Mathematical Models and Methods in Applied Sciences

Published by World Scientific Publishing
Artificial interface conditions parametrized by a complex number $\theta_{0}$ are introduced for 1D-Schr\"odinger operators. When this complex parameter equals the parameter $\theta\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $\theta_{0}$. Comment: 60 pages, 13 figures
We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global in time existence or finite-time blow-up of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also analyze conditions for global in time existence when the repulsion is modeled by the isothermal pressure law.
Evolution of local densities (solid line for S 1 and dashed line for S 2 ). Africa: top–left (a); 
We adopt an operatorial method based on the so-called creation, annihilation and number operators in the description of different systems in which two populations interact and move in a two-dimensional region. In particular, we discuss diffusion processes modeled by a quadratic hamiltonian. This general procedure will be adopted, in particular, in the description of migration phenomena. With respect to our previous analogous results, we use here fermionic operators since they automatically implement an upper bound for the population densities.
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and Vogelius in \cite{NguyenVogelius2} for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in \cite{NguyenVogelius2}, we revise and extend the work of Haddar, Joly, and Nguyen in \cite{HJNg1}, where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in \cite{NguyenVogelius2}, our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in \cite{NguyenVogelius2} concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in \cite{NguyenVogelius2} does not fit directly in our setting. Our proof makes use of a result by H\"ormander in \cite{Hor}. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.
We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinear one dimensional systems, namely the generation of transport with a preferred direction in the absence of a net driving force.
In this paper we analyze a time-reversal experiment in a random underwater acoustic channel. In this kind of waveguide with semi-infinite cross section a propagating field can be decomposed over three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. Using an asymptotic analysis based on a separation of scales technique we derive the asymptotic form of the the coupled mode power equation for the propagating modes. This approximation is used to compute the transverse profile of the refocused field and show that random inhomogeneities inside the waveguide deteriorate the spatial refocusing. This result, in an underwater acoustic channel context, is in contradiction with the classical results about time-reversal experiment in other configurations, for which randomness in the propagation medium enhances the refocusing.
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.
In this article we prove convergence of adaptive finite element methods for second order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation.
Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.
Many drug delivery systems suffer from undesirable interactions with the host immune system. It has been experimentally established that covalent attachment (irreversible adsorption) of suitable macromolecules to the surface of the drug carrier can reduce such undesirable interactions. A fundamental understanding of the adsorption process is still lacking. In this paper, the classical random irreversible adsorption model is generalized to capture certain essential processes involved in pharmacological applications, allowing for macromolecules of different sizes, partial overlapping of the tails of macromolecules, and the influence of reactions with the solvent on the adsorption process. Working in one dimension, an integro-differential evolution equation for the adsorption process is derived and the asymptotic behaviour of the surface area covered and the number of molecules attached to the surface is studied. Finally, equation-free dynamic renormalization tools are applied to study the asymptotically self-similar behaviour of the adsorption statistics.
1. Convergence in the L 2-norm plotted against the timestep τ = h, of the stabilized (dashed line for explicit and dotted line for implicit treatment of the stabilization operator) and unstabilized (dash-dot) Crank-Nicolson method. The full line shows optimal second order convergence.
We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the P\'eclet number and of the regularity of the exact solution.
We consider a linear integro-differential equation which arises to describe both aggregation-fragmentation processes and cell division. We prove the existence of a solution $(\lb,\U,\phi)$ to the related eigenproblem. Such eigenelements are useful to study the long time asymptotic behaviour of solutions as well as the steady states when the equation is coupled with an ODE. Our study concerns a non-constant transport term that can vanish at $x=0,$ since it seems to be relevant to describe some biological processes like proteins aggregation. Non lower-bounded transport terms bring difficulties to find $a\ priori$ estimates. All the work of this paper is to solve this problem using weighted-norms.
Notation on (0, +∞) × (0, +∞)
We study mathematically a system of partial differential equations arising in the modelling of an aging fluid, a particular class of non Newtonian fluids. We prove well-posedness of the equations in appropriate functional spaces and investigate the longtime behaviour of the solutions.
