Lyapunov Exponents for Passive Scalars in White-in-Time Flows

Abstract: We study a passive scalar on the two-dimensional torus driven by an arbitrary finite set of white-in-time divergence-free Fourier modes. We prove that the exponential $L<sup>2$ dissipation rate of the advection-diffusion equation is uniformly bounded as the diffusivity tends to zero as long as there exist two nonparallel forced modes. If not, then the dissipation rate can be arbitrarily fast depending on the initial data, but the top Lyapunov exponent remains uniformly bounded below. The proof uses a projection onto a finite set of modes which depends on the forcing set. Part of our argument follows the structure of the four-mode case investigated by Chemnitz and Chemnitz, but uses a strictly weaker condition than their matrix inequality between Frobenius and operator norms. Combined with an accessibility argument, this allows us to consider an arbitrary set of forcing modes and initial data.