Showing papers in "Siam Journal on Mathematical Analysis in 1990"
TL;DR: In this article, a general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory, and an effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix.
Abstract: A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter $\epsilon $ representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as $\epsilon $ tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as $\epsilon $ tends to zero.
595 citations
TL;DR: In this article, the flow of a nonhomogeneous viscous incompressible fluid that is known at an initial time $t = 0$ is considered and described by partial differential equations for the velocity u, the...
Abstract: The flow of a nonhomogeneous viscous incompressible fluid that is known at an initial time $t = 0$ is considered. Such a flow is described by partial differential equations for the velocity u, the ...
407 citations
TL;DR: In this paper, the dynamics and bifurcation theory of equivariant dynamical systems near relative equilibria, that is, group orbits invariant under the flow of an EIF, is discussed.
Abstract: This paper discusses the dynamics and bifurcation theory of equivariant dynamical systemsnear relative equilibria, that is, group orbits invariant under the flow of an equivariant vector field. The theory developed here applies, in particular, to secondary steady-state bifurcations from invariant equilibria. Let $\Gamma $ be a compact group of symmetries of $R^n $ and let $x_0 $ be in $R^n $. Suppose that f is a smooth $\Gamma $-equivariant vector field and $\Sigma $ the isotropy group of $x_0 $. It is shown that there exists a $\Sigma $-equivariant vector field $f_N $, defined on the space normal to X at $x_0 $, and that the local asymptotic dynamics of f are closely related to the local asymptotic dynamics of $f_N $. Next those bifurcations of X are studied which occur when an eigenvalue of $(df_N )_x $ crosses the imaginary axis. Properties of the vector field $f_N $ imply that branches of equilibria and periodic orbits of $f_N $ correspond to trajectories of f which are dense in tori. Field [Equivaria...
211 citations
TL;DR: In this article, a two-dimensional flow of polytropic gas with initial data being constant in each quadrant is considered and it is proved that only 16 combinations of initial data are reasonable.
Abstract: Two-dimensional flow of polytropic gas with initial data being constant in each quadrant is considered. Under the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves, or slip planes, it is proved that only 16 combinations of initial data are reasonable. For each combination, a conjecture on the structure of the solution in the whole space $t > 0$ is given.
210 citations
TL;DR: In this paper, the authors derived asymptotic expansions for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order when their order is purely imaginary and their argument is real and positive.
Abstract: Bessel functions of purely imaginary order are examined. Solutions of both the modified and unmodified Bessel equations are defined which, when their order is purely imaginary and their argument is real and positive, are pairs of real numerically satisfactory functions. Recurrence relations, analytic continuation formulas, power series representations, Wronskian relations, integral representations, behavior at singularities, and asymptotic forms of the zeros are derived for these numerically satisfactory functions. Also, asymptotic expansions in terms of elementary and Airy functions are derived for the Bessel functions when their order is purely imaginary and of large absolute value.Second-order linear ordinary differential equations having a large parameter and a simple pole are then examined, for the case where the exponent of the pole is complex. Asymptotic expansions are derived for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order.
169 citations
TL;DR: In this article, a singularly-perturbed, elliptic boundary value problem is considered in a square and a theory of corner singularities for a $90^ \circ $ angle that takes account of lower-order terms in the equation is presented.
Abstract: A singularly-perturbed, elliptic boundary value problem is considered in a square. A theory of corner singularities for a $90^ \circ $ angle that takes account of lower-order terms in the equation ...
139 citations
TL;DR: In this article, a new approach in the theory of homogenization is carried out on the variational boundary value problem of the stiff type that governs the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid.
Abstract: A new approach in the theory of homogenization is carried out on the variational boundary value problem of the stiff type that governs the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid. A convergence theorem is proved, which gives the behaviour of the solution and points out the role of the connectedness of phases in the mechanics of mixtures.
