Domains of Existence of Polymonogenic Functions

The paper addresses the Levi problem for a system of n Fueter equations in a domain in quaternionic space . This problem relates to various conditions of convexity and pseudoconvexity of the boundary of the domain. Received: October, 2007, Accepted: February, 2008.     

Existence of Periodic Solutions for the Quasi-Static Thermoelastic Thermistor Problem

We deal with a new model for the thermistor problem formulated as a coupled system of PDE’s involving nonlinear energy heat equation, stationary charge conservation equation of electrical current and thermoelastic equations of displacement. We establish the existence of weak periodic solutions rewriting our system as an abstract problem in order to utilize the maximal monotone mappings theory and a fixed point argument for a suitable operator equation.      

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.     

Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data

We obtain existence results for some strongly nonlinear Cauchy problems posed in and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on , they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.     

Stability of a Cartesian grid projection method for zero Froude number shallow water flows

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov–Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions. The key innovation compared to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection. The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method’s applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme.     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

The Nielsen number for free fundamental groups and maps without remnant

For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N(g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g _♯(u)g _♯(v) where g _♯ is the induced homomorphism and u, v ∈ π₁(Y) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π₁(Y) are solutions to the equation z = W ⁻¹ _x f _♯(z)W _y , with x and y fixed points that are represented in the fundamental group by W _x and W _y , respectively. The number of elements to be tested is profoundly decreased by using abelianization as well. This work is a significant extension of Wagner’s results involving maps with remnant and Wagner’s algorithm. Our proofs involve new concepts and techniques. We present an algorithm for N(g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. Dedicated to Edward Fadell for inspirational teaching and guidance as the thesis advisor of the first author     

Decay properties for the solutions of a partial differential equation with memory

Decay properties in energy norm for solutions of a class of partial differential equations with memory are studied by means of frequency domain methods. Our results are optimal for this class, as we are able to characterize polynomial as well as exponential decay rates. The results apply to models for viscoelastic materials. An extension to a semilinearly perturbed problem is also included. Received: 9 July 2008, Revised: 16 September 2008     

Regularity theory in Orlicz spaces for the parabolic polyharmonic equations

In this paper we generalize classical L ^p estimates to Orlicz spaces for the parabolic polyharmonic equations. Our argument is based on the iteration-covering procedure. Received: 10 September 2007     

Solution of nonlinear convection-diffusion problems by a conservative Galerkin-characteristics method

We consider a scheme for nonlinear (degenerate) convection dominant diffusion problems that arise in contaminant transport in porous media with equilibrium adsorption isotherm. This scheme is based on a regularization relaxation scheme that has been introduced by Jäger and Ka?ur (Numer Math 60:407–427, 1991; M²AN Math Model Numer Anal 29(N5):605–627, 1995) with a type of numerical integration by Bermejo (SIAM J Numer Anal 32:425–455, 1995) to the modified method of characteristics with adjusted advection MMOCAA that was recently developed by Douglas et al. (Numer Math 83(3):353–369, 1999; Comput Geosci 1:155–190, 1997). We present another variant of adjusting advection method. The convergence of the scheme is proved. An error estimate of the approximated scheme is derived. Computational experiments are carried out to illustrate the capability of the scheme to conserve the mass.     

Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods

We derive in this paper a posteriori error estimates for discretizations of convection–diffusion–reaction equations in two or three space dimensions. Our estimates are valid for any cell-centered finite volume scheme, and, in a larger sense, for any locally conservative method such as the mimetic finite difference, covolume, and other. We consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces. We allow for the cases of inhomogeneous and anisotropic diffusion–dispersion tensors and of convection dominance. The estimates are established in the energy (semi)norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type. They are fully computable (all occurring constants are evaluated explicitly) and locally efficient (give a local lower bound on the error times an efficiency constant), so that they can serve both as indicators for adaptive refinement and for the actual control of the error. They are semi-robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local Péclet number gets sufficiently small. Numerical experiments confirm their accuracy. This work was supported by the GdR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France. The main part of this work was carried out during the author’s post-doc stay at Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud and CNRS, Orsay, France.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

An algorithm for the fast solution of symmetric linear complementarity problems

This paper studies algorithms for the solution of mixed symmetric linear complementarity problems. The goal is to compute fast and approximate solutions of medium to large sized problems, such as those arising in computer game simulations and American options pricing. The paper proposes an improvement of a method described by Kocvara and Zowe (Numer Math 68:95–106, 1994) that combines projected Gauss–Seidel iterations with subspace minimization steps. The proposed algorithm employs a recursive subspace minimization designed to handle severely ill-conditioned problems. Numerical tests indicate that the approach is more efficient than interior-point and gradient projection methods on some physical simulation problems that arise in computer game scenarios. The research of J. L. Morales was supported by Asociación Mexicana de Cultura AC and CONACyT-NSF grant J110.388/2006. The research of J. Nocedal was supported by National Science Foundation grant CCF-0514772, Department of Energy grant DE-FG02-87ER25047-A004 and a grant from the Intel Corporation.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.      

A note on the uniqueness of mild solutions to the Navier-Stokes equations

In this note, we will give another proof of the uniqueness of mild solutions to the Navier-Stokes equations in the class C([0,∞); by a simple application of Giga-Shor’s L ^p − L ^q (time-space) estimates, i.e., integral norms in the time variable. The proof relies on a method introduced by S. Monniaux [9] to prove the same result. Received: 11 June 2006     

Convergence of finite volume schemes for triangular systems of conservation laws

We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type for these systems that exploit the triangular structure. We prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero. Some numerical examples are presented, one of which is related to flows in porous media. The research of K. H. Karlsen was supported by an Outstanding Young Investigators Award from the Research Council of Norway.     

Existence results for a class of porous medium type equations with a quadratic gradient term

In this paper we study the Dirichlet problem in Q _T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:

Large time behavior for the non-Newtonian flow in R3

This paper interests a system for the non-Newtonian flow in the whole space. [14] estimated decay of it as t tends to infinity. The aim of the paper is to investigate decay problem of it and to improve a result of [14].      

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form