The initial value problem is considered in the present paper for bipolar quantum hydrodynamic model for semiconductors (QHD) in $\mathbb{R}^3$. We prove that the unique strong solution exists globally in time and tends to the asymptotical state with an algebraic rate as $t\to+\infty$. And, we show that the global solution of linearized bipolar QHD system decays in time at an algebraic decay rate from both above and below. This means in general, we can not get exponential time-decay rate for bipolar QHD system, which is different from the case of unipolar QHD model (where global solutions tend to the equilibrium state at an exponential time-decay rate) and is mainly caused by the nonlinear coupling and cancelation between two carriers. Moreover, it is also shown that the nonlinear dispersion does not affect the long time asymptotic behavior, which by product gives rise to the algebraic time-decay rate of the solution of the bipolar hydrodynamical model in the semiclassical limit.
We consider optimization problems for complex systems in which the cost function has a multivalleyed landscape. We introduce a new class of dynamical algorithms which, using a suitable annealing procedure coupled with a balanced greedy-reluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the Sherrington-Kirkpatrick model of spin-glasses.
In this paper, we provide the $O(\epsilon)$ corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek & coauthors describing flocking biological agents. The parameter $\epsilon$ stands for the ratio of the microscopic to the macroscopic scales. The $O(\epsilon)$ corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first order derivatives of the density and velocity. The derivation method is based on the standard Chapman-Enskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.
In this paper, we present a two-species Vicsek model, that describes alignment interactions of self-propelled particles which can either move or not. The model consists in two populations with distinct Vicsek dynamics that interact only via the passage of the particles from one population to the other. The derivation of a macroscopic description of this model is performed using the methodology used for the Vicsek model: we find out a regime where alignment in the whole population occurs. We obtain a new macroscopic model for the densities of each populations and the common mean direction of the particles. The treatment of the non-conservativity of the interactions requires a detail study of the linearised interaction operator.
We present an individual-based model describing disk-like self-propelled particles moving inside parallel planes. The disk directions of motion follow alignment rules inside each layer. Additionally, the disks are subject to interactions with those of the neighboring layers arising from volume exclusion constraints. These interactions affect the disk inclinations with respect to the plane of motion. We formally de-rive a macroscopic model composed of planar Self-Organized Hydrodynamic (SOH) models describing the transport of mass and evolution of mean direction of motion of the disks in each plane, supplemented with transport equations for the mean disk inclination. These planar models are coupled due to the interactions with the neighboring planes. Numerical comparisons between the individual-based and macroscopic models are carried out. These models could be applicable, for instance, to describe sperm-cell collective dynamics.
In the present paper the macroscopic limits of the kinetic model for inter-acting entities (individuals, organisms, cells) are studied. The kinetic model is one-dimensional and entities are characterized by their position and orientation (+/-) with swarming interaction controlled by the sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various sensitivity parameters. In the former case the classical linear difusion equation results whereas in the latter a traveling wave solution does both in the zeroth (`Euler') and frst (`Navier-Stokes') order of approximation.
We consider the hydrodynamic limit of a collisionless and non-diffusive kinetic equation under strong local alignment regime. The local alignment is first considered by Karper, Mellet and Trivisa in [24], as a singular limit of an alignment force proposed by Motsch and Tadmor in [32]. As the local alignment strongly dominate, a weak solution to the kinetic equation under consideration converges to the local equilibrium, which has the form of mono-kinetic distribution. We use the relative entropy method and weak compactness to rigorously justify the weak convergence of our kinetic equation to the pressureless Euler system.
A modification of the parabolic Allen-Cahn equation, determined by the substitution of Fick's diffusion law with a relaxation relation of Cattaneo-Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.
Domain branching near the boundary appears in many singularly-perturbed models for microstructure in materials and was first demonstrated mathematically by Kohn and M\"uller for a scalar problem modeling the elastic behavior of shape-memory alloys. We study here a model for shape-memory alloys based on the full vectorial problem of nonlinear elasticity, including invariance under rotations, in the case of two wells in two dimensions. We show that, for two wells with two rank-one connections, the energy scales proportional to the power $2/3$ of the surface energy, in agreement with the scalar model. In a case where only one rank-one connection is present, we show that the energy exhibits a different behavior, proportional to the power $4/5$ of the surface energy. This lower energy is achieved by a suitable interaction of the two components of the deformations and hence cannot be reproduced by the scalar model. Both scalings are proven by explicit constructions and matching lower bounds.
This note 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.
An example of a constrained deformation.
