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  • 1.
    Bernhoff, Niclas
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Weak shock waves for the general discrete velocity model of the Boltzmann equation2007In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, no 4, p. 815-832Article in journal (Refereed)
    Abstract [en]

    We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed c, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive proof following a more straightforward way, suiting the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points to general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures

  • 2.
    Vo Anh, Khoa
    et al.
    Gran Sasso Science Institute, Italy; Hasselt University, Belgium.
    Muntean, Adrian
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Corrector homogenization estimates for a non-stationary Stokes-Nernst-Planck-Poisson system in perforated domains2019In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 17, no 3, p. 705-738Article in journal (Refereed)
    Abstract [en]

    We consider a non-stationary Stokes-Nernst-Planck-Poisson system posed in perforated domains. Our aim is to justify rigorously the homogenization limit for the upscaled system derived by means of two-scale convergence in [N. Ray, A. Muntean, and P. Knabner, J. Math. Anal. Appl., 390(1):374-393, 2012]. In other words, we wish to obtain the so-called corrector homogenization estimates that specify the error obtained when upscaling the microscopic equations. Essentially, we control in terms of suitable norms differences between the micro-and macro-concentrations and between the corresponding micro- and macro-concentration gradients. The major challenges that we face are the coupled flux structure of the system, the nonlinear drift terms and the presence of the microstructures. Employing various energy-like estimates, we discuss several scalings choices and boundary conditions.

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