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On Bardina and Approximate Deconvolution Models

Roger Lewandowski (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We first outline the procedure of averaging the incompressible Navier-Stokes equations when the flow is turbulent for various type of filters. We introduce the turbulence model called Bardina’s model, for which we are able to prove existence and uniqueness of a distributional solution. In order to reconstruct some of the flow frequencies that are underestimated by Bardina’s model, we next introduce the approximate deconvolution model (ADM). We prove existence and uniqueness of a “regular weak solution”...

On global solutions to a nonlinear Alfvén wave equation

XS. Feng, F. Wei (1995)

Annales Polonici Mathematici

We establish the global existence and uniqueness of smooth solutions to a nonlinear Alfvén wave equation arising in a finite-beta plasma. In addition, the spatial asymptotic behavior of the solution is discussed.

On optimal decay rates for weak solutions to the Navier-Stokes equations in R n

Tetsuro Miyakawa, Maria Elena Schonbek (2001)

Mathematica Bohemica

This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in ℝ n . Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound ∥ u ( t ) ∥ ≥ ( t + 1 ) - n + 4 2 .

On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities

Martin Lanzendörfer, Jan Stebel (2011)

Applications of Mathematics

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.

On the continuity of degenerate n-harmonic functions

Flavia Giannetti, Antonia Passarelli di Napoli (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ∫ 1 ∞ P ( t ) t 2 d t = ∞ . ∫ 1 ∞ P ( t ) t 2   d t = ∞ .

On the continuity of degenerate n-harmonic functions

Flavia Giannetti, Antonia Passarelli di Napoli (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ∫ 1 ∞ P ( t ) t 2   d t = ∞ .

On the global existence for a regularized model of viscoelastic non-Newtonian fluid

Ondřej Kreml, Milan Pokorný, Pavel Šalom (2015)

Colloquium Mathematicae

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like μ ( D ) ∼ | D | p - 2 (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

On the global existence for the Muskat problem

Peter Constantin, Diego Córdoba, Francisco Gancedo, Robert M. Strain (2013)

Journal of the European Mathematical Society

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L 2 ( ℝ ) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance f 1 ≤ 1 / 5 . Previous results of this...

On the local Cauchy problem for first order partial differential functional equations

Elżbieta Puźniakowska-Gałuch (2010)

Annales Polonici Mathematici

A theorem on the existence of weak solutions of the Cauchy problem for first order functional differential equations defined on the Haar pyramid is proved. The initial problem is transformed into a system of functional integral equations for the unknown function and for its partial derivatives with respect to spatial variables. The method of bicharacteristics and integral inequalities are applied. Differential equations with deviated variables and differential integral equations can be obtained...

On the Neumann problem with L¹ data

J. Chabrowski (2007)

Colloquium Mathematicae

We investigate the solvability of the linear Neumann problem (1.1) with L¹ data. The results are applied to obtain existence theorems for a semilinear Neumann problem.

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