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Macroscopic models of collective motion and self-organization

Pierre Degond, Amic Frouvelle, Jian-Guo Liu, Sebastien Motsch, Laurent Navoret (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view...

Magnetization switching on small ferromagnetic ellipsoidal samples

François Alouges, Karine Beauchard (2009)

ESAIM: Control, Optimisation and Calculus of Variations

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

Magnetization switching on small ferromagnetic ellipsoidal samples

François Alouges, Karine Beauchard (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

Malliavin method for optimal investment in financial markets with memory

Qiguang An, Guoqing Zhao, Gaofeng Zong (2016)

Open Mathematics

We consider a financial market with memory effects in which wealth processes are driven by mean-field stochastic Volterra equations. In this financial market, the classical dynamic programming method can not be used to study the optimal investment problem, because the solution of mean-field stochastic Volterra equation is not a Markov process. In this paper, a new method through Malliavin calculus introduced in [1], can be used to obtain the optimal investment in a Volterra type financial market....

Mathematical analysis of a two-phase parabolic free boundary problem derived from a Bingham-type model with visco-elastic core

A. Farina, L. Fusi (2005)

Bollettino dell'Unione Matematica Italiana

In this paper we consider a two-phase one-dimensional free boundary problem for the heat equation, arising from a mathematical model for a Bingham-like fluid with a visco-elastic core. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Existence of classical solution is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a technique based on a...

Mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere.

Stanislav Antontsev, Jesús Ildefonso Díaz (2007)

RACSAM

We study the boundary layer approximation of the, already classical, mathematical model which describes the discharge of a laminar hot gas in a stagnant colder atmosphere of the same gas. We start by proving the existence and uniqueness of solutions of the nondegenerate problem under assumptions implying that the temperature T and the horizontal velocity u of the gas are strictly positive: T ≥ δ > 0 and u ≥ ε > 0 (here δ and ε are given as boundary conditions in the external atmosphere)....

Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed-hybrid finite element solution

Kamyar Malakpoor, Enrique F. Kaasschieter, Jacques M. Huyghe (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory [J.M. Huyghe and J.D. Janssen, Int. J. Engng. Sci.35 (1997) 793–802; K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modeling of incompressible charged porous media. ESAIM: M2AN41 (2007) 661–678]. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic...

Mathematical models of tumor growth systems

Takashi Suzuki (2012)

Mathematica Bohemica

We study a class of parabolic-ODE systems modeling tumor growth, its mathematical modeling and the global in time existence of the solution obtained by the method of Lyapunov functions.

Mathematical study of a petroleum-engineering scheme

Robert Eymard, Raphaèle Herbin, Anthony Michel (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies...

Mathematical study of a petroleum-engineering scheme

Robert Eymard, Raphaèle Herbin, Anthony Michel (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies...

Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in  L p -spaces

Jan Prüss (2002)

Mathematica Bohemica

Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal L p -regularity is shown. By means of this purely operator theoretic approach, classical results on L p -regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface...

Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows

Jonas Sauer (2016)

Czechoslovak Mathematical Journal

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L p -regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group G : = ℝ n - 1 × ℝ / L ℤ to obtain an ℛ -bound for the...

Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Sahbi Boussandel, Ralph Chill, Eva Fašangová (2012)

Czechoslovak Mathematical Journal

Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and L 2 -maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented...

Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

Luchko, Yury (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional...

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