敛的速率。