131 citations
TL;DR: In this paper, the authors studied the number of solutions of the Abel differential equation for planar polynomial vector fields with homogeneous nonlinearities under the assumption that either $A(t)$ or $B(t)) does not change sign for $t \in [0, 1].
Abstract: The number of solutions of the Abel differential equation ${{dx(t)} /{dt}} = A(t)x(t)^3 + B(t)x(t)^2 + C(t)x(t)$ satisfying the condition $x(0) = x(1)$ is studied, under the hypothesis that either $A(t)$ or $B(t)$ does not change sign for $t \in [0,1]$. The main result obtained is that there are either infinitely many or at most three such solutions. This result is also applied to control the maximum number of limit cycles for some planar polynomial vector fields with homogeneous nonlinearities.
129 citations
TL;DR: Transitional shock waves are discontinuous solutions that possess viscous profiles but do not conform to the Lax characteristic criterion; they are sensitive to the precise form of the physical viscosity.
Abstract: A new class of fundamental waves arises in conservation laws that are not strictly hyperbolic These waves serve as transitions between wave groups associated with particular characteristic families Transitional shock waves are discontinuous solutions that possess viscous profiles but do not conform to the Lax characteristic criterion; they are sensitive to the precise form of the physical viscosity Transitional rarefaction waves are rarefaction fans across which the characteristic family changes from faster to slowerThis paper identifies an extensive family of transitional shock waves for conservation laws with quadratic fluxes and arbitrary viscosity matrices; this family comprises all transitional shock waves for a certain class of such quadratic models The paper also establishes, for general systems of two conservation laws, the generic nature of rarefaction curves near an elliptic region, thereby identifying transitional rarefaction waves The use of transitional waves in solving Riemann problems
123 citations
TL;DR: In this paper, the linear heat equation in materials with memory was studied by reducing it to an abstract Volterra equation, and results of regularity, asymptotic behavior, and positivity were given.
Abstract: The linear heat equation in materials with memory is studied by reducing it to an abstract Volterra equation. Results of regularity, asymptotic behavior, and positivity are given.
118 citations
TL;DR: In this article, it was shown that among three-dimensional systems that are competitive or cooperative, those satisfying the generic Kupka-Smale conditions also satisfy the Morse-scale conditions and are therefore s...
Abstract: It is shown that among three-dimensional systems that are competitive or cooperative, those satisfying the generic Kupka–Smale conditions also satisfy the Morse–Smale conditions and are therefore s...
TL;DR: An analogy existing between the concepts of approximate inertial manifolds in dynamical systems theory and multigrid methods in numerical analysis is presented, in view of the large-time approximation of dissipative evolution equations in a turbulent regime.
Abstract: This article presents an analogy existing between the concepts of approximate inertial manifolds in dynamical systems theory and multigrid methods in numerical analysis. In view of the large-time approximation of dissipative evolution equations in a turbulent regime, a new algorithm is proposed and studied that combines some ideas and concepts of inertial manifolds and multigrid methods. This article emphasizes theoretical questions. More practical (computational) questions will be investigated elsewhere.
TL;DR: In this article, the structure of the Reissner-Mindlin plate equations is investigated, emphasizing its dependence on the plate thickness. And the error bounds for the errors in the expansions in Sobolev norms are given.
Abstract: The structure of the solution of the Reissner–Mindlin plate equations is investigated, emphasizing its dependence on the plate thickness. For the transverse displacement, rotation, and shear stress, asymptotic expansions in powers of the plate thickness are developed. These expansions are uniform up to the boundary for the transverse displacement, but for the other variables there is a boundary layer. Rigorous error bounds are given for the errors in the expansions in Sobolev norms. As applications, new regularity results for the solutions and new estimates for the difference between the Reissner–Mindlin solution and the solution to the biharmonic equation are derived. Boundary conditions for a clamped edge are considered for most of the paper, and the very similar case of a hard simply-supported plate is discussed briefly at the end. Other boundary conditions will be treated in a forthcoming paper.
