Showing papers on "Domain (mathematical analysis) published in 1983"

Some mathematical questions related to the MHD equations

Abstract: Some questions relating to the large time behavior of the solutions of MHD equations for a viscous incompressible resistive fluid are investigated. The physical system is briefly described and the functional setting of the equations, a flow in a bounded domain or in whole space with a space periodicity property in all directions. The main existence and uniqueness results for weak and strong solutions of the MHD equations are recalled. Regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular. The squeezing property of the trajectories are stated and it is shown that any functional invariant set for the MHD equations, and in particular any attractor, has a finite Haussdorf dimension. The flow is found to be totally determined for large dimensions by a finite number of parameters. 26 references.

Necessary conditions for subellipticity of the $\overline\partial$-Neumann problem

Abstract: Let a be a a-closed form of type (p, q) with L2-coefficients on a smoothly bounded domain Q in C'. The problem of determining both the existence and regularity properties of the solution u of au = a, where u is orthogonal to the null space of a on (p, q l)-forms, is known as the a-Neumann problem. One of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates. Let U be a neighborhood of a point z0 in the boundary of Q. A subelliptic estimate is said to hold in U if the estimate

Everywhere-regularity for some quasilinear systems with a lack of ellipticity

Abstract: It is shown that the derivatives of the solutions of certain quasilinear degenerate elliptic systems are Holder-continuous, everywhere, in the interior of the domain. This work generalizes a result of K. Uhlenbeck [34].

Group explicit methods for parabolic equations

Abstract: In this paper, new explicit methods for the finite difference solution of a parabolic partial differential equation are derived. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.

Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain

Abstract: The problem of minimizing a concave function subject to linear inequality constraints may have many local solutions. Therefore, finding the global constrained minimum is a computationally difficult problem. A computational method is described which finds the global minimum of a smooth concave function over a polyhedron in Rn. The feasible domain is partitioned into a rectangular domain, which can be excluded from further consideration, and r ≤ 2n subdomains, at least one of which contains the global minimum. A known algorithm can be applied sequentially (or in parallel) to each of these r subdomains to compute the global minimum. A method is also presented (Appendix B) for the construction of nontrivial test problems for which the global minimum point is known. Given an arbitrary polyhedron and a selected vertex, it is shown how to determine a concave quadratic function (generally with many local minima) with its global minimum at the selected vertex.

Existence of Solutions to the Nonhomogeneous Steady Navier-Stokes Equations.

Abstract: : This paper concerns the existence of steady solutions to the Navier-Stokes equations in a bounded domain. The condition of solenoidality for the velocity field imposes a necessary condition on the boundary data. For a certain class of symmetrical domains, the authors show that this necessary condition implies the existence of a solution to the problem. The method consists of proving a priori bounds on solutions by assuming the contrary, rescaling the equations, and then arriving at a solution to the steady Euler equations in the limit. Examination of this equation leads to the desired contradiction. After one has suitable bounds on any solutions, one uses the Leray-Schauder theorem to prove existence. In addition, the authors remark on the problem of a general bounded domain, and suggest how certain maximum principles might yield the expected results.

Applications of variational-inequalities to the existence theorem on quadrature domains

Abstract: In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other. Introduction. In a previous paper [7], the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory. Let v be a finite positive measure on the two-dimensional Euclidean space R2. Let SL1(Q) be the class of subharmonic functions in an open set Q which are integrable with respect to the two-dimensional Lebesgue measure m. A nonempty open set Q is called a quadrature domain of v for class SL1 if (Qi) v is concentrated in Q, namely, v(QC) = 0, where Qc denotes the complement of Q, (Qii) Jus s dv 1 a.e. in a connected open set W with finite area, f = 0 a.e. in Wc and Jf dm > m(W), then Q(fm, SL1) #0 and there exists a minimum domain W in Q(fm, SL1), namely, Q E Q(Jm, SL1) if and only if W C Q and m(Q \ W) = O. Received by the editors October 5, 1981 and, in revised form, February 23, 1982. 1980 Mathematics Subject Classification. Primary 3 1AOS, 3 1BOS; Secondary 35A15.

