Now showing items 1-11 of 11

    • Asymptotic Behaviour of Ground States for Mixtures of Ferromagnetic and Antiferromagnetic Interactions in a Dilute Regime 

      Braides, Andrea; Causin, Andrea; Piatnitski, Andrey; Solci, Margherita (Journal article; Peer reviewed, 2018-04-30)
      We consider randomly distributed mixtures of bonds of ferromagnetic and antiferromagnetic type in a two-dimensional square lattice with probability 1−p 1−p and p, respectively, according to an i.i.d. random variable. We study minimizers of the corresponding nearest-neighbour spin energy on large domains in Z 2 Z2 . We prove that there exists p 0 p0 such that for p≤ p 0 p≤p0 such ...
    • Asymptotics of a spectral-sieve problem 

      Amirat, Youcef; Bodart, Olivier; Chechkin, Gregory; Piatnitski, Andrey (Peer reviewed, 2015-11-18)
      In a bounded domain with a thin periodically punctured interface we study the limit behavior of the bottom of spectrum for a Steklov type spectral problem, the Steklov boundary condition being imposed on the perforation surface. For a certain range of parameters we construct the effective spectral problem and justify the convergence of eigenpairs.
    • Homogenization of biomechanical models for plant tissues 

      Piatnitski, Andrey; Ptashnyk, Mariya (Journal article; Peer reviewed; Tidsskriftartikkel, 2017)
      In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical ...
    • Homogenization of nonisothermal immiscible incompressible two-phase flow in porous media 

      Amaziane, Brahim; Jurak, Murat; Pankratov, Leonid; Piatnitski, Andrey (Journal article; Peer reviewed, 2018-03-15)
      In this paper, we consider nonisothermal two-phase flows through heterogeneous porous media with periodic microstructure. Examples of such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste, thermally enhanced oil recovery and geothermal systems. The mathematical model is given by a coupled system of two-phase flow equations, and an ...
    • Homogenization of random Navier–Stokes-type system for electrorheological fluid 

      Piatnitski, Andrey; Zhikov, Vasily (Peer reviewed, 2015-11-19)
      The paper deals with homogenization of Navier–Stokes-type system describing electrorheological fluid with random characteristics. Under non-standard growth conditions we construct the homogenized model and prove the convergence result. The structure of the limit equations is also studied.
    • Osmosis for non-electrolyte solvents in permeable periodic porous media 

      Heintz, Alexei; Piatnitski, Andrey (Peer reviewed, 2016-08)
      The paper gives a rigorous description, based on mathematical homogenization theory, for flows of solvents with not charged solute particles under osmotic pressure for periodic porous media permeable for solute particles. The effective Darcy type equations for the flow under osmotic pressure distributed within the porous media are derived. The effective Darcy law contains an additional flux term ...
    • Periodic homogenization of nonlocal operators with a convolution-type kernel 

      Piatnitski, Andrey; Zhizhina, Elena (Journal article; Peer reviewed; Tidsskriftartikkel, 2017)
      The paper deals with a homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behavior of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators ...
    • Pointwise estimates for heat kernels of convolution-type operators 

      Grigor'yan, Alexander; Kondratiev, Yuri; Piatnitski, Andrey; Zhizhina, Elena (Journal article; Peer reviewed; Tidsskriftartikkel, 2018-04-16)
      We study the large‐time behaviour of the fundamental solution of parabolic equations with an elliptic part being non‐local convolution‐type operator. We assume that this operator is a generator of a Markov jump process, and that its convolution kernel decays at least exponentially at infinity. The fundamental solution shows rather different asymptotic behaviour depending on whether | x | ≲ t , or t ...
    • Resolvent bounds for jump generators 

      Kondratiev, Yuri; Molchanov, Stanislav; Piatnitski, Andrey; Zhizhina, Elena (Journal article; Peer reviewed; Tidsskriftartikkel, 2016-12-02)
      The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially, we prove a number of lower and upper bounds for the resolvent of such operators. In particular we focus on sharp estimates of the resolvent kernel for small values of the spectral parameter. We consider two applications of these results. First we obtain pointwise ...
    • Singularly perturbed spectral problems with Neumann boundary conditions 

      Piatnitski, Andrey; Rybalko, A; Rybalko, V (Peer reviewed, 2015-09-07)
      The paper deals with the Neumann spectral problem for a singularly perturbed second-order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation, we reduce the studied problem to an additive eigenvalue problem for a singularly perturbed Hamilton–Jacobi ...
    • Stationary convection-diffusion equation in an infinite cylinder 

      Pettersson, Irina; Piatnitski, Andrey (Journal article; Tidsskriftartikkel; Peer reviewed, 2017-12-21)
      We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we ...