TL;DR: In this paper, a system of reaction-diffusion equations that describes the evolution of population densities of a prey species u and a predator species v inhabiting the same bounded domain is presented.
Abstract: This paper concerns a system of reaction-diffusion equations that describes the evolution of population densities of a prey species u and a predator species v inhabiting the same bounded domain. Un...
TL;DR: Some functional inequalities satisfied by complete elliptic integrals of the first kind are obtained in this paper, and these inequalities generalize the functional identity of Landen, and a related inequality is given for certain quotients of such integrals.
Abstract: Some functional inequalities satisfied by complete elliptic integrals of the first kind are obtained. These inequalities are sharp and generalize the functional identity of Landen. A related inequality is given for certain quotients of such integrals.
TL;DR: In this paper, a general geometric approach for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations is given, which consists of a special normal form called admissible variables, exponential expansion, strong $\lambda $-lemma, and Lyapunov-Schmidt reduction for the Poincare maps under Sil'nikov variables.
Abstract: A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong $\lambda $-lemma, and Lyapunov–Schmidt reduction for the Poincare maps under Sil’nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.
TL;DR: In this paper, the authors studied the plate-bending problem with hard and soft simple support and showed that in the case of hard support, the plate paradox, which is known to occur in the Kirchhoff model, is also present in the threedimensional model and the Reissner-Mindlin model.
Abstract: This paper studies the plate-bending problem with hard and soft simple support. It shows that in the case of hard support, the plate paradox, which is known to occur in the Kirchhoff model, is also present in the three-dimensional model and the Reissner–Mindlin model. The paradox consists of the fact that, on a sequence of convex polygonal domains converging to a circle, the solutions of the corresponding plate-bending problems with a fixed uniform load do not converge to the solution of the limit problem. The paper also shows that the paradox is not present when soft simple support is assumed. Some practical aspects are briefly discussed.
TL;DR: In this paper, the authors considered a class of one-dimensional solidification problems, in which a kinetic undercooling is incorporated into the temperature condition at the interface, and showed that the presence of a kinetic term can prevent finite-time blowup even though supercooling (superheating) exists.
Abstract: This paper considers a class of one-dimensional solidification problems, in which a kinetic undercooling is incorporated into the temperature condition at the interface. A model problem with linear kinetic law is considered. This study indicates that the presence of a kinetic term at the interface can prevent finite-time blowup even though supercooling (superheating) exists. The mathematical erects of the kinetic term are discussed.
TL;DR: In this paper, the stability properties of the traveling front solutions to bistable reaction-diffusion systems in which there are big differences in both the diffusion rates and the reaction rates between two species are studied.
Abstract: The stability properties of the traveling front solutions to bistable reaction-diffusion systems in which there are big differences in both the diffusion rates and the reaction rates between two species are studied. In contrast to the scalar case, this bistable system has multiple existence of traveling waves in the appropriate region of parameters. Each wave can be constructed by using a singular perturbation method, and its stability or instability is determined by a simple algebraic quantity appearing in its construction: namely, the sign of the Jacobian of inner and outer matching conditions. The singular limit approach (which is quite different from formal limiting arguments) adopted in this paper is rigorous and very useful in the study of stability problems of singularly perturbed solutions.
TL;DR: In this paper, Hirsch's results concerning quasi-convergence of strongly monotone semi-low trajectories are derived under weaker assumptions adopted from Matano, which are based on a sequential limit set trichotomy, which follows from the nonordering principle and the limit set dichotomy.
Abstract: Hirsch’s results concerning quasi convergence of almost all trajectories of strongly monotone semiflows are derived under weaker assumptions adopted from Matano. The proofs are based on a sequential limit set trichotomy, which follows from the nonordering principle and the limit set dichotomy. The assumption excluding totally ordered arcs of equilibria, which is required for the set of asymptotically stable points to be dense, is verified for dynamical systems that are analytic on the state space.
