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Picone’s identity for a Finsler p -Laplacian and comparison of nonlinear elliptic equations

Jaroslav Jaroš (2014)

Mathematica Bohemica

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm H in ℝ n which is continuously differentiable for x ≠ 0 and such that H p is strictly convex for some p > 1 . Two important special cases are the p -Laplacian and the so-called pseudo p -Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...

Planar flows of incompressible heat-conducting shear-thinning fluids — existence analysis

Miroslav Bulíček, Oldřich Ulrych (2011)

Applications of Mathematics

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L ∞ -truncation method, used to obtain the strong convergence of the velocity...

Plane wave discontinuous Galerkin methods: Analysis of the h-version

Claude J. Gittelson, Ralf Hiptmair, Ilaria Perugia (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that ...

Plane wave stability of some conservative schemes for the cubic Schrödinger equation

Morten Dahlby, Brynjulf Owren (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

The plane wave stability properties of the conservative schemes of Besse [SIAM J. Numer. Anal.42 (2004) 934–952] and Fei et al. [Appl. Math. Comput.71 (1995) 165–177] for the cubic Schrödinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different. An energy preserving generalisation of the Fei method with improved stability is presented.

Poche de tourbillon pour Euler 2D incompressible dans un ouvert à bord

Nicolas Depauw (1998)

Journées équations aux dérivées partielles

Nous considérons l'équation d'Euler pour un fluide incompressible dans un domaine borné régulier du plan. Pour une donnée initiale avec un tourbillon de type poche, i.e valant 1 sur un ouvert lisse à bord höldérien et 0 en dehors, nous prouvons l'existence d'une solution de même type, pour tout temps si la poche initiale est décollée du bord du domaine et seulement localement en temps si la poche initiale est tangente au bord. Nous contrôlons l'influence du bord grâce à la théorie des problèmes...

Poches de tourbillon singulières dans un fluide faiblement visqueux.

Taoufik Hmidi (2006)

Revista Matemática Iberoamericana

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is C1+s outside a singular set Σ, so the velocity is, for all time, lipschitzian outside the image of Σ through the viscous flow. In addition, the correponding lipschitzian norm is independent of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

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