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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 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.

On the solvability of Dirichlet problem for the weighted p-Laplacian

Dominik Mielczarek, Jerzy Rydlewski, Ewa Szlachtowska (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we are concerned with the existence and uniqueness of the weak solution for the weighted p-Laplacian. The purpose of this paper is to discuss in some depth the problem of solvability of Dirichlet problem, therefore all proofs are contained in some detail. The main result of the work is the existence and uniqueness of the weak solution for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the solution belongs to the Sobolev space W ₀ 1 , p ( Ω ) .

On the solvability of the equation div u = f in L 1 and in C 0

Bernard Dacorogna, Nicola Fusco, Luc Tartar (2003)

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

We show that the equation div u = f has, in general, no Lipschitz (respectively W 1 , 1 ) solution if f is C 0 (respectively L 1 ).

On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients

Petr Harasim (2011)

Applications of Mathematics

We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution....

On weak-strong uniqueness property for full compressible magnetohydrodynamics flows

Weiping Yan (2013)

Open Mathematics

This paper is devoted to the study of the weak-strong uniqueness property for full compressible magnetohydrodynamics flows. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and an additional equation which describes the evolution of the magnetic field. Using the relative entropy inequality, we prove that a weak solution coincides with the...

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