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Andrej Zlatos

Publications and source records attributed to Andrej Zlatos.

53 records · Page 3Linked to original sources
arXiv2009

Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

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.

math.AP↗
arXiv2008

On the zero temperature limit of interacting corpora

We characterize the zero-temperature limits of minimal free energy states for interacting corpora -- that is, for objects with finitely many degrees of freedom, such as articulated rods. These limits are measures supported on zero-level-sets of the interaction potential. We describe a selection mechanism for the limits that is mediated by evanescent entropic contributions.

math.AP↗
arXiv2007

Pulsating Front Speed-up and Quenching of Reaction by Fast Advection

We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve this speed-up are precisely those which, when scaled properly, are able to quench any ignition reaction.

math.AP↗
arXiv2007

Relaxation Enhancement by Time-Periodic Flows

We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \cdot \nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $Γ+iAL(t)$ with a negative unbounded self-adjoint operator $Γ$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$.

math.AP↗
arXiv2007

KPP Pulsating Front Speed-up by Flows

We obtain a criterion for pulsating front speed-up by general periodic incompressible flows in two dimensions and in the presence of KPP nonlinearities. We achieve this by showing that the ratio of the minimal front speed and the effective diffusivity of the flow is bounded away from zero and infinity by constants independent of the flow. We also study speed-up of reaction-diffusion fronts by various examples of flows in two and three dimensions.

math.AP↗
arXiv2007

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

We consider the advection-diffusion equation \[ ϕ_t + Au \cdot \nabla ϕ= Δϕ, \qquad ϕ(0,x)=ϕ_0(x) \] on $\bbR^2$, with $u$ a periodic incompressible flow and $A\gg 1$ its amplitude. We provide a sharp characterization of all $u$ that optimally enhance dissipation in the sense that for any initial datum $ϕ_0\in L^p(\bbR^2)$, $p<\infty$, and any $τ>0$, \[ \|ϕ(\cdot,τ)\|_{L^\infty(\bbR^2)} \to 0 \qquad \text{as $A\to\infty$.} \] 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 $ϕ_0$ in the unit ball of $L^p(\mathbb{R}^2)$, $p<\infty$, and $\|\cdot\|_\infty$ can be replaced by any $\|\cdot\|_q$, $q>p$. Extensions to higher dimensions and applications to reaction-advection-diffusion equations are also considered.

math.AP↗
arXiv2005

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems

Let $μ$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $α_n$ its Verblunsky coefficients, and $Φ_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of $Φ_n$ in terms of the $α_n$. If the function $\log w$ is in $L^1(dθ)$, we do the same for its Fourier coefficients. As an application we prove that if $α_n \in \ell^4$ and $Q(z) = \sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) = \sum_{m=0}^N \bar q_m z^m$ and $S$ the left shift operator on sequences we have $|Q(e^{iθ})|^2 \log w(θ) \in L^1(dθ)$ if and only if $\{\bar Q(S)α\}_n \in \ell^2$. We also study relative ratio asymptotics of the reversed polynomials $Φ_{n+1}^*(μ)/Φ_n^*(μ)-Φ_{n+1}^*(ν)/Φ_n^*(ν)$ and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures $μ$ and $ν$ for this difference to converge to zero uniformly on compact subsets of $\bbD$.

math.CA↗
arXiv2005

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 $u_j$ 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.

math.AP↗
arXiv2005

Sharp Transition Between Extinction and Propagation of Reaction

We consider the reaction-diffusion equation \[ T_t = T_{xx} + f(T) \] on $\bbR$ with $T_0(x) \equiv χ_{[-L,L]} (x)$ and $f(0)=f(1)=0$. In 1964 Kanel' proved that if $f$ is an ignition non-linearity, then $T\to 0$ as $t\to\infty$ when $L L_1$. We answer the open question of relation of $L_0$ and $L_1$ by showing that $L_0=L_1$. We also determine the large time limit of $T$ in the critical case $L=L_0$, thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.

math.AP↗
arXiv2005

Quenching of combustion by shear flows

We consider a simple model describing premixed combustion in the presence of fluid flow: reaction diffusion equation with passive advection and ignition type nonlinearity. Strong advection can suppress flames - a process we call quenching. A flow is called quenching if any compactly supported initial data will become extinct provided that the amplitude of the flow is chosen sufficiently large. In this paper, we provide a sharp characterization of quenching shear flows.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 analytically behavior that has been observed earlier in numerical experiments. The technique involves probabilistic and PDE estimates, in particular applications of Malliavin calculus and central limit theorem for martingales.

math.AP↗
arXiv2004

Higher-Order Szego Theorems With Two Singular Points

We consider probability measures, $dμ=w(θ) \f{dθ}{2π} +dμ_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{α_j\}_{j=0}^\infty$. We prove for $θ_1\neqθ_2$ in $[0,2π)$ and $(δβ)_j=β_{j+1}$ that \[ \int [1-\cos(θ-θ_1)][1-\cos(θ-θ_2)] \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \bigl|\bigl\{(δ-e^{-iθ_2}) (δ-e^{-iθ_1}) α\bigr\}_j\bigr|^2 +\abs{α_j}^4 <\infty \] We also prove that \[ \int (1-\cosθ)^2 \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \abs{α_{j+2}-2α_{j+1} +α_j}^2 + \abs{α_j}^6 <\infty \]

math-ph↗
arXiv2004

Quenching and Propagation of Combustion Without Ignition Temperature Cutoff

We study a reaction-diffusion equation in the cylinder $Ω= \mathbb{R}\times\mathbb{T}^m$, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of $T$ larger than three as $T\to 0$ and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger for the no-flow case. We also consider shear flows with large amplitude and show that if the reaction power-law decay is larger than three and the flow has only small plateaux (connected domains where it is constant), then any compactly supported initial datum is quenched when the flow amplitude is large enough (which is not true if the power is smaller than three or in the presence of a large plateau). This extends results of Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our work carries over to the case $Ω= \mathbb{R}^n\times\mathbb{T}^m$, when the critical power is $1 + 2/n$, as well as to certain non-periodic flows.

math.AP↗
arXiv2004

Sum Rules for Jacobi Matrices and Divergent Lieb-Thirring Sums

Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to obtain various results on divergence of certain Lieb-Thirring sums of eigenvalues and Szego-type integrals of the a.c. part of the spectral measure.

math-ph↗
arXiv2002

The Szego Condition for Coulomb Jacobi Matrices

A Jacobi matrix with $a_n\to 1$, $b_n\to 0$ and spectral measure $ν'(x)dx + dν_{sing}(x)$ satisfies the Szeg\H o condition if $\int_{0}^π\ln \bigl[ ν'(2\cosθ) \bigr] dθ$ is finite. We prove that if $a_n = 1 + \frac α{n} + O(n^{-1-\eps})$ and $b_n = \frac β{n} + O(n^{-1-\eps})$ with $2α\ge |β|$ and $\eps>0$, then the corresponding matrix is Szeg\H o.

math-ph↗
arXiv2002

Sparse Potentials With Fractional Hausdorff Dimension

We construct non-random bounded discrete half-line Schr\" odinger 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.

math-ph↗
arXiv2002

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

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 $2α<\absβ$, then the Szegő condition fails.

math-ph↗