On Polynomial Approximation in Sobolev Spaces

Abstract: We obtain new bounds for the error in the polynomial approximation in Sobolev spaces in an open bounded set which is star-shaped with respect to every point in a set of positive measure $B \subset \Omega $. This estimate follows by applying the Hardy–Littlewood maximal function.

Cauchy Flux and Sets of Finite Perimeter

Abstract: : A Cauchy flux Q is a real-valued, additive, area-bounded function whose domain is the class of all Borel subsets of the reduced boundary of sets of finite perimeter. If the flux Q is also volume bounded, it is shown that Q can be represented as the integral of the normal component of some vector field. (Author)

Minimum domains for spatial patterns in a class of reaction diffusion equations.

Abstract: We study a general class of scalar reaction/interacting population diffusion equations in two space dimensions: convective terms, due to wind, are included. We consider boundary conditions which include a measure of the hostility to the species in the exterior of the domain. The main point of the paper is to obtain estimates for the minimum domain size which can sustain spatially heterogeneous structures and indicate the type of patterns which could appear.

An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains

Abstract: A numerical method, based on the integral equation formulation of Symm, is described for computing approximations to the mapping functions which accomplish the following conformal maps: (a) the mapping of a domain interior to a closed Jordan curve onto the interior of the unit disc, (b) the mapping of a domain exterior to a closed Jordan curve onto the exterior of the unit disc, (c) the mapping of a doubly-connected domain bounded by two closed Jordan curves onto a circular annulus. The numerical method is based on approximating the unknown source density by cubic splines and "singular" functions, and is particularly suited for the mapping of difficult domains having sharp corners.

On locating all zeros of an analytic function within a bounded domain by a revised Delves-Lyness method

Abstract: The method of Delves and Lyness [Math. Comp., 21 (1967), pp. 543–560; 561–577] on locating all the zeros of an analytic function in a bounded domain is revised such that no division of the region is needed. The method used here is the homotopy continuation method. Also discussed here is how the continuation curves can be followed efficiently in this particular setting.

On boundary integral equations of the first kind for the bi-laplacian in a polygonal plane domain

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On the Calculation of the Exact Number of Zeroes of a Set of Equations

Abstract: The number of simple zeroes common to a set of nonlinear equations is calculated exactly and analytically in terms of an integral taken over the boundary of the domain of interest. The integrand consists only of simple algebraic quantities containing the functions involved as well as their derivatives up to second order. The numerical feasibility is shown by some computed examples.

On the existence of hydrodynamic motion in a domain with free boundary type conditions

Abstract: In this work we prove existence theorems for generalized solutions of the Navier-Stokes equations for a newtonian incompressible fluid in a either bounded or exterior domain with part of the boundary free. In the case when the domain is bounded and the external forces are periodic in time we prove the existence of periodic solutions.

Topics in Energy and Potential Theory

Abstract: Energy is a frustratingly delicate item in modern Markov process theory. It is a subject which commands attention, since it is linked so closely with maximum principles and Hunt’s hypothesis (H). It entered potential theory in the work of Cartan and Deny, where it enabled them to prove various delicate principles about symmetric potential kernels. It has flourished in the modern theory of Dirichlet spaces and has added to the body of knowledge concerning symmetric Markov processes. But while energy is a natural and cooperative partner in the study of symmetric potential kernels, it becomes increasingly intractable as one attempts to study more asymmetric kernels and processes. Concrete results in this domain are few. We present some topics in energy and potential theory for Markov processes with nonsymmetric potential kernels which complement several results and articles by various authors.

Estimation Techniques for Transport Equations.

Abstract: We present convergence arguments for algorithms developed to estimate spatially and/or time dependent coefficients and boundary parameters in general transport (diffusion, advection, sink/source) models in a bounded domain Ω ⊂ R2. A brief summary of numerical results obtained using the algorithms is given.

