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  • 1.
    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.

  • 2.
    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.

  • 3.
    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.

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    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.

  • 8.
    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.

  • 9.
    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.

  • 10.
    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.

  • 11.
    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.

  • 12.
    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.

  • 13.
    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.

  • 14.
    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)
  • 15.
    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.

  • 16.
    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)
  • 17.
    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.

  • 18.
    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.

  • 19.
    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).

  • 20.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On the Hadamard formula for nonsmooth domains2005Report (Other academic)
  • 21.
    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.

  • 22.
    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.

  • 23.
    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.

  • 24.
    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.

  • 25.
    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)
  • 26.
    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.

  • 27.
    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.

  • 28.
    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.

  • 29.
    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.

  • 30.
    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.

  • 31.
    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.

  • 32.
    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.

  • 33.
    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.

  • 34.
    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.

  • 35.
    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.

  • 36.
    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.

  • 37.
    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.

  • 38.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academy of Science.
    Fundamental bounds for steady water waves2009In: MATHEMATISCHE ANNALEN, ISSN 0025-5831, Vol. 345, no 3, p. 643-655Article in journal (Refereed)
    Abstract [en]

    The two-dimensional free-boundary problem of steady gravity waves on water of finite depth is considered. Bounds on the free-surface profiles and on the values of Bernoullis constant are obtained under minimal assumptions about properties of solutions to the problem.

  • 39.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academic Science, St. Petersburg, Russia .
    No steady water waves of small amplitude are supported by a shear flow with a still free surface2013In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 717, p. 523-534Article in journal (Refereed)
    Abstract [en]

    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, that is, it is stagnant 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 quadratic ones with arbitrary positive coefficients.

  • 40.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetsov, Nikolay
    On the fundamental sloshing frequency in the ice-fishing problem2002In: Comptes rendus. Mecanique, ISSN 1631-0721, E-ISSN 1873-7234, Vol. 330, no 11, p. 723-728Article in journal (Refereed)
    Abstract [en]

    The sloshing problem is considered in a half-space covered by a rigid dock with apertures. The dependence of the fundamental sloshing frequency on the shape of the free surface region is studied. It is proved that the inequality holds between the fundamental eigenvalues corresponding to two different regions if some conditions are fulfilled. These conditions are verified for particular classes of regions of a fixed area in order to demonstrate that the disk yields the maximum of the fundamental eigenvalue for each of these classes. On the other hand, examples of regions are constructed for which the fundamental eigenfrequency is larger than that for the circular aperture of the same area and even as large as one wishes.

  • 41.
    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 .
    Steady water waves with vorticity: spatial Hamiltonian structure2013In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 733Article in journal (Refereed)
    Abstract [en]

    Spatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow-force invariant.

  • 42.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academy of Science.
    The Benjamin-Lighthill Conjecture for Near-Critical Values of Bernoullis Constant2010In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 197, no 2, p. 433-488Article in journal (Refereed)
    Abstract [en]

    In 1954, Benjamin and Lighthill made a conjecture concerning the classical nonlinear problem of steady gravity waves on water of finite depth. According to this conjecture, a point of some cusped region on the (r, s)-plane (r and s are the non-dimensional Bernoullis constant and the flow force, respectively), corresponds to every steady wave motion described by the problem. Conversely, at least one steady flow corresponds to every point of the region. In the present paper, this conjecture is proved for near-critical flows (when r attains values close to one), under the assumption that the slopes of wave profiles are bounded. Another question studied here concerns the uniqueness of solutions, and it is proved that for every near-critical value of r only the following waves do exist: (i) a unique (up to translations) solitary wave; (ii) a family of Stokes waves (unique up to translations), which is parametrised by the distance from the bottom to the wave crest. The latter parameter belongs to the interval bounded below by the depth of the subcritical uniform stream and above by the distance from the bottom to the crest of solitary wave corresponding to the chosen value of r.

