Andrej Zlatos

Existence and Non-Existence of Fisher-KPP Transition Fronts

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 the 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.

Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line

AbstractWe study the Case sum rules, especially C0, 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(an−1)=α and lim nbn=β exist and 2α<|β|, then the Szegő condition fails.

Regular embeddings of complete bipartite graphs

Abstract We prove that for any prime number p the complete bipartite graph K p , p has, up to isomorphism, precisely one regular embedding on an orientable surface—the well-known embedding with faces bounded by hamiltonian cycles.

ON THE LOSS OF CONTINUITY FOR SUPER-CRITICAL DRIFT-DIFFUSION EQUATIONS

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts, we prove that solutions satisfy a modulus of continuity depending only on the local L1 norm of the drift, which is a super-critical quantity.

On discrete models of the Euler equation

We consider two discrete models for the Euler equation describing incompressible fluid dynamics. These models are infinite coupled systems of ODEs for the functions

Sparse potentials with fractional Hausdorff dimension

u_j

Diffusion in Fluid Flow: Dissipation Enhancement by Flows in 2D

which can be thought of as wavelet coefficients of the fluid velocity. The first model has been proposed and studied by Katz and Pavlovic. The second has been recently discussed by Waleffe and goes back to Obukhov studies of the energy cascade in developed turbulence. These are the only basic models of this type satisfying some natural scaling and conservation conditions. We prove that the Katz-Pavlovic model leads to finite time blowup for any initial datum, while the Obukhov model has a global solution for any sufficiently smooth initial datum.

Quenching of combustion by shear flows

Abstract We construct non-random bounded discrete half-line Schrodinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.

Higher-order Szegő theorems with two singular points

We consider the advection-diffusion equation on ℝ2, with u a periodic incompressible flow and A ≫ 1 its amplitude. We provide a sharp characterization of all u that optimally enhance dissipation in the sense that for any initial datum φ0 ∈ L p (ℝ2), p < ∞, and any τ > 0, Our characterization is expressed in terms of simple geometric and spectral conditions on the flow. Moreover, if the above convergence holds, it is uniform for φ0 in the unit ball in L p (ℝ2), and ‖·‖∞ can be replaced by any ‖·‖ q , with q > p. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.

Quenching and propagation of combustion without ignition temperature cutoff

We consider a model describing premixed combustion in the presence of fluid flow: reaction diffusion equation with passive advection and ignition type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, provided the velocity’s amplitude is adequately large? Even for shear flows, the solution turns out to be surprisingly subtle. In this paper, we provide a sharp characterization of quenching for shear flows: the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant. The efficiency of quenching depends strongly on the geometry and scaling of the flow. We discuss the cases of slowly and quickly varying flows, proving rigorously scaling laws that have been observed earlier in numerical experiments. The results require new estimates on the behavior of the solutions to advectionenhanced diffusion equation (also known as passive scalar in physical literature), a classical model describing a wealth of phenomena in nature. The technique involves probabilistic and PDE estimates, in particular applications of Malliavin calculus and central limit theorem for martingales.