A new mathematical model based on differential equations and deep learning is proposed to study the propagation mechanism of southern corn rust (SCR). Equilibrium and stability analyses of system are conducted in detail. Compared with the quantitative assessment of SCR, numerical parameter estimation and the corresponding interpretation are given.

The 2D/3D interaction systems consisting of the incompressible Navier-Stokes equations in a bounded domain and the classical full von Kármán shallow shell equation were investigated in this paper, where the fluid and the shell were coupled via both the transverse and longitudinal displacements. The global existence of weak solutions in 2D/3D was proved by using a special choice of basis functions for the related elliptic equations, which allows one to tackle the delicate estimates for the interaction between the fluid and the shell. Moreover, the uniqueness of the weak solution in 2D was also established via a localization technique.