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

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

2010-11-02

arxiv.org/abs/1011.0562v1

Nonlinear Anal. 74 (2011), 7543--7561

Mathematics

Analysis of PDEs

29 pages

Scientific paper

10.1016/j.na.2011.08.018

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.

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