# American Institute of Mathematical Sciences

June  2020, 40(6): 3715-3736. doi: 10.3934/dcds.2020028

## A functional approach towards eigenvalue problems associated with incompressible flow

 1 Institute for Mathemaical Sciences, Renmin University of China, Beijing 100872, China 2 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

*Corresponding author: Shuang Liu

Received  September 2018 Revised  April 2019 Published  October 2019

Fund Project: SL was partially supported by the NSFC grant No. 11571364 and the Outstanding Innovative Talents Cultivation Funded Programs 2018 of Renmin Univertity of China. YL was partially supported by the NSF grants DMS-1411176 and DMS-1853561

We propose a certain functional which is associated with principal eigenfunctions of the elliptic operator $L_{A} = -\mathrm{div}(a(x)\nabla )+A\mathbf{V}\cdot\nabla +c(x)$ and its adjoint operator for general incompressible flow $\mathbf{V}$. The functional can be applied to establish the monotonicity of the principal eigenvalue $\lambda_1(A)$, as a function of the advection amplitude $A$, for the operator $L_{A}$ subject to Dirichlet, Robin and Neumann boundary conditions. This gives a new proof of a conjecture raised by Berestycki, Hamel and Nadirashvili [5]. The functional can also be used to prove the monotonicity of the normalized speed $c^{*}(A)/A$ for general incompressible flow, where $c^{*}(A)$ is the minimal speed of traveling fronts. This extends an earlier result of Berestycki [3] for steady shear flow.

Citation: Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028
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