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.

No abstract

It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars which interpolates between these two equations, via a parameter α ∈ [0, 1 2 ] appearing in the kernels of their Biot-Savart laws. The values α = 0 and α = 1 2 correspond to the 2D Euler and SQG cases, respectively. We prove global in time regularity for the 2D Euler patch model, even if the patches initially touch the boundary of the half-plane. On the other hand, for any sufficiently small α > 0, we exhibit initial data which lead to a singularity in finite time. Thus, these results show a phase transition in the behavior of solutions to these equations, and provide a rigorous foundation for classifying the 2D Euler equations as critical.

We study the Case sum rules, especially C 0 , for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if lim n(a n − 1) = α and lim nb n = β exist and 2α < |β|, then the Szegő condition fails.2. The full theorem (Theorem 4.1) does not require the limit (1.3) to exist, but is more complicated to state in that case.3. If the three quantities are finite, many additional sum rules hold. 4. This is what Killip-Simon [10] call the C 0 sum rule. 5. Peherstorfer-Yuditskii [16] (see their remark after Lemma 2.1) prove that if Z(J) < ∞, E 0 (J) = ∞, then the limit in (1.3) is also infinite.

We use a new method in the study of Fisher-KPP reaction-diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction-diffusion equations in several spatial dimensions. Our method is based on the construction of sub-and super-solutions to the non-linear PDE from solutions of its linearization at zero. RésuméOn utilise une nouvelle méthode pour l'étude d'équations de réaction-diffusion de type Fisher-KPP, afin de démontrer l'existence de fronts de transition pour des non linéarités hétérogènes de type KPP en dimension 1 d'espace. On obtient également de nouvelles estimations sur les solutions entières d'équations de réaction-diffusion de type KPP en dimensions d'espace supérieures. Notre méthode repose sur la construction de sous-et sur-solutions de l'EDP non linéaires à partir de solutions de sa linéarisation en 0.

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.

Abstract. We prove that there are solutions to the Euler equation on the torus with C 1,α vorticity and smooth except at one point such that the vorticity gradient grows in L ∞ at least exponentially as t → ∞. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev andŠverák [5].

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.