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Mathematical and Numerical Analysis of an Alternative Well-Posed Two-Layer Turbulence Model

Bijan Mohammadi, Guillaume Puigt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we wish to investigate the behavior of a two-layer k - ε turbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations. First, we explain the difficulties inherent in the model. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical...

Mathematical modelling of rock bolt systems. I

Josef Malík (1998)

Applications of Mathematics

The main goal of the paper is to give a variational formulation of the behaviour of bolt systems in rock mass. The problem arises in geomechanics where bolt systems are applied to reinforce underground openings by inserting steel bars or cables. After giving a variational formulation, we prove the existence and uniqueness and some other properties.

Mathematical modelling of rock bolt systems. II

Josef Malík (2000)

Applications of Mathematics

The main goal of the paper is to describe a reinforcement consisting of fully grouted bolts, which is applied to stabilizing underground openings and tunnels. After a variational formulation is given, the existence and uniqueness is proved. Some asymptotic results that make it possible to replace the real system with a continuous one more suitable for discretization are presented. Some other types of reinforcements and properties are studied.

Multiple solutions for nonlinear discontinuous elliptic problems near resonance

Nikolaos Kourogenis, Nikolaos Papageorgiou (1999)

Colloquium Mathematicae

We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when λ → λ 1 from the left, λ 1 being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

New variational principle and duality for an abstract semilinear Dirichlet problem

Marek Galewski (2003)

Annales Polonici Mathematici

A new variational principle and duality for the problem Lu = ∇G(u) are provided, where L is a positive definite and selfadjoint operator and ∇G is a continuous gradient mapping such that G satisfies superquadratic growth conditions. The results obtained may be applied to Dirichlet problems for both ordinary and partial differential equations.

Nonhomogeneous boundary value problem for a semilinear hyperbolic equation

Andrzej Nowakowski (2008)

Applicationes Mathematicae

We discuss the solvability of a nonhomogeneous boundary value problem for the semilinear equation of the vibrating string x t t ( t , y ) - Δ x ( t , y ) + f ( t , y , x ( t , y ) ) = 0 in a bounded domain and with a certain type of superlinear nonlinearity. To this end we derive a new dual variational method.

Nonlinear fourth order problems with asymptotically linear nonlinearities

Abir Amor Ben Ali, Makkia Dammak (2024)

Mathematica Bohemica

We investigate some nonlinear elliptic problems of the form Δ 2 v + σ ( x ) v = h ( x , v ) in Ω , v = Δ v = 0 on ∂ Ω , ( P ) where Ω is a regular bounded domain in ℝ N , N ≥ 2 , σ ( x ) a positive function in L ∞ ( Ω ) , and the nonlinearity h ( x , t ) is indefinite. We prove the existence of solutions to the problem (P) when the function h ( x , t ) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

Nonlocal variational problems arising in long wave propagatioN

Orlando Lopes (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the existence of minimizer for certain constrained variational problems given by functionals with nonlocal terms. This type of functionals are first integrals of evolution equations describing long wave propagation and the existence of minimizer gives the existence and the stability of traveling waves for these equations. Due to loss of compactness, the major problem is to prevent dichotomy of minimizing sequences. Our approach is an alternative to the concentration-compactness...

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