TL;DR: In this paper, boundedness and decay results for semilinear parabolic systems of partial differential equations were obtained for m-component systems of the form \[ u_t = D\Delta u + f(u)
Abstract: Boundedness and decay results are obtained for semilinear parabolic systems of partial differential equations. m-component systems of the form \[ u_t = D\Delta u + f(u)\quad {\text{on }}\Omega \tim...
TL;DR: In this article, a linear functional on the space of real polynomials is defined, and the following linear functional is defined: Θ( √ 1/1/1) √ √ k = 1, \cdots,
u ^ *, \varepsilon _k \in \{ - 1,1,1\}
Abstract: Let $l \in \mathbb{N}$, $a_1 < a_2 < \cdots < a_{2l} $, $E_l = \bigcup _{k = 1}^l [a_{2k - 1} ,a_{2k} ]$, $H(x) = \prod _{k = 1}^{2l} (x - a_k )$ and let $\rho _
u (x) = c\prod _{k = 1}^{
u ^ * } (x - w_k )^{
u _k } $ be a real polynomial with $w_k
otin \operatorname{int} (E_l )$ for $k = 1, \cdots ,
u ^ * $ and $
u _k = 1$ if $w_k $ is a boundary point of $E_l $. For given $\rho _
u $ and $\varepsilon = (\varepsilon _1 , \cdots ,\varepsilon _{
u ^ * } )$, $\varepsilon _k \in \{ - 1,1\} $, the following linear functional on $\mathbb{P}$, $\mathbb{P}$ denoting the space of real polynomials, is defined: \[ \begin{gathered} \Psi _{H,\rho _
u ,\varepsilon } (p) = \int_{E_l } {p(x)} \frac{{\sqrt { - H(x)} }}{{\rho _
u (x)}}\operatorname{sgn} \left( { - \mathop \prod \limits_{k = 1}^l \left( {x - a_{2k - 1} } \right)} \right)dx \hfill \\ \qquad \qquad \qquad + \sum_{k = 1}^{
u ^ * } {\left( {1 - \varepsilon _k } \right)} \sum\limits_{j = 1}^{
u _k } {\mu _{j,k} } p^{(j - 1)} \left( {w_k } \right) \...
TL;DR: In this paper, a system of two hyperbolic equations describing the interaction of a complex nucleon field with a real meson field is considered in a domain of $n \leqq 3.
Abstract: A system of two hyperbolic equations describing the interaction of a complex nucleon field with a real meson field is considered in a domain of $\mathbb{R}^n $, $n \leqq 3$. The global finite-dimen...
TL;DR: In this article, the existence of strictly positive solutions for elliptic partial differential equations subject to Dirichlet boundary conditions is discussed. But the authors focus on the situation in which a predator feeds on two-prey species, and use degree theory in cones, positive operators, and sub-and supersolution techniques.
Abstract: This paper discusses the existence of strictly positive solutions (in all three components) of the three-dimensional system of elliptic partial differential equations subject to Dirichlet boundary conditions, and models the situation in which a predator feeds on two-prey species. Results are obtained by the use of degree theory in cones, positive operators, and sub- and supersolution techniques.
TL;DR: In this article, it was shown that the linear functionals of the dual basis are local, hence making it possible to contruct local linear projectors onto the space spanned by its integer translates.
Abstract: Given a multivariate compactly supported function $\phi $, linear projectors to the space $S(\phi )$ spanned by its integer translates are discussed here. These projectors are constructed with the aid of a dual basis for the integer translates of $\phi $, hence under the assumption that these translates are linearly independent. The main result shows that the linear functionals of the dual basis are local, hence making it possible to contruct local linear projectors onto $S(\phi )$. A scheme for the construction of such local projectors is then discussed for a general compactly supported function.In the second part of the paper these observations are applied to piecewise-polynomials and piecewise-exponentials to obtain a necessary and sufficient condition for a quasi interpolant to be a projector. The results of that part extend and refine recent constructions of dual bases and linear projectors for polynomial and exponential box splines.
