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  • 1.
    Andersson, Jonathan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Radosavljevic, Sonja
    Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Density-Dependent Feedback in Age-Structured Populations2019In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 242, no 1, p. 2-24Article in journal (Refereed)
    Abstract [en]

    The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

  • 2.
    Avdonin, S
    et al.
    University of Alaska at Fairbanks.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maxwell, D
    University of Alaska at Fairbanks.
    Truffer , M
    University of Alaska at Fairbanks.
    Iterative methods for solving a nonlinear boundary inverse problem in glaciology2009In: JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, ISSN 0928-0219 , Vol. 17, no 3, p. 239-258Article in journal (Refereed)
    Abstract [en]

    We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

  • 3.
    Avdonin, Sergey
    et al.
    University of Alaska Fairbanks, USA.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Stability estimate for an inverse problem in glaciology2012In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 2, no 4, p. 367-387Article in journal (Refereed)
    Abstract [en]

    We consider the problem of reconstruction of the basal velocity of a glacier by measurements of the velocity on glacier’s surface. The main result is a stability estimate in a near-surface region, which represents a multiplicative inequality and shows that small errors in measurements produce small errors in the velocity in this region.

  • 4.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ghosh, Arpan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, S. A.
    St Petersburg State Univ, Russia; Inst Problems Mech Engn RAS, Russia.
    A one dimensional model of blood flow through a curvilinear artery2018In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 63, p. 633-643Article in journal (Refereed)
    Abstract [en]

    We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

    The full text will be freely available from 2020-07-17 13:46
  • 5.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
    A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients2018Report (Other academic)
    Abstract [en]

    One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

    In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

    The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

  • 6.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey A.
    St Petersburg State University, St Petersburg State Polytechnical University, and Institute of Problems of Mechanical Engineering RAS, Russia..
    A one-dimensional model of a false aneurysm2017In: International Journal of Research in Engineering and Science (IJRES), ISSN 2320-9356, Vol. 5, no 6, p. 61-73Article in journal (Refereed)
    Abstract [en]

     A false aneurysm is a hematoma, i.e. collection ofblood outside of a blood vessel, that forms due to a hole  in the wall of an artery . This represents a serious medical condition that needs to be monitored and, under certain conditions, treatedurgently. In this work a one-dimensional model of a false aneurysm isproposed. The new model is based on a one-dimensional model of anartery previously presented by the authors and it takes into accountthe interaction between the hematoma  and the surrounding musclematerial. The model equations are derived  using rigorous asymptoticanalysis for the case of a simplified geometry.   Even though the model is simple it still supports a realisticbehavior for the system consisting of the vessel and the  hematoma. Using numerical simulations we illustrate the behavior ofthe model. We also investigate the effect  of changing the size of the hematoma. The simulations show that ourmodel can reproduce realistic solutions. For instance we show thetypical strong pulsation of an aneurysm by blood entering the hematoma during the work phase of the cardiac cycle, and the blood returning tothe vessel during the resting phase. Also we show that the aneurysmgrows  if the pulse rate is increased due to, e.g., a higher work load. 

  • 7.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia; RAS, Russia.
    A one-dimensional model of viscous blood flow in an elastic vessel2016In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 274, p. 125-132Article in journal (Refereed)
    Abstract [en]

    In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel. (C) 2015 Elsevier Inc. All rights reserved.

  • 8.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Kozlov, Vladimir A.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. University of Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Numerical Solution of the Cauchy Problem for the Helmholtz Equation2014Report (Other academic)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

  • 9.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mpinganzima, L.
    University of Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation2017In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 73, no 1, p. 163-172Article in journal (Refereed)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved.

  • 10.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation2014In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, no 1-2, p. 44-60Article in journal (Refereed)
    Abstract [en]

    In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.

  • 11.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    An alternating iterative procedure for the Cauchy problem for the Helmholtz equation2014In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, p. 45-62Article in journal (Refereed)
    Abstract [en]

    We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.

  • 12.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mpinganzima, Lydie
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2014Manuscript (preprint) (Other academic)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.

