# Showing papers in "Archive for Rational Mechanics and Analysis in 1976"

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2,329 citations

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Lund University

^{1}TL;DR: In this article, the authors discuss the determination of the figure of the earth and its gravity field from astrogeodetic and gravimetric data (By gravity we mean the resultant of the attractive force of the masses of the Earth, also called gravitation, and the centrifugal force of earth's rotation).

Abstract: The purpose of this paper is to discuss the determination of the figure of the earth and its gravity field from astrogeodetic and gravimetric data (By gravity we mean the resultant of the attractive force of the masses of the earth, also called gravitation, and the centrifugal force of the earth's rotation) Let us recall that at the surface of the earth: (a) astronomic observations allow one to determine the direction of the gravity vector G; (b) gravimetric measurements give the length I GI of the gravity vector; (c) levelling combined with gravimetric measurements gives the differential of the gravity potential W, and thus yields W apart from an additive constant We assume the measured data G and W corrected for, say, the gravitational interaction with the moon, the sun and the planets, for the precession of the earth, and so on, so that we have the following idealized situation:

309 citations

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225 citations

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TL;DR: In this paper, necessary and sufficient conditions for strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid were established in terms of local principal stretches.

Abstract: In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.

215 citations

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TL;DR: In this paper, the existence of the Moore-Penrose inverse for elements of a *-regular ring R is discussed, and a technique is developed for computing conditional and reflexive inverses for matrices in R 2×2, which is then used to calculate the Moore Penrose inverse.

Abstract: The existence of the Moore-Penrose inverse is discussed for elements of a *-regular ring R. A technique is developed for computing conditional and reflexive inverses for matrices in R2×2, which is then used to calculate the Moore-Penrose inverse for these matrices. Several applications are given, generalizing many of the classical results; in particular, we shall emphasize the cases of bordered matrices, Schur complements, block-rank formulae and EP elements.

145 citations

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143 citations

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TL;DR: A detailed analysis of the flow of a Newtonian fluid in the annular region between two infinitely long circular cylinders with parallel axes, resulting from the uniform rotation of one, or both, of the cylinders about their axes, is carried out in this paper.

Abstract: A detailed analysis is carried out of the flow of a Newtonian fluid in the annular region between two infinitely long circular cylinders with parallel axes, resulting from the uniform rotation of one, or both, of the cylinders about their axes. No restriction is placed on the geometry of the system and results are obtained both with the neglect of inertial effects and for the linearized inertial approximation. In both cases, the resultant of the forces exerted by the fluid on the cylinders and the distribution of their normal and tangential components over the cylinders are calculated, and the stream-line patterns are analyzed in some detail. A number of conditions, under which stagnation points, separation points and eddies can exist, are established.

142 citations

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121 citations

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TL;DR: In this article, it was shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.

Abstract: Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain a class of conservation laws associated with linear elastodynamics. These laws represent dynamical generalizations of certain path-independent integrals in elastostatics which have been of considerable recent interest. It is shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.

117 citations

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TL;DR: In this paper, the total potential of unsaturated flows is expressed as the sum of a hydrostatic potential due to capillary suction 0(0) and a gravitational potential −3, 10, 11.

Abstract: Here K(O) is the hydraulic conductivity, ~b is the total potential and t denotes time. If adsorption and chemical, osmotic and thermal effects are neglected, then for unsaturated flows 9 may be expressed as the sum of a hydrostatic potential due to capillary suction 0(0) and a gravitational potential [-3, 10, 11]. Thus if we choose our (x, y, z) coordinate system in such a way that the z-coordinate is vertical, and pointing upwards, we may write

99 citations

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TL;DR: In this paper, SERRIN discusses the existence and regularity of solutions u of the problem (0.1) divv=to(x,u) on t2 (v-Du/1/l+lDul 2) v.r/=cos 0 on ~?t2, where t2 is a bounded domain in IR", n > 2, tO and 0 are given functions on ~ x IR and Qf2 respectively, and r/is the outward unit normal to Of 2.

Abstract: CGmmunicated by J. SERRIN In this paper we discuss the existence and regularity of solutions u of the problem (0.1) divv=to(x,u) on t2 (v-Du/1/l+lDul 2) (0.2) v.r/=cos 0 on ~?t2, where t2 is a bounded domain in IR", n > 2, tO and 0 are given functions on ~ x IR and Qf2 respectively, and r/is the outward unit normal to Of 2. Recently URAL'TSEVA [8] has shown that (0.1), (0.2) has a C2(~)) solution if 0 = constant E(0, n), if tO(x, z) has HiSlder continuous partial derivatives with respect to (x, z)e~ x P,, if (0.3) inf 8tO(x,z) >0,

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TL;DR: In this paper, the effects of lift and Brownian motion in basic parallel flows are considered in order to determine the importance and the consequences of these effects, and the relation of the Brownians motion model involving momentum balance to the diffusive model of particle motions is studied.

Abstract: Constitutive relations for the lift force on the particulate phase and the effect of Brownian motion are presented. These constitutive relations are derived subject to three new principles of constitutive equations. The effects of lift and Brownian motion in basic parallel flows are considered in order to determine the importance and the consequences of these effects. The relation of the Brownian motion model involving momentum balance to the diffusive model of particle motions is studied. Dimensional and scaling arguments are given.

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TL;DR: In this article, a rearrangement inequality for the integral of a product of functions of one variable is further extended to the case of function of several variables, which is known as the Luttinger and Friedberg rearrange inequality.

