Existence and Uniqueness of Solutions to Nonlinear Evolution Equations with Locally Monotone Operators

TL;DR

This paper proves the existence and uniqueness of solutions for a broad class of nonlinear evolution equations with locally monotone operators, extending classical results and applying to various PDEs.

Contribution

It generalizes the classical monotone operator theory to locally monotone operators, establishing new existence and uniqueness results for nonlinear evolution equations.

Findings

Local monotonicity implies pseudo-monotonicity.

Results apply to PDEs like Navier-Stokes and p-Laplace equations.

Extended classical theory to broader operator classes.

Abstract

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$\alpha$ model and $p$-Laplace equation with non-monotone perturbations.