  • 13.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mpinganzima, Lydie
    Univ Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Robin-Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2018In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, no 7, p. 1062-1078Article in journal (Refereed)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

  • 14.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wokiyi, Dennis
    Makerere Univ, Uganda.
    Solvability of a non-linear Cauchy problem for an elliptic equation2019In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96, no 12, p. 2317-2333Article in journal (Refereed)
    Abstract [en]

    We study a non-linear operator equation originating from a Cauchy problem for an elliptic equation. The problem appears in applications where surface measurements are used to calculate the temperature below the earth surface. The Cauchy problem is ill-posed and small perturbations to the used data can result in large changes in the solution. Since the problem is non-linear certain assumptions on the coefficients are needed. We reformulate the problem as an non-linear operator equation and show that under suitable assumptions the operator is well-defined. The proof is based on making a change of variables and removing the non-linearity from the leading term of the equation. As a part of the proof we obtain an iterative procedure that is convergent and can be used for evaluating the operator. Numerical results show that the proposed procedure converges faster than a simple fixed point iteration for the equation in the the original variables.

  • 15.
    Ghersheen, Samia
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Dynamical behaviour of SIR model with coinfection: The case of finite carrying capacity2019In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 42, no 17, p. 5805-5826Article in journal (Refereed)
    Abstract [en]

    Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection are important in this case. We formulated an susceptible infected recovered (SIR) model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.

  • 16.
    Ghersheen, Samia
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Mathematical analysis of complex SIR model with coinfection and density dependence2019In: Computational and Mathematical Methods, ISSN 2577-7408, Vol. 1, no 4Article in journal (Refereed)
    Abstract [en]

    An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed, and several threshold conditions are obtained, which describes the establishment of diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease-free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

  • 17.
    Ghosh, Arpan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A Two Dimensional Model of the Thin Laminar Wall of a Curvilinear Flexible Pipe2018Report (Other academic)
    Abstract [en]

    We present a two dimensional model describing the elastic behaviour of the wall of a curved pipe to model blood vessels in particular. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the vessel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with the surrounding material and the fluid flowing inside into account and is obtained via a dimension reduction procedure. The curvature and twist of the vessel axis as well as the anisotropy of the laminate wallpresent the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of vessels and their walls are supplied with explicit systems of dierential equations at the end.

  • 18.
    Ghosh, Arpan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. St Petersburg State Univ, Russia; RAS, Russia.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE2018In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 71, no 3, p. 349-367Article in journal (Refereed)
    Abstract [en]

    We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

  • 19.
    Herberthson, Magnus
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Johansson, KarinLinköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.Kozlov, VladimirLinköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.Ljungkvist, EmmaLinköping University, National Supercomputer Centre (NSC).Singull, MartinLinköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Proceedings from Workshop: Mathematics in Biology and Medicine, 11-12 May 2017, Linköping University2017Conference proceedings (editor) (Refereed)
  • 20.
    Johansson, B Tomas
    et al.
    University of Birmingham.
    Kozlov , Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium2009In: IMA JOURNAL OF APPLIED MATHEMATICS, ISSN 0272-4960 , Vol. 74, no 1, p. 62-73Article in journal (Refereed)
    Abstract [en]

    Kozlov & Mazya (1989, Algebra Anal., 1, 144-170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

  • 21.
    Johansson, Björn
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    Elfving, Tommy
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Censor, Y.
    Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 31905, Israel.
    Forssén, Per-Erik
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    Granlund, Gösta
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Computer Vision.
    The application of an oblique-projected Landweber method to a model of supervised learning2006In: Mathematical and computer modelling, ISSN 0895-7177, E-ISSN 1872-9479, Vol. 43, no 7-8, p. 892-909Article in journal (Refereed)
    Abstract [en]

    This paper brings together a novel information representation model for use in signal processing and computer vision problems, with a particular algorithmic development of the Landweber iterative algorithm. The information representation model allows a representation of multiple values for a variable as well as an expression for confidence. Both properties are important for effective computation using multi-level models, where a choice between models will be implementable as part of the optimization process. It is shown that in this way the algorithm can deal with a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. While the algorithm has been applied to the solution of such problems, it has so far been used heuristically. In this paper we describe the properties and some of the peculiarities of the channel representation and optimization, and put them on firm mathematical ground. We consider the optimization a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective with currently available projected Landweber methods. An application to supervised learning is described, and the method is evaluated in an experiment involving function approximation, as well as application to transient signals. © 2006 Elsevier Ltd. All rights reserved.