TL;DR: In this paper, the Enskog equation with large initial data in full physical space and bounded velocities was studied in a box with periodic boundary conditions under the restriction of bounded velocity.
Abstract: This paper is concerned with the Enskog equation with large initial data in $L^1 $, where the high density factor is constant. As a preliminary step, existence and uniqueness is first studied in full physical space and in a box with periodic boundary conditions under the restriction of bounded velocities, by the use of a priori estimates in the norm $\int {(\sup _{0 \leqq t \leqq T} {f(x + tv,v,t)} |)dx\,dv} $. Global existence and uniqueness for small data and unbounded velocities is an easy consequence of this step. The rest of the paper is devoted to the central topic: global existence, regularity, and uniqueness for large initial data in full physical space for the case of unbounded velocities, provided all v-moments are initially finite. Here the more detailed structure of the collision operator is exploited in the a priori estimates.
TL;DR: In this article, the authors provide coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde.
Abstract: The purpose of this paper is to present a first step toward providing coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde. This is achieved by correlating: (i) Numerical results on the bifurcations to temporal chaos with spatial coherence of the damped, periodically forced sine-Gordon equation with periodic boundary conditions; (ii) An interpretation of the spatial and temporal bifurcation structures of this perturbed integrable system with regard to the exact structure of the sine-Gordon phase space; (iii) A model dynamical systems problem, which is itself a perturbed integrable Hamiltonian system, derived from the perturbed sine-Gordon equation by a finite mode Fourier truncation in the nonlinear Schrodinger limit; and (iv) The bifurcations to chaos in the truncated phase space.In particular, a potential source of chaos in both the pde and ...
TL;DR: In this paper, a mathematical formulation for the guided modes of an optical fiber is derived from Maxwell's equations: this formulation leads to an eigenvalue problem for a family of self-adjoint noncompact operators.
Abstract: A mathematical formulation for the guided modes of an optical fiber is derived from Maxwell's equations: this formulation leads to an eigenvalue problem for a family of self-adjoint noncompact operators. The main spectral properties of these operators are established. Then the min-max principle provides an expression of the nonlinear dispersion relation, which connects the propagation constants of guided modes to the frequency. Various existence results are finally proved and a complete description of the dispersion curves (monotonicity, asymptotic behavior, existence of cutoff values) is carried out.
TL;DR: In this article, it was shown that in general such expansions can be obtained directly from an eigenfunction of a singular second-order differential operator, and with additional assumptions they converge in some interval.
Abstract: In a recent work, Frenzen and Wong [Canad. J. Math., 37 (1985), pp. 979–1007] have obtained a uniform asymptotic expansion for the Jacobi polynomials in terms of Bessel functions. An analogous expansion for the Jacobi functions had been given earlier by Stanton and Tomas [Acta Math., 140 (1978), pp. 251–276]. The common starting point of these papers is an integral representation.In this paper it is shown that in general such expansions can be obtained directly from an eigenfunction of a singular second-order differential operator, and with additional assumptions they converge in some interval. This leads to an expansion for the eigenfunction of an integral representation of Mehler type with good information on the kernel.
TL;DR: In this article, the stability of planar wave solutions to a one-step combustion process was studied from the point of view of matched asymptotics in the limit of infinite activation energy.
Abstract: A system of reaction diffusion equations which arise as a model for a one-step combustion process is considered. The primary concern is with the stability of planar wave solutions to this model. This problem has been studied extensively from the point of view of matched asymptotics in the limit of infinite activation energy. The asymptotic analysis has demonstrated that for a large range of parameters, the planar wave solution is unstable. As a particular parameter is varied, the planar wave solution may undergo either a Hopf or steady state bifurcation. This paper gives a rigorous mathematical justification of some of the asymptotic results.