In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By constraining the deformed configurations to special 1D atomic chains with position-dependent elongation vectors for the vertical direction, we show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configurations with additional boundary layers localized near the twinning interfaces. In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizing sequences and rigorously derive the corresponding limiting form of the surface energy.
The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g., a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Amp\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\`ere equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: It enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge-Amp\`ere equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.
We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.
We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability $\mu$ and the permittivity $\varepsilon$ are symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We show that a unique continuation result for globally $W^{1,\infty}$ coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise $W^{1,\infty}$ with respect to a suitable countable collection of sub-domains with $C^{0}$ boundaries. Such suitable collections include any bounded finite collection. The proof relies on a general argument, not specific to Maxwell's equations. This result is then extended to the case when within these sub-domains the permeability and permittivity are only $L^\infty$ in sets of small measure.
We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.
The Self-Organized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steady-state is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the low-order modes are provided. The fully non-linear model is numerically solved and nonlinear mode-coupling is then analyzed. Finally, the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated.
We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution.
This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schr\"odinger system. A combination of tools, proofs and results are presented in the framework of the concentration--compactness method.
We prove that, for a smooth two-body potentials, the quantum mean-field approximation to the nonlinear Schroedinger equation of the Hartree type is stable at the classical limit h \to 0, yielding the classical Vlasov equation.
Two finite element approximations of the Oldroyd-B model for dilute polymeric fluids are considered, in bounded 2- and 3-dimensional domains, under no flow boundary conditions. The pressure and the symmetric conformation tensor are aproximated by either (a) piecewise constants or (b) continuous piecewise linears, the velocity by (a) continuous piecewise quadratics or a reduced version with linear tangential component on each edge, and (b) by continuous piecewise quadratics or the mini-element. Both schemes (a) and (b) satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler type time discretization. This extends the results of [Boyaval et al. M2AN 43 (2009) 523--561], where a piecewise constant approximation of the conformation tensor was necessary to treat the advection term in the stress equation, and a restriction on the time step, based on the initial data, was required to ensure that the approximation to the conformation tensor remained positive definite. Furthermore, for (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms like in [Barrett and S\"uli, M3AS 18 (2008) 935--971] for the FENE dumbbell model, we show (subsequence) convergence towards global-in-time weak solutions (when d=2, cut-offs can be replaced with a time step restriction dependent on the spatial discretization parameter). Hence, we prove existence of global-in-time weak solutions to these regularized models. Comment: 52 pages, 1 figure
We establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended T-mesh of an analysis-suitable T-spline, is contained in the corresponding analysis-suitable T-spline space. This is accomplished through the theory of perturbed analysis-suitable T-spline spaces and a simple topological dimension formula. Second, we establish the theory of analysis-suitable local refinement and describe the conditions under which two analysis-suitable T-spline spaces are nested. Last, we demonstrate that these results can be used to establish basic approximation results which are critical for analysis.
In this paper, we will give approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the mesh size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.
We present a posteriori error estimates for a recently developed atomistic/continuum coupling method, the Consistent Energy-Based QC Coupling method. The error estimate of the deformation gradient combines a residual estimate and an a posteriori stability analysis. The residual is decomposed into the residual due to the approximation of the stored energy and that due to the approximation of the external force, and are bounded in negative Sobolev norms. In addition, the error estimate of the total energy using the error estimate of the deformation gradient is also presented. Finally, numerical experiments are provided to illustrate our analysis.
We deal with local density approximations for the kinetic and exchange energy terms of a periodic Coulomb model. For the kinetic energy, we give a rigorous derivation of the usual combination of the von-Weizsäcker term and the Thomas-Fermi term in the “high density” limit. Furthermore, we justify the inclusion of the Dirac term for the exchange energy and the Slater term for the local exchange potential. Our method is based on deformations (local scaling transformat ions) of plane waves in a periodic box.
In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a "cohesive" energy, that is, whose cost depends on the actual opening of the discontinuity.
The configurational distribution function, solution of an evolution (diffusion) equation of the Fokker-Planck-Smoluchowski type, is (at least part of) the corner stone of polymer dynamics: it is the key to calculating the stress tensor components. This can be reckoned from \cite{bird2}, where a wealth of calculation details is presented regarding various polymer chain models and their ability to accurately predict viscoelastic flows. One of the simplest polymer chain idealization is the Bird and Warner's model of finitely extensible nonlinear elastic (FENE) chains. In this work we offer a proof that the steady state configurational distribution equation has unique solutions irrespective of the (outer) flow velocity gradients (i.e. for both slow and fast flows).