  • 22.
    Johansson, Björn
    et al.
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Elfving, Tommy
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Censor, Yair
    Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 31905, Israel.
    Granlund, Gösta
    Linköping University, Department of Electrical Engineering. Linköping University, The Institute of Technology.
    The Application of an Oblique-Projected Landweber Method to a Model of Supervised Learning2004Report (Other academic)
    Abstract [en]

    This report brings together a novel approach to some computer vision problems and a particular algorithmic development of the Landweber iterative algorithm. The algorithm solves a class of high-dimensional, sparse, and constrained least-squares problems, which arise in various computer vision learning tasks, such as object recognition and object pose estimation. The algorithm has recently been applied to these problems, but it has been used rather heuristically. In this report we describe the method and put it on firm mathematical ground. We consider a convexly constrained weighted least-squares problem and propose for its solution a projected Landweber method which employs oblique projections onto the closed convex constraint set. We formulate the problem, present the algorithm and work out its convergence properties, including a rate-of-convergence result. The results are put in perspective of currently available projected Landweber methods. The application to supervised learning is described, and the method is evaluated in a function approximation experiment.

  • 23.
    Kozlov , Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains2009In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, ISSN 0360-5302 , Vol. 34, no 1, p. 24-51Article in journal (Refereed)
    Abstract [en]

    We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.

  • 24.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotic representation for solutions to the Dirichlet problem for elliptic systems with discontinuos coefficients near the boundary2005Report (Other academic)
  • 25.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotic representation of solutions to the Dirichlet problem for elliptic systems with discontinuous coefficients near the boundary2006In: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, Vol. 2006Article in journal (Refereed)
    Abstract [en]

    We consider variational solutions to the Dirichlet problem for elliptic systems of arbitrary order. It is assumed that the coefficients of the principal part of the system have small, in an integral sense, local oscillations near a boundary point and other coefficients may have singularities at this point. We obtain an asymptotic representation for these solutions and derive sharp estimates for them which explicitly contain information on the coefficients. ©2006 Texas State University - San Marcos.

  • 26.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions2005Report (Other academic)
  • 27.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Domain dependence of eigenvalues of elliptic type operators2013In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 357, no 4, p. 1509-1539Article in journal (Refereed)
    Abstract [en]

    The dependence on the domain for the Dirichlet eigenvalues of elliptic operators considered in bounded domains is studied. The proximity of domains 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 eigenvalues corresponding to 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. Applications of this result are given. The results are new for the Laplace operator.

  • 28.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lq-perturbations of leading coefficients of elliptic operators: Asymptotics of eigenvalues2006In: Abstract and Applied Analysis, ISSN 1085-3375, E-ISSN 1687-0409, Vol. 2006Article in journal (Refereed)
    Abstract [en]

    We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L8 -norm. Copyright © 2006 Vladimir Kozlov.

  • 29.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On bounded solutions of the Emden-Fowler equation in a semi-cylinder2002In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 179, no 2, p. 456-478Article in journal (Refereed)
    Abstract [en]

    Bounded solutions of the Emden-Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions. © 2002 Elsevier Science (USA).

  • 30.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On the Hadamard formula for nonsmooth domains2005Report (Other academic)
  • 31.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On the Hadamard formula for nonsmooth domains2006In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 230, no 2, p. 532-555Article in journal (Refereed)
    Abstract [en]

    We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1, 1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications. © 2006 Elsevier Inc. All rights reserved.

  • 32.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Traction boundary value problem for anisotropic elasticity in polyhedral domains2001In: Russian journal of mathematical physics, ISSN 1061-9208, E-ISSN 1555-6638, Vol. 8, no 3, p. 275-286Article in journal (Refereed)
    Abstract [en]

    The traction boundary value problem for anisotropic elasticity is considered. For polyhedral domains in R-3, it is proved that the displacements are Holder continuous. In the n-dimensional case, n > 3, the Holder continuity is proved for domains with conic points on the boundary. The proof is based on the study of spectrum of operator pencils associated with singularities of the boundary, which is of independent interest.

  • 33.
    Kozlov, Vladimir A
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St.Petersburg State University, Mathematics and Mechanics Faculty, St. Petersburg, Russia.
    A simple one-dimensional model of a false aneurysm in the femoral artery2016In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 214, no 3, p. 287-301Article in journal (Refereed)
    Abstract [en]

    Using the dimension reduction procedure, a one-dimensional model of a periodic blood flow in the artery through a small hole in a thin elastic wall to a spindle-shaped hematoma, is constructed. This model is described by a system of two parabolic and one hyperbolic equations provided with mixed boundary and periodicity conditions. The blood exchange between the artery and the hematoma is expressed by the Kirchhoff transmission conditions. Despite the simplicity, the constructed model allows us to describe the damping of a pulsating blood flow by the hematoma and to determine the condition of its growth. In medicine, the biological object considered is called a false aneurysm.

