Cauchy Conditions Research Articles

Large‐scale field‐aligned plasma flows

Stationary, large scale plasma flows along the field lines are analyzed using the first three moments of the collisionless Boltzmann equation. The number of integrals of motion which can be found is restricted not only by the cutoff of the hierarchy of the moment equations but also by the forms of the available first‐order partial differential equations which often do not satisfy Cauchy condition of integrability. Only three integrals of motion exist for the plasma continuum described by plasma density, velocity, two components of the pressure tensor and two components of the heat flow tensor. The equation for the field‐aligned current, which is algebraic in j∥, is derived. It expresses the field‐aligned current as function of the perpendicular electric field and perpendicular gradients of the pressure components and of the magnetic field strength determined at the investigated point of the field line.

On the consistency conditions for force systems

In analogy with the classical Cauchy conditions, this work presents conditions so that a force system, the assignment of a force to each subbody of a given body, can be represented by a stress. The setting in which the theory is formulated is more general than that of classical continuum mechanics as stresses can be as irregular as measures, equilibrium is not assumed, it applies to continuum mechanics of order higher than one and it may be extended to the case where the body and space are modelled by general differentiable manifolds. The consistency conditions presented are that of additivity of the force system on pairs of disjoint subbodies, continuity and boundedness.

Elasticity, shear‐mode softening and high‐pressure polymorphism of wüstite (Fe1‐xO)

Elastic wave travel times have been determined as functions of hydrostatic pressure to 3 GPa for five modes of propagation in synthetic single‐crystal wüstite Feo.943O by ultrasonic phase comparison. The measured travel times, corrected for transducer‐bond phase shifts, constrain very accurately the zero‐pressure elastic moduli (GPa) and, for the first time, their first pressure derivatives (dimensionless) as follows: C11∶218.4, dC11/dP∶9.65, C12∶123.0, dC12/dP∶2.77, C44∶45.5, dC44/dP∶−1.03. The zero‐pressure moduli are in good agreement with the results of previous determinations by ultrasonic wave propagation but not with all of the moduli determined by resonance techniques. The variation of bulk modulus with pressure calculated from the Cij (P) is extrapolated to much higher pressures via third‐order Eulerian isotherms and isentropes based on K0S = 154.9 GPa and (dKs/dP)0T = 4.90. The resulting isothermal and shock compression curves satisfactorily reproduce the experimental data to ∼70 GPa, thereby providing a unified description of essentially all data bearing on the compressibility of wüstite. At higher pressures, published shock compression studies provide clear evidence for the existence of a different phase of much greater density and incompressibility. Metallic values of electrical conductivity have been reported for pressures >70 GPa under conditions of shock and high‐temperature static loading. Polyhedral face‐sharing in either the B8(NiAs) or B2(CsCl) (or derivative) structures would result in shorter Fe‐Fe distances, allowing greater 3d orbital overlap conducive to metallic conductivity. However, none of these possibilities satisfactorily accounts for the large inferred increase (14–20%) in zero‐pressure density unless the Fe‐O distance is also reduced by 3–5% by electron delocalization or spin‐pairing. The marked violation of the Cauchy condition associated with the very low value of C44 and its unusual temperature and pressure derivatives are attributable mainly to exchange coupling between nearest and next‐nearest neighbor spins.

Simulation of Biodegradation and Sorption Processes in Ground Water

ABSTRACTA one‐dimensional, finite‐difference model for simulating biodegradation and sorption processes in saturated porous media is developed. The model formulation is general and capable of accommodating a variety of boundary conditions and process theories. Advective and dispersive transport of a substrate and an electron acceptor are considered. Aerobic biodegradation is modeled using a modified Monod function; anaerobic biodegradation is modeled using Michaelis‐Menten kinetics. In addition, first‐order degradation is allowed for both substances. Sorption can be incorporated using linear, Freundlich, or Langmuir equilibrium isotherms for either substance. Dirichlet, Neumann, or Cauchy boundary conditions are allowed, where they are constant in time, or include first‐order degradation. The resulting nonlinear problem is solved using a Newton‐Raphson iterative technique. Model verification is achieved through a series of simulations, two of which arc included. In addition, three applications demonstrate the utility of the model.

Ein abstraktes nichtlineares Cauchy-Kowalewskaja-Theorem mit singulären Koeffizienten II

New qualitative and quantitative properties of abstract nonlinear Cauchy-Kowalewskaja problems with singular coefficients are represented. By applying the method of the Banach space scales the existence and uniqueness of the solution for a Cauchy problem of an operator equation in the scale can be shown. The Cauchy condition is given outside the singularity t=0 . In a subscale there exists the continuous solution up to singularity.

