Abstract: In this talk, I will present two recent results on non-uniqueness in two-dimensional fluid equations. First, we show the existence of two distinct global smooth (for t>0) solutions evolving from a common initial datum in $BMO^{-1}(T^2)$. The three-dimensional case was recently settled by Coiculescu and Palasek  (Invent. Math., 2025); however, their method does not extend to the two-dimensional problem due to geometric complications arising from the intersections of two-dimensional Mikado flows. We overcome these difficulties by introducing a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows. Second, for the two-dimensional viscous and resistive magnetohydrodynamic equations, we prove sharp non-uniqueness of weak solutions in the class $L^2_t L^p(R^2) \cap L^1_t W^{1,p}(R^2)$ for any $1 \le p < \infty$, showing the sharpness of the Ladyzhenskaya-Prodi-Serrin condition at the endpoint (2,$\infty$). These results are joint with Changxing Miao and Weikui Ye (arXiv:2602.19074, arXiv:2605.25097).