Nonmonotonicity of Principal Eigenvalues in Diffusion Rate for Some Non-Self-Adjoint Operators with Large Advection
Shuang Liu · SIAM Journal on Mathematical Analysis · 2024
Abstract. The paper is concerned with the monotonicity of the principal eigenvalues with respect to diffusion rate for two classes of non-self-adjoint operators: time-periodic parabolic operators and elliptic operators with shear flow. These operators behave similarly to some averaged self-adjoint elliptic operators when the frequency or flow amplitude, referred to as advection rate, is sufficiently large. It was conjectured in [ S. Liu and Y. Lou, J. Funct. Anal., 282 (2022), 109338 ] that the principal eigenvalues are monotone in diffusion rate for large advection, similarly to those self-adjoint elliptic operators. We provide some counterexamples to the conjecture by establishing the high order expansion of the principal eigenvalues for sufficiently large diffusion and advection rates.