  • 34.
    Kozlov, Vladimir A.
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St. Petersburg State University, St. Petersburg, Russia .
    Asymptotic Models of the Blood Flow in Arteries and Veins2013In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 194, no 1, p. 44-57Article in journal (Refereed)
    Abstract [en]

    Asymptotic analysis is applied for obtaining one-dimensional models of the blood flow in narrow, thin-walled, elastic vessels. The models for arteries and veins essentially distinguish from each other, and the reason for this is the structure of their walls, as well as the operationing conditions. Although the obtained asymptotic models are simple, they explain various effects known in medical practice, in particular, describe the mechanism of vein-muscle pumping of blood.

  • 35.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Avdonin, Sergei
    Department of Mathematics and Statistics University of Alaska Fairbanks.
    Maxwell, David
    Department of Mathematics and Statistics University of Alaska Fairbanks.
    Truffer, Martin
    Department of Mathematics and Statistics University of Alaska Fairbanks.
    Iterative Methods for Solving a Nonlinear Boundary Inverse Problem in Glaciology2008Report (Other academic)
  • 36.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Baravdish, George
    Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.
    Johansson, Tomas
    Lesnic, Daniel
    An alternating method for the stationary Stokes system2006In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 86, no 4, p. 268-280Article in journal (Refereed)
    Abstract [en]

    An alternating procedure for solving a Cauchy problem for the stationary Stokes system is presented. A convergence proof of this procedure and numerical results are included. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  • 37.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Johansson, B. Tomas
    University of Birmingham, UK.
    Solvability and asymptotics of the heat equation with mixed variable lateral conditions and applications in the opening of the exocytotic fusion pore in cells2014In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 79, no 2, p. 377-392Article in journal (Refereed)
    Abstract [en]

    We investigate a mixed problem with variable lateral conditions for the heat equation that arises in modelling exocytosis, i.e. the opening of a cell boundary in specific biological species for the release of certain molecules to the exterior of the cell. The Dirichlet condition is imposed on a surface patch of the boundary and this patch is occupying a larger part of the boundary as time increases modelling where the cell is opening (the fusion pore), and on the remaining part, a zero Neumann condition is imposed (no molecules can cross this boundary). Uniform concentration is assumed at the initial time. We introduce a weak formulation of this problem and show that there is a unique weak solution. Moreover, we give an asymptotic expansion for the behaviour of the solution near the opening point and for small values in time. We also give an integral equation for the numerical construction of the leading term in this expansion.

  • 38.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Khludnev, A. M.
    Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions2008In: Zeitschrift für Angewandte Mathematik und Physik, ISSN 0044-2275, E-ISSN 1420-9039, Vol. 59, no 2, p. 264-280Article in journal (Refereed)
    Abstract [en]

    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. © 2007 Birkhaeuser.

  • 39.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Khludnev, Alexander
    Russian Ac. Sci. L.
    Asymptotic behavior of the solution to the Poisson equation near a crack tip with nonlinear boundary conditions on the crack faces2006In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 74, no 3, p. 865-868Article in journal (Refereed)
    Abstract [en]

    The asymptotic behavior of the solution to the Poisson equations near the crack tip with nonlinear boundary conditions and second derivatives of the in weighted Sobolev spaces were determined. The internal regularity results for the equation showed that the second derivative of u belonged to L2 in the neighborhood of interior points of the crack. The results demonstrated analogy between the properties of the solution to the linear problem and the nonlinear problem. It was also found that an asymptotic representation to the problem can be constructed near the crack tip.

  • 40.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kuznetsov, N.
    Russian Acad Sci, Russia.
    A COMPARISON THEOREM FOR SUPER- AND SUBSOLUTIONS OF del(2)u + f(u)=0 AND ITS APPLICATION TO WATER WAVES WITH VORTICITY2019In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 30, no 3, p. 471-483Article in journal (Refereed)
    Abstract [en]

    A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form f (u) with f belonging to the class L-loc(p) and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.

  • 41.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetsov, N.
    On two-dimensional water waves in a canal2003In: Comptes rendus. Mecanique, ISSN 1631-0721, E-ISSN 1873-7234, Vol. 331, no 7, p. 489-494Article in journal (Refereed)
    Abstract [en]

    This Note deals with an eigenvalue problem that contains a spectral parameter in a boundary condition. The problem for the two-dimensional Laplace equation describes free, time-harmonic water waves in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is shown that there exists a domain for which at least one of eigenfunctions has a nodal line with both ends on the free surface. Since Kuttler essentially used the non-existence of such nodal lines in his proof of simplicity of the fundamental sloshing eigenvalue in the two-dimensional case, we propose a new variational principle for demonstrating this latter fact. ⌐ 2003 AcadΘmie des sciences.

