二维非定常Prandtl边界层方程组在倒流点出现前后的适定性分析

本项目将研究二维非定常不可压流体边界层中倒流点出现前后边界层方程组在Sobolev空间中的适定性问题。对于逆压梯度作用下的二维非定常不可压单调边界层流，随着时间的演化流场中将会出现倒流，进而可能发生边界层的分离。本项目将研究二维非定常Prandtl边界层方程组在倒流点出现前是否在Sobolev空间中适定的问题，在一定条件下建立Prandtl边界层方程组在倒流点出现前的全局适定性理论；此外，研究边界层在倒流点出现后的稳定性，在不同外流压力梯度条件下分别建立二维非定常Prandtl边界层方程组在Sobolev空间中的适定性和不适定性理论。此研究将给出逆压梯度条件下二维非定常Prandtl边界层方程组在Sobolev空间中的解的性态刻画，对非定常Prandtl边界层方程组在Sobolev空间中非Oleinik单调类里的适定性问题有所突破，同时促进非定常边界层分离问题数学理论的发展。

This proposal focuses on the well-posedness and ill-posedness of the two-dimensional unsteady incompressible boundary layer equations in Sobolev spaces before and after the appearance of back-flow point in the boundary layer. For the two-dimensional unsteady incompressible boundary layer with monotonicity structure under an adverse pressure gradient, as time involves, there will be back flow forming in the boundary layer, then boundary layer separation may happen. In this proposal, the well-posedness of the two-dimensional unsteady Prandtl boundary layer equations in Sobolev spaces before the appearance of back-flow point will be studied, global well-posedness theories for the Prandtl boundary layer equations before the appearance of back-flow point will be established under certain conditions. In addition, the stability of boundary layer after the appearance of back-flow point will be investigated in this proposal, well-posedness and ill-posedness of the two-dimensional unsteady Prandtl equations in Sobolev spaces are considered under different pressure gradients of outflow respectively. This proposal will give a characterization of the behavior of the solution to the two-dimensional unsteady incompressible boundary layer equations, make a breakthrough in the issue of well-posedness of unsteady Prandtl boundary layer equations in Sobolev spaces without Oleinik monotonicity structure, and promote the development of mathematical theories for unsteady boundary layer separation.