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Vladimir Kozlov

Publications and source records attributed to Vladimir Kozlov.

45 records · Page 3Linked to original sources
arXiv2015

New perspectives: stability and complex dynamics of food webs via Hamiltonian methods

We investigate global stability and dynamics of large ecological networks by classical methods of the dynamical system theory, including Hamiltonian methods, and averaging. Our analysis exploits the network topological structure, namely, existence of strongly connected nodes (hubs) in the networks. We reveal new relations between topology, interaction structure and network dynamics. We describe mechanisms of catastrophic phenomena leading to sharp changes of dynamics and investigate how these phenomena depend on ecological interaction structure. We show that a Hamiltonian structure of interaction leads to stability and large biodiversity.

math.AP↗
arXiv2014

Hadamard Type Asymptotics for Eigenvalues of the Neumann Problem for Elliptic Operators

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. We consider two applications: the Laplacian in both $C^{1,α}$ and Lipschitz domains. For the $C^{1,α}$ case, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result. For the Lipschitz case, the proximity of eigenvalues is estimated.

math.AP↗
arXiv2014

On bounds and non-existence in the problem of steady waves with vorticity

For the problem describing steady, gravity waves with vorticity on a two-dimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoulli's constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli's constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

math-ph↗
arXiv2014

Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: for waves with fixed Bernoulli's constant and fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Sufficient conditions guaranteeing the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate general results.

math-ph↗
arXiv2013

Oblique derivative problem for non-divergence parabolic equations with discontinuous in time coefficients

We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in $t$ coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class ${\cal C}^{1,δ}$, $δ\in (0,1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.

math.AP↗
arXiv2012

No steady water waves of small amplitude are supported by a shear flow with still free surface

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadric ones with arbitrary positive coefficients.

math-ph↗
arXiv2012

Domain dependence of eigenvalues of elliptic type operators

The dependence on the domain is studied for the Dirichlet eigenvalues of an elliptic operator considered in bounded domains. Their proximity is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. As an application, the stability of eigenvalues is estimated by virtue of integrals of squares of the gradients of eigenfunctions for elliptic problems in different domains. It occurs that these stability estimates imply well-known inequalities for perturbed eigenvalues.

math.SP↗
arXiv2011

The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge

We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders $Ω\times(0,T)$, where $Ω$ is a bounded domain with an edge or with a conical point.

math.AP↗