Narrow Results

We provide a rigorous mathematical framework to establish the limit of a nonlocal model of cell–cell adhesion system to a local model. When the parameter of the nonlocality goes to 0, the system tends to a Cahn–Hilliard system with degenerate mobility and cross-interaction forces. Our analysis relies on a-priori estimates and compactness properties.

In this paper, we study the singular limit c → ∞ of the family of Euler–Nordström systems indexed by the parameters κ² and c, where κ² > 0 is the cosmological constant and c is the speed of light. Using Christodoulou's techniques to generate energy currents, we develop Sobolev estimates that show initial data belonging to an appropriate Sobolev space launch unique solutions to the system that converge to corresponding unique solutions of the Euler–Poisson system with the cosmological constant κ² as c tends to infinity.

We consider a linear evolution problem with memory arising in the theory of hereditary electromagnetism. Under general assumptions on the memory kernel, all single trajectories are proved to decay to zero, but the decay rate is not uniform in dependence of the initial data, so that the system is not exponentially stable. Nonetheless, if the kernel is rapidly fading and close to the Dirac mass at zero, then the solutions are close to exponentially stable trajectories.

This paper investigates front propagation in random media for a free boundary problem arising in combustion theory. We show the existence of asymptotic traveling waves solutions with effective speed depending only on the essential infimum of the combustion rate. This result generalizes a previous result of the same authors in the periodic case.

We consider a pattern-forming system in two space dimensions defined by an energy . The functional models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter ε is the ratio between the intrinsic, material length-scale and the scale of the domain Ω. We show that in the limit ε → 0 any sequence u_ε of patterns with uniformly bounded energy becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(ε). In the limit the stripes become uniform in width and increasingly straight.

Our results are formulated as a convergence theorem, which states that the functional Gamma-converges to a limit functional . This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L²-norm of the divergence of the projection field, or equivalently the L²-norm of the curvature of the field.

At the level of patterns the converging objects are the jump measures |∇_uε| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and Röger, Arch. Rational Mech. Anal.193 (2009) 475–537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.

This paper discusses a time global existence, asymptotic behavior and a singular limit of a solution to the initial boundary value problem for a quantum drift-diffusion model of semiconductors over a one-dimensional bounded domain. Firstly, we show a unique existence and an asymptotic stability of a stationary solution for the model. Secondly, it is shown that the time global solution for the quantum drift-diffusion model converges to that for a drift-diffusion model as the scaled Planck constant tends to zero. This singular limit is called a classical limit. Here these theorems allow the initial data to be arbitrarily large in the suitable Sobolev space. We prove them by applying an energy method.

We examine a hydrodynamic model of the motion of ions in plasma in the regime of small Debye length, a small ratio of the ion/electron temperature, and high Reynolds number. We analyze the associated singular limit and identify the limit problem — the incompressible Euler system. The result leans on careful analysis of the oscillatory component of the solutions by means of Fourier analysis.

We study the low Mach low Freude numbers limit in the compressible Navier–Stokes equations and the transport equation for evolution of an entropy variable — the potential temperature $\mathrm{\Theta }$. We consider the case of well-prepared initial data on “flat” torus and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier–Stokes system and the transport equation for the second-order variation of $\mathrm{\Theta }$.

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes the relaxation dynamics for an energy which takes lipid–phase separation and lipid–cholesterol interaction energy into account. It takes the form of an (extended) Cahn–Hilliard equation. Different laws for the exchange term represent equilibrium and nonequilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the nonequilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.

In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third-order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second-order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third-order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments.

A system of drift-diffusion equations for the electron, hole, and oxygen vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet–Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. In the fast-relaxation limit, the system reduces to a drift-diffusion system for the oxygen vacancy density and electric potential, which is often used in neuromorphic applications. The following results are proved: the global existence of weak solutions to the full system in any space dimension; the uniform-in-time boundedness of the solutions to the full system and the fast-relaxation limit in two space dimensions; the global existence and weak–strong uniqueness analysis of the reduced system. Numerical experiments in one space dimension illustrate the behavior of the solutions and reproduce hysteresis effects in the current–voltage characteristics.

We show that spatial patterns (“hotspots”) may form in the crime model

We do this by first constructing classical solutions to the nonlocal scalar problem

We present a model for biological network formation originally introduced by Cai and Hu [Adaptation and optimization of biological transport networks, Phys. Rev. Lett.111 (2013) 138701]. The modeling of fluid transportation (e.g., leaf venation and angiogenesis) and ion transportation networks (e.g., neural networks) is explained in detail and basic analytical features like the gradient flow structure of the fluid transportation network model and the impact of the model parameters on the geometry and topology of network formation are analyzed. We also present a numerical finite-element based discretization scheme and discuss sample cases of network formation simulations.

We investigate the system of compressible Navier–Stokes equations with hyperbolic heat conduction, i.e. replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time $\tau$, global smooth solution exists for small initial data. Moreover, as $\tau$ goes to zero, we obtain the uniform convergence of solutions of the relaxed system to that of the classical compressible Navier–Stokes equations.

We consider a singular limit problem from the damped wave equation with a power type nonlinearity (NLDW) to the corresponding heat equation (NLH). We call our singular limit problem non-delay limit. We show that the solution of NLDW goes to the one of NLH in $H^1$ topology under the both $H^1$ regularity solutions. We also obtain the positive convergence rate in the weaker topology $L^2$. Moreover, with restriction of the range of power, if the solution to NLH is global and decays to zero, then we get the global-in-time uniform convergence of the non-delay limit.

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:={𝟙}_{(-\infty ,0]}(\cdot )exp(\cdot )\ast \rho$ satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V^{\prime }(W)W{\partial }_{x}W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.