The quasi-stationary Maxwell equations as singular limit of the complete equations: The quasi-linear case

The quasi-stationary Maxwell equations are considered as the time-singular limit of the complete equations at the vanishing of the dielectric constant. Uniformly stable solutions of the complete equations are constructed, and their convergence to a solution of the quasi-stationary equations is proved and estimated.     

A simple justification of the singular limit for equatorial shallow‐water dynamics

The equatorial shallow‐water equations at low Froude number form a symmetric hyperbolic system with large variable‐coefficient terms. Although such systems are not covered by the classical Klainerman‐Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long‐wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory. © 2008 Wiley Periodicals, Inc.     

Maximal attractors for the phase-field equations of penrose-fife type

In this paper, the dynamics for the phase-field equations of Penrose-Fife type arising from the study of phase transitions is investigated. One of important features of this problem is that the metric space H we work with is incomplete.     

The singular limit of a vector-valued reaction-diffusion process

We study the asymptotic behaviour of the solution to the vector-valued reaction-diffusion equation

where . We assume that the the potential depends only on the modulus of and vanishes along two concentric circles. We present a priori estimates for the solution , and, in the spatially radially symmetric case, we show rigorously that in the singular limit as , two phases are created. The interface separating the bulk phases evolves by its mean curvature, while evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.

     

Singular limit of a transmission problem for the parabolic phase-field model

A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the Caginalp-Fix phase field model. Dirichlet (or Neumann) and Cauchy conditions are required. A regular solution is obtained by means of approximation techniques for parabolic systems. Then, an asymptotic study of the problem is carried out as the time relaxation parameter for the phase field tends to 0 in one of the domains. It is also proved that the limit formulation admits a unique solution in a suitable weak sense.     

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation of the form ut=Δu+ε⁻²fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u₀ that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε²|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form

     

The approximate solution of singular integral equations

A computational scheme of collocation type is proposed for a singular linear integral equation with a power singularity in the regular integral and the justification is given. The results obtained are used to justify the approximate solution of the singular integral equation $$Kx \equiv a(t)x(t) + \frac{{b(t)}}{{\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{x(\tau )d\tau }}{{\tau - t}} + \frac{1}{{2\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{h|t,\tau )x(\tau )}}{{\left| {\tau - t} \right|^\delta }}d\tau = f(t)$$ by a modification of the method of minimal residuals.     

On anisotropic caginalp phase-field type models with singular nonlinear terms

Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.     

The strong relaxation limit of the multidimensional Euler equations

This paper is devoted to the analysis of global smooth solutions to the multidimensional isentropic Euler equations with stiff relaxation. We show that the asymptotic behavior of the global smooth solution is governed by the porous media equation as the relaxation time tends to zero. The results are proved by combining some classical energy estimates with the so-called Shizuta–Kawashima condition.     

Computing the singular behavior of solutions of the cauchy singular integral equations

Stenger's formula for numerical computation of principal value integrals isused to determine the singular behavior of solutions of homogeneous Cauchy singular integral equations near the end-points of the domain of integration.     

On the singular limit for slightly compressible fluids

In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.     

Lp-solutions of singular integro-differential equations

We study a variety of scalar integro-differential equations with singular kernels including linear, nonlinear, and resolvent equations. The first result involves a type of existence theorem which uses a fixed point mapping defined by the integro-differential equation itself and produces a unique solution with a continuous derivative in a very simple way. We then construct a Liapunov functional yielding qualitative properties of solutions. The work answers questions raised by Volterra in 1928, by Levin in 1963, and by Grimmer and Seifert in 1975. Previous results had produced bounded solutions from bounded perturbations. Our results mainly concern integrable solutions from integrable perturbations.     

The Forsythe-Motzkin method for singular linear operator equations

Convergence of the gradient method of Forsythe and Motzkin to a generalized solution of a singular linear operator equation is established.     

On the singular limit of a model transport semigroup

We consider the diffusion limit of a model transport equation on the torus or the whole space, as a scaling parameter ε (the mean free path), tends to zero. We show that, for arbitrary initial data $u_0(x,v)$\nopagenumbers\end , the solution converges in norm topology for each $t>0$\nopagenumbers\end , to the solution of a diffusion equation with initial data \def\d{{\rm d}}$u_D^0(x)=\int u_0(x,v)\,\d v$\nopagenumbers\end . The proof relies on Fourier analysis which diagonalizes the transport operator, a Dunford functional calculus and the analysis of the behaviour of the transport spectrum as ε tends to zero. Copyright © 2000 John Wiley & Sons, Ltd.     

On the scaling limit of a singular integral operator

The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems. Research supported in part by National Science Foundation grants DMS-02-45371 and DMS-04-05519.     

The tree property at the successor of a singular limit of measurable cardinals

Assume \(\lambda \) is a singular limit of \(\eta \) supercompact cardinals, where \(\eta \le \lambda \) is a limit ordinal. We present two methods for arranging the tree property to hold at \(\lambda ^{+}\) while making \(\lambda ^{+}\) the successor of the limit of the first \(\eta \) measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \(\aleph _{\eta ^2+1}\) with the failure of SCH at \(\aleph _{\eta ^2}\). This extends results of Neeman and Sinapova. The second method is also used to get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova.     

The strong relaxation limit of the multidimensional isothermal Euler equations

We construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation. When the relaxation time tends to zero, we show that the density converges towards the solution to the heat equation.