NONMONOTONICITY OF PRINCIPAL EIGENVALUES IN DIFFUSION RATE FOR SOME NON-SELF-ADJOINT OPERATORS WITH LARGE ADVECTION

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.