Accuracy of finite-difference methods in recirculating flows

Abstract: The accuracy of six numerical approximations of the convection terms in the conservation equations is examined for a steady, recirculating flow Quadratic upwind, central, nine-point, third-order, and power-law approximations are tested as alternatives to the widely used upwind I central hybrid method Forced flow in a heated cavity is chosen as a reasonably severe test problem An exact analytical solution is used to evaluate truncation errors and solution errors Expressions for the leading truncated terms, including velocity derivatives, provide insight into why errors in the convection terms dominate errors in the diffusion terms for high grid Peclet numbers If an average solution error of less than 10% is desired, higher order methods are clearly superior to the first-order upwind/hybrid method One must have at least one finite domain within a wall gradient layer to reduce flux errors to 10% with the second-order central-difference method, whereas one must have at least two finite domains

Limit states for modified Navier-Stokes equations in three-dimensional space

Abstract: One gives the description of the limit set (when t→∞) for all, the trajectories (solutions) of the system where satisfying the boundary condition at the boundary of the bounded domain Ω and emanating at t=0 from the points of the sphere . In particular, it is proved that for R greater than some Ro, the semigroup Vt,t⩾0, which corresponds to this problem can be extended to a group Vt,teR1, possessing a series of interesting properties.

Some singularities in the behavior of solutions of equations of minimal-surface type in unbounded domains

Abstract: In this paper the behavior of the solutions of equations of minimal-surface type is studied in unbounded domains. It is established that if the domain is sufficiently narrow in the neighborhood of the point at infinity of R2, then any solution having zero Dirichlet or Neumann data on the boundary must be identically constant. A condition on the narrowness of the domain is found under which the solution cannot change sign in the domain. An estimate of the form ?k i(ak) ? c is proved, where i(ak) is the topological index of the solution at the point ak, c is a constant depending only on the equation, the domain and the number of points of local extremum of the boundary function, and the summation is taken over all critical points of the solution. Bibliography: 11 titles.

A stream function-vorticity finite element formulation for Navier-Stokes equations in multi-connected domain

Abstract: In this paper we present a stream function-vorticity finite element formulation for Navier-Stokes equations which does not require an iterative procedure for satisfying the boundary conditions, and show that this formulation is of great advantage in solving the flow problems in a multi-connected domain.

An approach to nonlinear elliptic boundary value problems

Abstract: Let D ⊂ Rn be a bounded domain and L: dom L ⊂ L2{D) → L2(D) be a self-adjoint operator of finite dimensional kernel. Let f: D × R → R be a function satisfying the Caratheodory condition. Assume that there are constants λ > 0 and δ ∊ [0,1) such that Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there exists u ∊ L2(D) such that L(u)(x) = f(x, u(x)) for ∀x ∊ D. This result is then used to prove the existence of weak solutions of nonlinear elliptic boundary value problems. Other abstract results applicable to ordinary and partial differential equations have also been proved.

The Equatorial Waves of Balanced Models

Abstract: An analysis is made of the linear waves of the Balance Equations and the global Balance Equations on an equatorial β-plane. We consider both finite and infinite meridional domains and show the effect of different choices of boundary conditions in a finite domain. The infinite domain is similar to a complete spherical domain, a problem studied by Moura. We find analogies to several of his results: for example, the Balance Equations have no eastward traveling waves, whereas the global Balance Equations do. We also make an extensive study of the long-wave limit, which is relevant for ocean domains whose width greatly exceeds the Rossby radius of deformation. This limit is singular for many of the wave solutions. In general, however, the balanced models provide reasonably good approximations to the low-frequency waves of the primitive equations. The global Balance Equations do have high-frequency waves, but they are very different from those of the primitive equations.

Qualitative Behavior and Bounds in a Nonlinear Plasma Problem

Abstract: Bounds for the boundary values of the variational solutions of a plasma problem are constructed, estimates for the distance between the boundary of the domain and the region filled with plasma are given and an isoperimetric inequality for the area of this region is derived.