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Diffusive limit for finite velocity Boltzmann kinetic models.

Pierre Louis Lions, Giuseppe Toscani (1997)

Revista Matemática Iberoamericana

We investigate, in the diffusive scaling, the limit to the macroscopic description of finite-velocity Boltzmann kinetic models, where the rate coefficient in front of the collision operator is assumed to be dependent of the mass density. It is shown that in the limit the flux vanishes, while the evolution of the mass density is governed by a nonlinear parabolic equation of porous medium type. In the last part of the paper we show that our method adapts to prove the so-called Rosseland approximation...

Direct approach to mean-curvature flow with topological changes

Petr Pauš, Michal Beneš (2009)

Kybernetika

This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves Γ ( t ) : S → ℝ 2 , t ≧ 0 . The curves are driven by the normal velocity v which is the function of curvature κ and the position. The evolution law reads as: v = - κ + F . The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved...

Directional and scale-dependent statistics of quasi-static magnetohydrodynamic turbulence

Naoya Okamoto, Katsunori Yoshimatsu, Kai Schneider, Marie Farge (2011)

ESAIM: Proceedings

Anisotropy and intermittency of quasi-static magnetohydrodynamic (MHD) turbulence in an imposed magnetic field are examined, using three-dimensional orthonormal wavelet analysis. Wavelets are an efficient tool to examine directional scale-dependent statistics, since they are based on well-localized functions in space, scale and direction. The analysis is applied to two turbulent MHD flows computed by direct numerical simulation with 5123 grid points...

Dirichlet control of unsteady Navier–Stokes type system related to Soret convection by boundary penalty method

S. S. Ravindran (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the boundary penalty method for optimal control of unsteady Navier–Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the...

Disclinations and hedgehogs in nematic liquid crystals with variable degree of orientation

Epifanio G. Virga (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

There is enough evidence to re-examine disclinations and hedgehogs, the singularities often observed in nematic liquid crystals, in the light of a new theory that allows for local changes in the degree of orientation.

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

Oto Havle, Vít Dolejší, Miloslav Feistauer (2010)

Applications of Mathematics

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation...

Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson

N. Crouseilles, M. Mehrenberger, F. Vecil (2011)

ESAIM: Proceedings

We present a discontinuous Galerkin scheme for the numerical approximation of the one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.

Discrete coagulation-fragmentation system with transport and diffusion

Stéphane Brull (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove the existence of solutions to two infinite systems of equations obtained by adding a transport term to the classical discrete coagulation-fragmentation system and in a second case by adding transport and spacial diffusion. In both case, the particles have the same velocity as the fluid and in the second case the diffusion coefficients are equal. First a truncated system in size is solved and after we pass to the limit by using compactness properties.

Dissipation d’énergie pour des solutions faibles des équations d’Euler et Navier-Stokes incompressibles

Jean Duchon, Raoul Robert (1999/2000)

Séminaire Équations aux dérivées partielles

On étudie l’équation locale de l’énergie pour des solutions faibles des équations d’Euler et Navier-Stokes incompressibles tridimensionnelles. On explicite un terme de dissipation provenant de l’éventuel défaut de régularité de la solution. On donne au passage une preuve simple de la conjecture d’Onsager, améliorant un peu l’hypothèse de [1]. On propose une notion de solution dissipative pour de telles solutions faibles.

Dissipative Euler flows and Onsager's conjecture

Camillo De Lellis, László Székelyhidi (2014)

Journal of the European Mathematical Society

Building upon the techniques introduced in [15], for any θ < 1 10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent θ . A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent θ < 1 3 . Our theorem is the first result in this direction.

Distributed control for multistate modified Navier-Stokes equations

Nadir Arada (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to establish necessary optimality conditions for optimal control problems governed by steady, incompressible Navier-Stokes equations with shear-dependent viscosity. The main difficulty derives from the fact that equations of this type may exhibit non-uniqueness of weak solutions, and is overcome by introducing a family of approximate control problems governed by well posed generalized Stokes systems and by passing to the limit in the corresponding optimality conditions.

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