基于旧偏微分方程组解求解新PDE方程组的研究困境咨询

Hey everyone, I'm currently stuck in my research and could really use some help unpacking this problem. Let me lay it out clearly: I'm working with a system of 6 variables — u, v, p, h₁₁, h₁₂, h₂₂ — and here's the set of partial differential equations I'm trying to analyze:

$$
g^2\frac{\partial u}{\partial X}+\frac{\partial v}{\partial Y}=\frac{h_{11}+h_{22}}{2}
$$

$$
\frac{dh_{11}}{dt}=2\gamma h_{11}+\beta\left(2\frac{\partial u}{\partial X}-2g^2\frac{\partial v}{\partial Y}-g^2h_{11}+g2h_{22}\right)
$$

$$
\frac{dh_{12}}{dt}=2\gamma h_{12}+\beta\left[2\frac{\partial u}{\partial Y}+2g^2\frac{\partial v}{\partial X}-\left(g^2+\frac{1}{g2}\right)h_{12}\right]
$$

$$
\frac{dh_{22}}{dt}=2\gamma h_{22}+\beta\left(2g^2\frac{\partial v}{\partial Y}-2\frac{\partial u}{\partial X}+\frac{1}{g^{2}h_{11}-\frac{1}{g}2}h_{22}\right)
$$

Quick heads-up: The final equation in my original draft got truncated, but the equations above represent the core of the system I'm grappling with. I'm trying to find ways to solve or analyze this system — whether that's through symmetry reductions, steady-state solution analysis, or linking it to simpler PDE systems I've studied previously. Any insights, tips, or references to similar systems would be incredibly helpful!

备注：内容来源于stack exchange，提问作者IntegralLover