[再寄小读者之数学篇](2014-06-21 微分不等式)

Assume that $a$ is a positive constant, $x(t),y(t)$ are two nonnegative $C^1(\bbR^+)$ functions, and $D(t)$ is a nonnegative function, satisfying $$\bex \cfrac{\rd}{\rd t} (x^2+y^2)+D \leq a(x^2+y^2+x+y)D. \eex$$ If additionally, the initial data satisfy $$\bex x^2(0)+y^2(0)+\sqrt{2(x^2(0)+y^2(0))}<\cfrac{1}{a}, \eex$$ then, for any $t>0$, one has $$\bex x^2(t)+y^2(t)+x(t)+y(t)<x^2(0)+y^2(0)+\sqrt{2(x^2(0)+y^2(0))}<\cfrac{1}{a}. \eex$$ see [D. Chae, P. Degond, J.G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. I. H. Poincar\'e-AN, 31 (2014),555--565].