In this paper, we give explicit estimates that insure the existence of solutions for first order partial differential operators on compact manifolds, using a viscosity method. In the linear case, an explicit integral formula can be found, using the characteristics curves. The solution is given explicitly on the critical points and the limit cycles of the vector field of the first order term of the operator. In the nonlinear case, a generalization of the Weitzenbock formula provides pointwise estimates that insure the existence of a solution, but the uniqueness question is left open. Nevertheless we prove that uniqueness is stable under a C^{1} perturbation. Finally, we give some examples where the solution fails to exist globally, justifying the need to impose conditions that warrant global existence. The last result reveals that the zero order term in the first order operator is necessary to obtain generically bounded solutions.
A stochastic invariantization method for Itô stochastic perturbations of differential equations
一般来说，只要使用伊藤形式主义，向具有不变流形的微分方程添加随机扰动就会破坏不变性。在这篇文章中，我们提出了一个不变化方法的扰动伊藤的情况下，可以用来恢复不变性。然后，我们应用我们的研究结果开发一个随机版本的Landau-Lifshitz方程。我们特别讨论了Etore等人在[P. Étoré，S.Labbé，J. Lelong，Long time behavior of a stochastic nanoparticle，J. Differential Equations 257（2014），2115-2135]中获得的先前结果。
