Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2004
We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem fo... more We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two-dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.
Journal of Mathematical Analysis and Applications, 2014
This paper considers to the equation S U (Q) |P − Q| N−1 dS(Q) = F (P), P ∈ S, where the surface ... more This paper considers to the equation S U (Q) |P − Q| N−1 dS(Q) = F (P), P ∈ S, where the surface S is the graph of a Lipschitz function ϕ on R N , which has a small Lipschitz constant. The integral in the left-hand side is the single layer potential corresponding to the Laplacian in R N+1. Let Λ(r) be a Lipschitz constant of ϕ on the ball centered at the origin with radius 2r. Our analysis is carried out in local L p-spaces and local Sobolev spaces, where 1 < p < ∞, and results are presented in terms of Λ. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behaviour of the solutions near a point on the surface. The estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.
Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidim... more Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.
We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dim... more We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.
Using the dimension reduction procedure for a three-dimensional elasticity system, we derive a tw... more Using the dimension reduction procedure for a three-dimensional elasticity system, we derive a two-dimensional model for elastic laminate walls of a blood vessel. In the case of a sufficiently small wall thickness, we derive a system of limit equations coupled with the Navier-Stokes equations through the stress and velocity, i.e., dynamic and kinematic conditions on the interior surface of the wall. We deduce explicit formulas for the effective rigidity tensor of the wall in two natural cases. We show that if the blood flow remains laminar, then the cross-section of the orthotropic homogeneous blood vessel becomes circular. Bibliography: 30 titles. Illustrations: 2 figures.
Function Spaces, Differential Operators and Nonlinear Analysis, 2003
We consider the Dirichlet problem for elliptic equations of arbitrary order and prove an asymptot... more We consider the Dirichlet problem for elliptic equations of arbitrary order and prove an asymptotic formula for a singular solution near a boundary point. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.
Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharm... more Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity γ that describes the depth O(ε γ) of irregularity (ε is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.
Journal de Mathématiques Pures et Appliquées, 2008
The paper deals with the classical non-linear problem of steady two-dimensional waves on water of... more The paper deals with the classical non-linear problem of steady two-dimensional waves on water of finite depth. The problem is formulated so that it describes all waves without stagnation points on the free-surface profiles that are bounded themselves and have bounded slopes. By virtue of reducing the problem to an integro-differential equation the following three results are proved. First, there are no waves when the flow is critical. Second, there are no waves having profiles totally above the upper boundary of the uniform subcritical stream. Finally, only two types of the free-surface behaviour are possible at positive (or/and negative) infinity: the profile either oscillates infinitely many times around the upper boundary of the subcritical uniform stream or asymptotes the upper level of a uniform stream (subcritical or supercritical). The latter assertion is proved under additional assumption that the slope of the free surface is a uniformly continuous function.
Zeitschrift für angewandte Mathematik und Physik, 2008
The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrica... more The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrical domain has a cut (crack) where inequality type boundary conditions are imposed. A behavior of the solution near the crack tips is analyzed. In particular, estimates for the second derivatives in a weighted Sobolev space are obtained and asymptotics of the solution near crack tips is established.
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2004
We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem fo... more We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two-dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is demonstrated that there exist domains such that at least one of the eigenfunctions has a nodal line or lines with both ends on the free surface (earlier, Kuttler tried to prove that there are no such nodal lines for all domains but his proof is erroneous). It is also shown that the fundamental eigenvalue is simple, and for the corresponding eigenfunction the behaviour of the nodal line is characterized. For this purpose, a new variational principle is proposed for an equivalent statement of the sloshing problem in terms of the conjugate stream function.
Journal of Mathematical Analysis and Applications, 2014
This paper considers to the equation S U (Q) |P − Q| N−1 dS(Q) = F (P), P ∈ S, where the surface ... more This paper considers to the equation S U (Q) |P − Q| N−1 dS(Q) = F (P), P ∈ S, where the surface S is the graph of a Lipschitz function ϕ on R N , which has a small Lipschitz constant. The integral in the left-hand side is the single layer potential corresponding to the Laplacian in R N+1. Let Λ(r) be a Lipschitz constant of ϕ on the ball centered at the origin with radius 2r. Our analysis is carried out in local L p-spaces and local Sobolev spaces, where 1 < p < ∞, and results are presented in terms of Λ. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behaviour of the solutions near a point on the surface. The estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.
Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidim... more Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.
We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dim... more We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.
Using the dimension reduction procedure for a three-dimensional elasticity system, we derive a tw... more Using the dimension reduction procedure for a three-dimensional elasticity system, we derive a two-dimensional model for elastic laminate walls of a blood vessel. In the case of a sufficiently small wall thickness, we derive a system of limit equations coupled with the Navier-Stokes equations through the stress and velocity, i.e., dynamic and kinematic conditions on the interior surface of the wall. We deduce explicit formulas for the effective rigidity tensor of the wall in two natural cases. We show that if the blood flow remains laminar, then the cross-section of the orthotropic homogeneous blood vessel becomes circular. Bibliography: 30 titles. Illustrations: 2 figures.
Function Spaces, Differential Operators and Nonlinear Analysis, 2003
We consider the Dirichlet problem for elliptic equations of arbitrary order and prove an asymptot... more We consider the Dirichlet problem for elliptic equations of arbitrary order and prove an asymptotic formula for a singular solution near a boundary point. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.
Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharm... more Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity γ that describes the depth O(ε γ) of irregularity (ε is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.
Journal de Mathématiques Pures et Appliquées, 2008
The paper deals with the classical non-linear problem of steady two-dimensional waves on water of... more The paper deals with the classical non-linear problem of steady two-dimensional waves on water of finite depth. The problem is formulated so that it describes all waves without stagnation points on the free-surface profiles that are bounded themselves and have bounded slopes. By virtue of reducing the problem to an integro-differential equation the following three results are proved. First, there are no waves when the flow is critical. Second, there are no waves having profiles totally above the upper boundary of the uniform subcritical stream. Finally, only two types of the free-surface behaviour are possible at positive (or/and negative) infinity: the profile either oscillates infinitely many times around the upper boundary of the subcritical uniform stream or asymptotes the upper level of a uniform stream (subcritical or supercritical). The latter assertion is proved under additional assumption that the slope of the free surface is a uniformly continuous function.
Zeitschrift für angewandte Mathematik und Physik, 2008
The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrica... more The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrical domain has a cut (crack) where inequality type boundary conditions are imposed. A behavior of the solution near the crack tips is analyzed. In particular, estimates for the second derivatives in a weighted Sobolev space are obtained and asymptotics of the solution near crack tips is established.
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Papers by V. Kozlov