Perturbation Methods and First Order Partial Differential Equations

TL;DR

This paper develops explicit estimates and formulas for solutions of first order PDEs on compact manifolds, addressing existence, uniqueness, and stability issues using viscosity methods and characteristic curves.

Contribution

It introduces explicit integral formulas and pointwise estimates for first order PDEs, extending to nonlinear cases and analyzing stability and conditions for global solutions.

Findings

Explicit integral formulas for linear first order PDEs.

Existence of solutions guaranteed under certain estimates.

Stability of uniqueness under C^{1} perturbations.

Abstract

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.