Narrow Results

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

We consider a model for bioremediation of a pollutant by bacteria in a well-stirred bioreactor. A key feature is the inclusion of dormancy for bacteria, which occurs when the critical nutrient level falls below a critical threshold. This feature gives a discrete component to the system due to the change in dynamics (governed by a system of ordinary differential equations between transitions) at switches to/from dormancy. After setting the problem in an appropriate state space, the control is the rate of injection of the critical nutrient and the functional to be minimized is the pollutant level at the final time and the amount of nutrient added. The existence of an optimal control and a discussion of the transitions between dormant and active states are given.

The design problem for semiconductor devices is studied via an optimal control approach for the standard drift–diffusion model. The solvability of the minimization problem is proved. The first-order optimality system is derived and the existence of Lagrange-multipliers is established. Further, estimates on the sensitivities are given. Numerical results concerning a symmetric n–p-diode are presented.

In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

In this paper we consider the Rational Large Eddy Simulation model recently introduced by Galdi and Layton. We briefly present this model, which (in principle) is similar to others commonly used, and we prove the existence and uniqueness of a class of strong solutions. Contrary to the gradient model, the main feature of this model is that it allows a better control of the kinetic energy. Consequently, to prove existence of strong solutions, we do not need subgrid-scale regularization operators, as proposed by Smagorinsky. We also introduce some breakdown criteria that are related to the Euler and Navier–Stokes equations.

In this paper, we present a tsunami model based on the displacement of the lithosphere and the mathematical and numerical analysis of this model. More precisely, we give an existence and uniqueness result for a problem which models the flow and formation of waves at the time of a submarine earthquake in the vicinity of the coast. We propose a model which describes the behavior of the fluid using a bi-dimensional shallow-water model by means of a depth-mean velocity formulation. The ocean is coupled to the Earth's crust whose movement is assumed to be controlled on a large scale by plate equations. Finally, we give some numerical results showing the formation of a tsunami.

We give the formulation of the boundary layer problem of triple deck type with a known displacement in von Mises variables. The condition associated with the displacement is transformed into a nonlocal condition. We introduce an appropriate method to prove the existence of a solution. It relies on a semi-discrete problem in which the pressure gradients are considered as a parameter. We prove the existence of a solution of the von Mises problem for Lipschitzian nondecreasing displacements. We can apply the inverse von Mises transform using an original expression of y and we prove that the functions u, v, p satisfy the system in physical variables except v(x, 0) = 0 because of a lack of regularity. We obtain all the asymptotic behaviors when y → +∞.

In this paper, we address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior. Models of this kind arise in, e.g., plasticity, solid phase transformations, and several other problems in non smooth mechanics. Existence of solutions is proved via passage to the limit in a time-discretization scheme, whereas uniqueness results are obtained by means of convex analysis techniques.

When solving numerically approximations of the Vlasov–Maxwell equations, the source terms in Maxwell's equations coming from the numerical solution of the Vlasov equation do not generally satisfy the continuity equation which is required for Maxwell's equations to be well-posed. Hence it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. Different such formulations have been introduced previously. The aim of this paper is to perform their mathematical analysis and verify the existence and uniqueness of the solution.

We consider the following nonlinear parabolic system

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L² norm of the space derivative of the density, via a new kind of entropy.

We establish new lower semicontinuity results for energy functionals containing a very general volume term of polyconvex type and a surface term depending on the spatial variable in a discontinuous way.

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

The multi-configuration methods are widely used by quantum physicists/chemists for numerical approximation of the many electron Schrödinger equation. Recently, first mathematically rigorous results were obtained on the time-dependent models, e.g. short-in-time well-posedness in the Sobolev space H² for bounded interactions²⁰ with initial data in H², in the energy space for Coulomb interactions with initial data in the same space,^25,5 as well as global well-posedness under a sufficient condition on the energy of the initial data.^4,5 The present contribution extends the analysis by setting an L² theory for the MCTDHF for general interactions including the Coulomb case. This kind of results is also the theoretical foundation of ad hoc methods used in numerical calculation when modification ("regularization") of the density matrix destroys the conservation of energy property, but keeps the mass invariant.

The process of invasion of tissue by cancer cells is crucial for metastasis — the formation of secondary tumours — which is the main cause of mortality in patients with cancer. In the invasion process itself, adhesion, both cell–cell and cell–matrix, plays an extremely important role. In this paper, a mathematical model of cancer cell invasion of the extracellular matrix is developed by incorporating cell–cell adhesion as well as cell–matrix adhesion into the model. Considering the interactions between cancer cells, extracellular matrix and matrix degrading enzymes, the model consists of a system of reaction–diffusion partial integro–differential equations, with nonlocal (integral) terms describing the adhesive interactions between cancer cells and the host tissue, i.e. cell–cell adhesion and cell–matrix adhesion. Having formulated the model, we prove the existence and uniqueness of global in time classical solutions which are uniformly bounded. Then, using computational simulations, we investigate the effects of the relative importance of cell–cell adhesion and cell–matrix adhesion on the invasion process. In particular, we examine the roles of cell–cell adhesion and cell–matrix adhesion in generating heterogeneous spatio-temporal solutions. Finally, in the discussion section, concluding remarks are made and open problems are indicated.

We present a three-dimensional phenomenological model for the magneto-mechanical behavior of magnetic shape memory materials such as Ni₂MnGa. Moving from micromagnetic considerations, we advance to some thermodynamically consistent constitutive relations describing the magnetically-induced cubic-to-tetragonal martensitic transformation in a single crystal. We present an existence analysis for both the constitutive relation problem and the three-dimensional quasi-static evolution problem. Finally, we discuss the reduction of this model to some simpler one by means of a rigorous Γ-convergence analysis.

This paper focuses on a three-dimensional phenomenological model for the isothermal evolution of a polycrystalline shape memory alloy. The model, originally proposed by Auricchio, Taylor, and Lubliner in 1997, is thermodynamically consistent and reproduces the crucial martensitic reorientation effect as well as the tension-compression asymmetric behavior of the material. We prove the existence of a weak solution of the corresponding quasistatic evolution problem by passing to the limit within a time-discretization procedure.

We recall a recently introduced mixed formulation of thin film magnetization problems for type-II superconductors written in terms of two variables, the electric field and the magnetization function, see [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. A finite element approximation, , based on this mixed formulation, involving the lowest-order Raviart–Thomas element for approximating the electric field, was also introduced in [Electric field formulation for thin film magnetization problems, Supercond. Sci. Technol.25 (2012) 104002]. Here h, τ are the spatial and temporal discretization parameters, and with p-1 the value of power in the current–voltage relation characterizing the superconducting material. In this paper, we establish well-posedness of , and prove convergence of the unique solution of to a solution of the power law model (Q_r), for a fixed r > 1, as h, τ → 0. In addition, we prove convergence of a solution of (Q_r) to a solution of the critical state model (Q), as r → 1. Hence, we prove existence of solutions to (Q_r), for a fixed r > 1, and (Q). Finally, numerical experiments are presented.