Analytical solutions for radial dispersion with cauchy boundary at injection well

In dealing with solute transport from an injection well into an aquifer a macroscopic boundary condition of the Cauchy type (the third type) can be formulated at the well‐aquifer interface if the mass balance principle is invoked. This Cauchy boundary condition includes effects of radial advection and longitudinal dispersion as molecular diffusion is neglected in the problem. Analytical solutions related to a continuous and a pulse injection are determined for the radial dispersion problem concerning such a Cauchy boundary condition. These solutions give resident concentrations in the radial flow system. Results of the continuous injection solution and of that related to a Dirichlet boundary condition (the first type) are different for short injection periods but converged for long injection periods. The deviation caused by using two different boundary conditions is further studied with appropriate asymptotic solutions valid only for short injection periods. It is found that significant concentration gradients may exist across the well‐aquifer interface for short injection periods. Under this condition, neglect of the local dispersion process at the well‐aquifer interface (as assumed in the Dirichlet boundary condition)amounts to forcing more solute mass into the aquifer. As injection is prolonged, the concentration gradient across the interface decreases as the associated local aquifer concentration is increased to the injected concentration level. For relatively long injection periods a local concentration equilibrium at the well‐aquifer interface is reached and thereafter advection dominates the local transport. The Dirichlet boundary condition is a special case neglecting local dispersion at the well bore to the Cauchy boundary condition. For the radial dispersion equation, solutions regarding a Dirichlet boundary condition do not give the flux concentrations due to the nonconstant groundwater velocity. Nevertheless, solutions of flux concentrations for a continuous and a pulse injection are acquired here by applying the flux concentration transformation to appropriate solutions regarding the Cauchy boundary condition.

Acoustic resonance scattering by viscoelastic objects

In this paper acoustic scattering from a viscoelastic sphere accurately modeled by means of the Kelvin–Voigt model is studied. The approach based on a single impedance-type or Cauchy boundary condition used in the past to account for losses in the body, is reconciled with the exact approach of viscoelasticity, which accounts for material losses via complex field equations, complex propagation vectors, and a set of three realistic boundary conditions on the surface of the sphere. Using the exact approach of viscoelasticity theory, the effect of viscoelastic losses on the various quantities of interest is determined. The Resonance Scattering Theory (RST) is ideally suited to isolate features of the acoustic spectrum which are dependent upon material composition. In order to use the RST for the combination of a viscoelastic object in a liquid medium, an impedance-matched background is required, and it is developed here for the first time. Subtraction of this background successfully isolates the resonances in the present case. Finally, an exact expression for the specific surface impedance of the sphere, which depends in a complicated way on frequency, on mode order, and on the four parameters controlling the viscoelastic properties of the sphere is derived. The effect of all these quantities on the sonar cross section of the sphere, or on the modal contributions contained within it is studied, and many pertinent results are displayed.

Calculation of elasticity and high pressure instabilities in corundum and stishovite with the Potential Induced Breathing Model

Direct calculation of elasticity in the Potential Induced Breathing (PIB) model is developed. This allows calculation of elastic constants and acoustic velocities of oxides and silicates without any experimental data other than the values of fundamental constants, such as Planck's constant. The PIB model allows for spherical charge relaxation in response to the long‐range electrostatic potential. This feature is not present in any other lattice dynamical model, and leads to many‐body forces that break the Cauchy conditions for centrosymmetric crystals (i.e. C44=C12) without the introduction of explicit angle bending forces. The elastic properties are calculated here for corundum and stishovite. High pressure elastic instabilities are found for both corundum (α‐Al2O3) and stishovite (SiO2). Softening of C44 for corundum is calculated to begin at approximately 120 GPa. This may require reconsideration of the ruby fluorescence pressure scale and recent ultra‐high pressure diamond cell experiments. The high pressure instability in stishovite leads to a low pressure dependence for the aggregate shear wave velocity.

Boundary conditions in the modeling of injection mold‐filling of thin cavities

AbstractA governing equation for injection mold‐filling of thin cavities with a power‐law fluid is derived. The interaction between upstream delivery channel flow and cavity flow results in a continuously changing gate condition as the total viscous dissipation of the delivery channel‐cavity assembly is minimized. Depending upon the relative magnitude of pressure drops or viscous dissipation across the channel and the cavity, the boundary conditions which determine the cavity filling process will lie between the following two limiting cases: a Cauchy type gate condition such that the location of the melt front is completely determined by the upstream flow; a Cauchy type melt front condition in which the gate condition is controlled by the downstream flow. For most injection molding cases this may be manifested as equilibration of dissipation density on the melt front. Experimentally observed melt front locations from isothermal, Newtonian filling of a constant gap rectangular cavity and of a bi‐gap rectangular cavity are reported and the validity of the limiting cases are tested.