In this paper we deduce by {\Gamma}-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by {\epsilon} the thickness of the plate, we study the case where the scaling factor of the elasto- plastic energy is of order {\epsilon}^ (2{\alpha}-2), with {\alpha}>=3. We show that solutions to the three- dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on {\alpha}.
Initial data, numerical solution at T = 0, scattering state for (5.1) with σ = 2 and λ = 1 (left), λ = 5 (right)
The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schrodinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proven rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed.
Examples of geometry configurations with interior inclusions. 
Example of disconnected background. Here D 0 = A ∪ C and D 1 = B 
Four different configurations with high-conductivity 1 /ǫ (red) and low-conductivity ǫ (blue) regions. Expansion (52) can be obtained in these cases. 
In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in \cite{ge09_1} where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the paper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high and low conductivity inclusions.
We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper \cite{limits}, we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is \emph{asymptotic}, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter.
We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, counterpropagating waves, each one satisfying such an equation. One also recovers homologous equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green-Naghdi type model, its simplified version in the so-called Camassa-Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.
We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.
Direct comparison between the 2D (full line) and 1D (dashed line) numerical predictions for 8 different values of the diffusion κ. Concentration distribution: full line stands for the average over a cross-section of concentration distribution for the 2D geometry; dashed line and dots stand for the 1D concentration distribution.
In the paper the flow in a thin tubular structure is considered. The velocity of the flow stands for a coefficient in the diffusion-convection equation set in the thin structure. An asymptotic expansion of solution is constructed. This expansion is used further for justification of an asymptotic domain decomposition strategy essentially reducing the memory and the time of the code. A numerical solution obtained by this strategy is compared to the numerical solution obtained by a direct FEM computation.
We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model $u_t=\Delta u + f(u) \1_{|y|\leq R} - \alpha u \1_{|y|>R}$, with $f$ a bistable nonlinearity taking effect only on the domain $\Rm\times [-R,R]$, which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of possible travelling fronts. For this purpose, we present dynamical systems technics and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution $u$ after stimulation, depending of the thickness $R$ of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials.
We investigate analytically and numerically the existence of stationary solutions converging to zero at infinity for the incompressible Navier-Stokes equations in a two-dimensional exterior domain. More precisely, we find the asymptotic behaviour for such solutions in the case where the net force on the boundary of the domain is non-zero. In contrast to the three dimensional case, where the asymptotic behaviour is given by a scale invariant solution, the asymptote in the two-dimensional case is not scale invariant and has a wake. We provide an asymptotic expansion for the velocity field at infinity, which shows that, within a wake of width $|\boldsymbol{x}|^{2/3}$, the velocity decays like $|\boldsymbol{x}|^{-1/3}$, whereas outside the wake, it decays like $|\boldsymbol{x}|^{-2/3}$. We check numerically that this behaviour is accurate at least up to second order and demonstrate how to use this information to significantly improve the numerical simulations. Finally, in order to check the compatibility of the present results with rigorous results for the case of zero net force, we consider a family of boundary conditions on the body which interpolate between the non-zero and the zero net force case.
In this paper, we analyze a mathematical model focusing on key events of the cell invasion process. The three equations of the corresponding coupled system describe the behavior of the invasive cells, the extracellular matrix and the degradative enzymes. We employ a fix-point method and a priori estimates to prove local and global existence, uniqueness and regularity properties of the solutions. Our approach enable us to find estimates that are uniform in time. This is essential in proving, in the last part of the paper, new results that establish the asymptotic behavior of the solutions.
The non-isentropic compressible Euler-Maxwell system is investigated in $R^3$ in the present paper, and the $L^q$ time decay rate for the global smooth solution is established. It is shown that the density and temperature of electron converge to the equilibrium states at the same rate $(1+t)^{-11/4}$ in $L^q$ norm.
Top-cited authors
Nicola Bellomo
  • University of Granada
Franco Brezzi
  • University School for Advanced Studies IUSS Pavia
Alessandro Russo
  • Università degli Studi di Milano-Bicocca
Abdelghani Bellouquid
  • Cadi Ayyad University
Yuri Bazilevs
  • University of California, San Diego