We derive and analyze transport equations for the energy density o f w aves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization e ects, coupling of di erent t ypes of waves, etc. We also show that di usive behavior occurs on long time and distance scales and we determine the di usion coe cients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.

We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form Γ + iAL with a negative unbounded self-adjoint operator Γ, a self-adjoint operator L, and parameter A 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered.

We consider the Fisher-KPP equation with a nonlocal saturation effect defined through an interaction kernel φ(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transformφ(ξ) is positive or if the length σ of the nonlocal interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Our second concern is the study of traveling waves. We prove that this equation admits traveling wave solutions that connect u = 0 to an unknown positive steady state u ∞ (x), for all speeds c ≥ c *. The traveling wave connects to the standard state u ∞ (x) ≡ 1 under the aforementioned conditions:φ(ξ) > 0 or σ is sufficiently small. However, the wave is not monotonic for σ large.

In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions u along their level sets to the profile of the minimal travelling front.

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.