  • 43.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academy of Science.
    The Benjamin-Lighthill Conjecture for Steady Water Waves (revisited)2011In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 201, no 2, p. 631-645Article in journal (Refereed)
    Abstract [en]

    The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin-Lighthill conjecture is proved for these waves provided Bernoullis constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoullis constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoullis constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433-488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.

  • 44.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetsov, Nikolay
    The ice-fishing problem: The fundamental sloshing frequency versus geometry of holes2004In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 27, no 3, p. 289-312Article in journal (Refereed)
    Abstract [en]

    We study an eignevalue problem with a spectral parameter in a boundary condition. This problem for the Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a half-space covered by a rigid dock with some apertures (an ice sheet with fishing holes). The dependence of the fundamental eigenvalue on holes' geometry is investigated. We give conditions on a plane region guaranteeing that the fundamental eigenvalue corresponding to this region is larger than the fundamental eigenvalue corresponding to a single circular hole. Examples of regions satisfying these conditions and having the same area as the unit disk are given. New results are also obtained for the problem with a single circular hole. On the other hand, we construct regions for which the fundamental eigenfrequency is larger than the similar frequency for the circular hole of the same area and even as large as one wishes. In the latter examples, the hole regions are either not connected or bounded by a rather complicated curves.

  • 45.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    The two-dimensional problem of steady waves on water of finite depth: Regimes without waves of small amplitude2005In: Comptes rendus. Mecanique, ISSN 1631-0721, E-ISSN 1873-7234, Vol. 333, no 10, p. 733-738Article in journal (Refereed)
    Abstract [en]

    The two-dimensional problem of steady waves on water of finite depth is considered without assumptions about periodicity and symmetry of waves. A new form of Bernoulli's equation is derived, and it involves a new bifurcation parameter which is the product of the Froude number μ and the rate of flow ω. The main result obtained from this equation is the absence of waves, having sufficiently small amplitude, provided |μω| > 1. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.

  • 46.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St Petersburg.
    Motygin, Oleg
    Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St Petersburg.
    Correction: On the two-dimensional sloshing problem (vol. 460 pp. 2427–2430, 2011)2011In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 467, no 2132, p. 2427-2430Article in journal (Other academic)
    Abstract [en]

    A correct proof is given for the following assertions about the two-dimensional sloshing problem. The fundamental eigenvalue is simple and the corresponding stream function may be chosen to be non-negative in the closure of the water domain. New proof is based on stricter assumptions about the water domain; namely, it must satisfy John’s condition.                 

  • 47.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Kuznetzov, N.
    On behaviour of free-surface profiles for bounded steady water waves2008In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 90, no 1Article in journal (Refereed)
    Abstract [en]

    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. © 2008 Elsevier Masson SAS. All rights reserved.

  • 48.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Langer, Mikael
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Rand , Peter
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Asymptotic analysis of solutions to parabolic systems2009In: MATHEMATISCHE NACHRICHTEN, ISSN 0025-584X , Vol. 282, no 3, p. 430-458Article in journal (Refereed)
    Abstract [en]

    We study asymptotics as t -> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0, infinity), where Omega is a bounded domain. On partial derivative Omega x (0, infinity) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t -> infinity. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if kappa epsilon L-1 (0, infinity), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

  • 49.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Langer, Mikael
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rand, Peter
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Asymptotic analysis of solutions to parabolic systems2008In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916, Vol. 40, no 01-Mar, p. 369-374Article in journal (Refereed)
    Abstract [en]

    We study asymptotics as t --> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0,infinity), where Omega is a bounded domain. On partial derivative Omega x (0,infinity) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t --> infinity. The main result is an asymptotic representation of solutions for large t. A consequence is that for kappa is an element of L-1(0,infinity), the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

  • 50.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Differential Equations with Operator Coefficients: with Applications to Boundary Value Problems for Partial Differential Equations1999Book (Refereed)
12 1 - 50 of 90
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