Abstract: In this paper we prove a rearrangement inequality that generalizes inequalities given in the b∞k by Hardy, Littlewood and Polya1 and by Luttinger and Friedberg.2 The inequality for an integral of a product of functions of one variable is further extended to the case of functions of several variables.

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TL;DR: In this paper, the authors consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t→∞ and prove that similar solutions of such an equation can be represented as solutions of Cauchy problems.

Abstract: In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t→∞. We prove that similar solutions of the equation u
t = (u
λ
)xx asymptotically represent solutions of the Cauchy problem for the full equation u
t = [φ(u)]xx if ⩼φ(u) is “close” to u
λ
for small u.

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TL;DR: The Perkelis differential operator (PDE) as discussed by the authors is defined in terms of positive constants and is used in the study of acoustic wave propagation in a layer of water having sound speedc1 and densityϱ1 which overlays a bottom having soundspeedc2 and density ϱ2.

Abstract: ThePekeris differential operator is defined by
$$Au = - c^2 (x_n )\rho (x_n )
abla \cdot \left( {\frac{1}{{\rho (x_n )}}
abla u} \right),$$
wherex=(x1,x2,...xn)∈Rn,∇=(∂/∂x1, ∂/∂x2,...∂/∂xn), and the functionsc(xn),σ(xn) satisfy
$$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
and
$$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$
wherec1,c2,ϱ1,ϱ2, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc1 and densityϱ1 which overlays a bottom having sound speedc2 and densityϱ2.

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TL;DR: In this paper, the authors considered the linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation and showed that if G satisfies the assumptions, the rest state of the fluid is stable in an appropriate fading memory norm.

Abstract: The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation
$$S + p\delta = 2\int\limits_0^\infty {m(s)(E(t - s} ) - E(t))ds$$
is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as
$$G(s) = \int\limits_\infty ^s {m(\xi )d\xi .}$$
(i)
In this paper it is shown that if G satisfies the assumptions
$$G \in C^2 [0,\infty ),{\text{ }}G(s) \to 0{\text{ as }}s \to \infty,$$
(i)
$$( - 1)^k \frac{{d^k G(s)}}{{ds^k }} > 0,{\text{ }}k = 0,1,$$
(ii)
$$G''(s) \geqq 0,$$
(iii)
then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption
$$ - \int\limits_0^\infty {G'} (s)s^2 ds < \infty$$
(iv)
yields asymptotic stability.

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TL;DR: In this paper, a boundary value problem which yields exact solutions of the Navier-Stokes equations for the flow between two infinite, coaxial, and permeable disks is studied.

Abstract: A boundary value problem which yields exact solutions of the Navier-Stokes equations for the flow between two infinite, coaxial, and permeable disks is studied here. This problem was introduced in [1], and various conclusions were drawn on the basis of approximations to the solution. In what follows the existence, uniqueness, and asymptotic behavior of a solution are rigorously established. These results are of interest in themselves in providing a complete discussion of an exact solution of the Navier-Stokes equations. On the other hand, the flow problem is closely related to the VON K.~RMAN problem of flow between rotating, coaxial disks, and, if suction or injection is allowed on the disks in that problem, the one studied here may be thought of as the special case in which the disks do not rotate. Many questions remain unanswered for the VON KARM,~N problem ([2], [3] give recent work on existence of a solution), and in particular nothing is known about uniqueness. The boundary value problem dealt with here is much simpler than the one which arises from the rotating disk flow problem, but it is of a similar type, and it may be hoped that these results will shed some light on that problem.

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TL;DR: In this paper, the qualitative behavior of mixed problem solutions in the case when a>0 and Ω⊂IRn is bounded is studied. And the authors show that if amax is sufficiently small, depending on Ω, then no overdamping occurs, with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping.

Abstract: The qualitative behavior of solutions of the mixed problem utt = Δu-a(x)ut in IR x Ω, u=0 on IR x ∂Ω, is studied in the case when a>0 and Ω⊂IRn is bounded. Roughly speaking, if a≧amin>0, then solutions decay at least as fast as exp t(ɛ −1/2amin), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If amax is sufficiently small, depending on Ω, then no overdamping occurs.

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TL;DR: In this paper, conditions under which the formally self-adjoint elliptic differential operator in R σ m σ σ = σ 1 σ 2 σ √ σ 3 σ 4 σ 5 σ 6 σ 0.

Abstract: The paper discusses conditions under which the formally self-adjoint elliptic differential operator in R
m
given by 1
$$\tau {\text{ }}u = \sum\limits_{j,{\text{ }}k = 1}^m {[i\partial _j + b_j (x)]} {\text{ }}a_{jk} (x){\text{ }}[i\partial _k + b_k (x)]{\text{ }}u + q(x){\text{ }}u$$
has a unique self-adjoint extension. The novel feature is that the major conditions on the coefficients have to be imposed only in an increasing sequence of shell-like regions surrounding the origin. On the other hand it is shown that if these shells are broken so as to allow a tube extending to infinity in which the conditions on the coefficients are too weak, then, regardless of the coefficients elsewhere, there may not be a unique self-adjoint extension. The mathematical theorems are linked to the quantum-mechanical interpretation of essential self-adjointness (in the case that τ is the Schrodinger operator), that there is a unique self-adjoint extension if the particle cannot escape to infinity in a finite time.

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