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

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.

29 pages

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Cited by in corpus (9)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• Random attractors for a class of stochastic partial differential equations driven by general additive noise
• Local and global well-posedness of SPDE with generalized coercivity conditions
• Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives
• Well-posedness of stochastic partial differential equations with fully local monotone coefficients
• Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes
• Coercivity condition for higher order moments for nonlinear SPDEs and existence of solution under local monotonicity
• Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients
• Note on the existence theory for evolution equations with pseudo-monotone operators