Accelerating diffusion by incompressible drift on the two-dimensional torus


Abstract: In this paper we construct an explicit sequence of divergence-free vector fields $\rm{b}_{n}$ that pushes the spectral gap of the nonself-adjoint operator $A_{\rm{b}_{n}}=\Delta +\rm{b}_{n}\cdot\nabla $ to infinity. The spectral gap is an indicator for the speed at which this diffusion converges toward its equilibrium, which corresponds to the uniform distribution.

Keywords: Nonself-adjoint operator, spectral gap, divergence-free vector fields, rearrangement, comparison manifold, Faber-Krahn inequality

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