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Communications on Pure & Applied Analysis

March 2002 , Volume 1 , Issue 1

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A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation
Hector D. Ceniceros
2002, 1(1): 1-18 doi: 10.3934/cpaa.2002.1.1 +[Abstract](804) +[PDF](410.7KB)
Abstract:
An efficient adaptive moving mesh method for investigation of the semi-classical limit of the focusing nonlinear schrödinger equation is presented. The method employs a dynamic mesh to resolve the sea of solitons observed for small dispersion parameters. A second order semi-implicit discretization is used in conjunction with a dynamic mesh generator to achieve a cost-efficient, accurate, and stable adaptive scheme. This method is used to investigate with highly resolved numerics the solution's behavior for small dispersion parameters. Convincing evidence is presented of striking regular space-time patterns for both analytic and non-analytic inital data.
An application of homogenization techniques to population dynamics models
B. E. Ainseba, W. E. Fitzgibbon, M. Langlais and J. J. Morgan
2002, 1(1): 19-33 doi: 10.3934/cpaa.2002.1.19 +[Abstract](556) +[PDF](195.1KB)
Abstract:
We are interested in partial differential equations and systems of partial differential equations arising in some population dynamics models, for populations living in heterogeneous spatial domains. Discontinuities appear in the coefficients of divergence form operators and in reaction terms as well. Global posedness results are given. For models offering a great a degree of heterogeneity we derive simpler models with constant coefficients by applying homogenization method. Long term behavior is then analyzed.
Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains
Daniel Coutand, J. Peirce and Steve Shkoller
2002, 1(1): 35-50 doi: 10.3934/cpaa.2002.1.35 +[Abstract](658) +[PDF](201.5KB)
Abstract:
In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains. The LANS-$\alpha$ equations are able to accurately reproduce the large-scale motion (at scales larger than $\alpha >0$) of the Navier-Stokes equations while filtering or averaging over the motion of the fluid at scales smaller than α, an a priori fixed spatial scale.
We prove the global well-posedness of weak $H^1$ solutions for the case of no-slip boundary conditions in three dimensions, generalizing the periodic-box results of [8]. We make use of the new formulation of the LANS-$\alpha$ equations on bounded domains given in [20] and [14], which reveals the additional boundary conditions necessary to obtain well-posedness. The uniform estimates yield global attractors; the bound for the dimension of the global attractor in 3D exactly follows the periodic box case of [8]. In 2D, our bound is $\alpha$-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.
Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits
K. T. Joseph and Philippe G. LeFloch
2002, 1(1): 51-76 doi: 10.3934/cpaa.2002.1.51 +[Abstract](725) +[PDF](247.3KB)
Abstract:
This paper is concerned with the boundary layers that arise in solutions of a nonlinear hyperbolic system of conservation laws in presence of vanishing diffusion. We consider self-similar solutions of the Riemann problem in a half-space, following a pioneering idea by Dafermos for the standard Riemann problem. The system is strictly hyperbolic but no assumption of genuine nonlinearity is made; moreover, the boundary is possibly characteristic, that is, the wave speed do not have a specific sign near the (stationary) boundary.
First, we generalize a technique due to Tzavaras and show that the boundary Riemann problem with diffusion admits a family of continuous solutions that remain uniformly bounded in the total variation norm. Careful estimates are necessary to cope with waves that collapse at the boundary and generate the boundary layer.
Second, we prove the convergence of these continuous solutions toward weak solutions of the Riemann problem when the diffusion parameter approaches zero. Following Dubois and LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible boundary traces. Following Part I of this work, we identify and rigorously analyze the boundary set associated with the zero-diffusion method. In particular, our analysis fully justifies the use of the scaling $1/\varepsilon$ near the boundary (where $\varepsilon$ is the diffusion parameter), even in the characteristic case as advocated in Part I by the authors.
Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics
Congming Li and Eric S. Wright
2002, 1(1): 77-84 doi: 10.3934/cpaa.2002.1.77 +[Abstract](617) +[PDF](138.3KB)
Abstract:
The carbonate system is an important reaction system in natural waters because it plays the role of a buffer, regulating the pH of the water. We present a global existence result for a system of partial differential equations that can be used to model the combined dynamics of diffusion, advection, and the reaction kinetics of the carbonate system.
Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems
Y. Kabeya, Eiji Yanagida and Shoji Yotsutani
2002, 1(1): 85-102 doi: 10.3934/cpaa.2002.1.85 +[Abstract](545) +[PDF](199.3KB)
Abstract:
We propose a method to investigate the structure of positive radial solutions to semilinear elliptic problems with various boundary conditions. It is already shown that the boundary value problems can be reduced to a canonical form by a suitable change of variables. We show structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
A new approach to study the Vlasov-Maxwell system
Sergiu Klainerman and Gigliola Staffilani
2002, 1(1): 103-125 doi: 10.3934/cpaa.2002.1.103 +[Abstract](648) +[PDF](237.7KB)
Abstract:
We give a new proof based on Fourier Transform of the classical Glassey and Strauss [6] global existence result for the 3D relativistic Vlasov-Maxwell system, under the assumption of compactly supported particle densities. Though our proof is not substantially shorter than that of [6], we believe it adds a new perspective to the problem. In particular the proof is based on three main observations, see Facts 1-3 following the statement of Theorem 1.4, which are of independent interest.
Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields
H. M. Yin
2002, 1(1): 127-134 doi: 10.3934/cpaa.2002.1.127 +[Abstract](674) +[PDF](127.5KB)
Abstract:
In this paper we prove a fundamental estimate for the weak solution of a degenerate elliptic system: $\nabla\times [\rho(x)\nabla\times H]=F$, $\nabla\cdot H=0$ in a bounded domain in $R^3$, where $\rho(x)$ is only assumed to be in $L^{\infty}$ with a positive lower bound. This system is the steady-state of Maxwell’s system for the evolution of a magnetic field $H$ under the influence of an external force $F$, where $\rho(x)$ represents the resistivity of the conductive material. By using Campanato type of techniques, we show that the weak solution to the system is Hölder continuous, which is optimal under the assumption. This result solves the regularity problem for the system under the minimum assumption on the coefficient. Some applications arising in inductive heating are presented.

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