  • 42.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, N
    Russian Academy of Sciences.
    Steady Free-Surface Vortical Flows Parallel to the Horizontal Bottom2011In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 64, no 3, p. 371-399Article in journal (Refereed)
    Abstract [en]

    Steady, free-surface, vortical flows of an inviscid, incompressible, heavy fluid over a horizontal, rigid bottom are considered. All flows of constant depth are described for any Lipschitz vorticity distribution. It is shown that the values of Bernoullis constant, for which such flows exist, are greater than or equal to some critical value depending on the vorticity. For the critical value, only one flow exists and it is unidirectional. Supercritical flows exist for all values of Bernoullis constant greater than the critical one; every such flow is also unidirectional and its depth is smaller than that of the critical flow. Furthermore, at least one flow other than supercritical does exist for every value of Bernoullis constant greater than the critical one. It is found that for some vorticity distributions, the number of constant depth flows increases unrestrictedly as Bernoullis constant tends to infinity. However, all these flows, except for one or two, have counter-currents; their number depends on Bernoullis constant and increases by at least two every time when this constant becomes greater than a critical value (the above mentioned is the smallest of them), belonging to a sequence defined by the vorticity. A classification of vorticity distributions is presented; it divides all of them into three classes in accordance with the behaviour of some integral of the distribution on the interval [0, 1]. For distributions in the first class, a unidirectional subcritical flow exists for all admissible values of Bernoullis constant. For vorticity distributions belonging to the other two classes such a flow exists only when Bernoullis constant is less than a certain value. If Bernoullis constant is greater than this value, then at least one flow with counter-currents does exist along with the unidirectional supercritical flow. The second and third classes of vorticity distributions are distinguished from one another by the character of the counter-currents. If a distribution is in the second class, then a near-bottom counter-current is always present for sufficiently large values of Bernoullis constant. For distributions in the third class, a near-surface counter-current is always present for such values of the constant. Several examples illustrating the results are considered.

  • 43.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, N.
    Russian Academic Science, Russia.
    Lokharu, Evgeniy
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On bounds and non-existence in the problem of steady waves with vorticity2015In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 765, no R1Article in journal (Refereed)
    Abstract [en]

    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 are found for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

  • 44.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kuznetsov, N.
    Russian Academic Science, Russia.
    Lokharu, Evgeniy
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    On the Benjamin-Lighthill conjecture for water waves with vorticity2017In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 825, p. 961-1001Article in journal (Refereed)
    Abstract [en]

    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 Bernoullis 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.

  • 45.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, N.
    Russian Academy of Sciences, St Petersburg, Russia.
    Lokharu, Evgeniy
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Steady water waves with vorticity: an analysis of the dispersion equation2014In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 751Article in journal (Refereed)
    Abstract [en]

    Two-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm-Liouville problem considered by Constantin and Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481-527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133-175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm-Liouville problem mentioned in (i) has an eigenvalue is obtained.

  • 46.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetsov, N.
    Lab. Math. Modelling Wave Phenomena, Inst. for Prob. in Mech. Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St Petersburg 199178, Russian Federation.
    Motygin, O.
    Lab. Math. Modelling Wave Phenomena, Inst. for Prob. in Mech. Engineering, Russian Academy of Sciences, Bol'shoy pr. 61, St Petersburg 199178, Russian Federation.
    On the two-dimensional sloshing problem2004In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 460, no 2049, p. 2587-2603Article in journal (Refereed)
    Abstract [en]

    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. © 2004 The Royal Society.

  • 47.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetsov, Nikolay
    Russian Academy of Sciences.
    Bounds for Arbitrary Steady Gravity Waves on Water of Finite Depth2009In: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 11, no 3, p. 325-347Article in journal (Refereed)
    Abstract [en]

      Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.

  • 48.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kuznetsov, Nikolay
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    BOUNDS FOR SOLUTIONS TO THE PROBLEM OF STEADY WATER WAVES WITH VORTICITY2017In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 70, no 4, p. 497-518Article in journal (Refereed)
    Abstract [en]

    The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the assumption that the vorticity distribution is a locally Lipschitz function. It is also shown that wave flows have countercurrents in the case when the infimum of the free surface profile exceeds a certain critical value.

  • 49.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academy of Science, St. Petersburg.
    Bounds for steady water waves with vorticity2012In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 1, p. 663-691Article in journal (Refereed)
    Abstract [en]

    The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.

  • 50.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academic Science, Russia.
    Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents2014In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 214, no 3, p. 971-1018Article in journal (Refereed)
    Abstract [en]

    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: one for waves with fixed Bernoullis constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.

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