Line Method Approximations to the Cauchy Problem for Nonlinear Parabolic Differential Equations.
Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
Liouville type condition and the interior regularity of quasilinear parabolic system (the case of BMO-solutions)
Lipschitz stability of solutions to parametric optimal control problems for parabolic equations.
Local existence, uniqueness and regularity for a class of degenerate parabolic systems arising in biological models
Local solution of parabolic equations with strongly increasing nonlinearity by the Rothe method
Local solvability of diagonal semilinear parabolic systems
Local stability of spike steady states in a simplified Gierer-Meinhardt system.
Localization and Blow-Up of Thermal Waves in Nonlinear Heat Conduction with Peaking.
Log-concavity in some parabolic problems.
Long-Range Numerical Solution of Mildly Non-Linear Parabolic Equations.
Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
The nonlinear heat equation with a fractional Laplacian , is considered in a unit ball . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.
Lyapunov functions and -estimates for a class of reaction-diffusion systems
We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, , x ∈ Ω, t > 0, for , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform -estimates.
Macroscopic models of collective motion and self-organization
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
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
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.
Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed-hybrid finite element solution
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 study of a petroleum-engineering scheme
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
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...
Method of Rothe and nonlinear parabolic boundary value problems of arbitrary order