Identifiability in the large of a linear heat-conduction equation subject to cauchy boundary conditions

The feasibility of simultaneously determining constant values of the specific heat, thermal conductivity, and heat-transfer coefficient from observations of a unique temperature field is studied.

Cauchy Conditions on Symmetrics

Cauchy Conditions on Symmetrics

Cauchy conditions on symmetrics

We call a symmetric d d on a space X X a wC {\text {wC}} symmetric if whenever A ⊆ X A \subseteq X and there exists ε > 0 \varepsilon > 0 such that d ( x , y ) ⩾ ε d(x,y) \geqslant \varepsilon for all x x , y ∈ A y \in A , then A A is relatively discrete. We show that there are no L L -spaces which admit wC {\text {wC}} symmetries. The wC {\text {wC}} notion is extended to certain weaker structures such as F \mathcal {F} -spaces with similar results.

Necessary conditions for optimality of cauchy problems for parabolic partial delay-differential equations

In this paper, we consider the question concerning the necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations indivergence form with Cauchy conditions. All the coefficients of the system are assumed measurable and contain controls and delays in their arguments. An integral maximum principle and its pointwise version for the corresponding controlled system are given.

An approach to rapid plasma shape diagnostics in tokamaks

A study is made of a diagnostic procedure which allows one to estimate the plasma boundary shape and location in a tokamak by using numerical values of the poloidal magnetic flux function ψ in the vicinity of the plasma. In the case of the Impurity Study Experiment (ISX-B) tokamak, the magnetic sensor coils located around the periphery of the vacuum chamber provide information on ∂BR/∂t and ∂Bz/∂t, which can be processed to obtain data on ψ (and ∂ψ/∂n) through the relation BR = - R-1∂ψ/∂Z and Bz = R-1∂ψ/∂R. This leads to a Cauchy boundary condition for the Grad-Shafranov equation Δ*ψ = 0 in the region between the contour of the sensor coil locations and the plasma boundary flux surface. Numerical equilibria calculated for the ISX-B with different shapes are used to simulate ψ data along the coil locations. Random errors are added to the data to test the efficacy of two different approaches: global fitting and local fitting. Reasonably accurate results are obtained by the method of global fitting of the boundary values, which is based on the expansion of ψ in terms of eigenfunctions of Δ*ψ = 0 in a toroidal ring co-ordinate system. This approach is found to permit a relatively large random error in poloidal B field values at the sensor coils and appears to have the potential of rapidly displaying the plasma shape and location.

Existence of optimal controls for systems governed by parabolic partial differential equations with Cauchy boundary conditions

This paper considers the existence of optimal controls for systems governed by a second order parabolic partial differential equation in divergence form with Cauchy conditions. As preliminary results, theorems concerning the convergence of the sequence of weak solutions corresponding to a sequence of admissible controls are proved. Two general forms of criteria are considered. The first one is taken as a function of the weak solution of the system, and the other is taken as a function of the solution of the system and control. Several theorems and corollaries on the existence of optimal controls are then presented.

Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

Theoretical properties of cubic crystals at arbitrary pressure—II. Shear moduli

G eneral expressions are derived for calculating the shear moduli μ and μ' at arbitrary pressure for cubic crystals in which the interatomic interaction energies are modelled by pairwise functions Ø. Under hydrostatic loading, k−2/3μ = μ' + 2P (where k is the bulk modulus and P is the pressure); this relation is, in effect, a generalization of the Cauchy condition ( C 12 = C 44), applicable for such crystals at non-zero pressure. Calculations of the shear moduli are carried out with the aid of a particular family of functions Ø, and a systematic investigation is made of the influence of the crystal structure and the characteristics of the potential upon the relative elasticity at arbitrary pressure. In particular, detailed computations are made (i) for the three cubic Bravais lattices, (ii) for lattices with varying degrees of anharmonicity (as characterized by a “potential range parameter” β, and (iii) over a wide range of all-round stretch λ. For large values of β, the elastic behaviour of the lattice approaches that of a collection of “hard spheres” with nearest neighbour interactions only; there exists a lower limit to the value of β for which the behaviour of the crystal, in a sense, approaches that of a continuum. Calculations are also made of derivatives d N μ(λ)/ dλ N and d N μ'(λ)/ dλ N, at zero pressure, and these “higher order moduli” are used in forming series expansion approximations to the functions μ(λ) and μ'(λ) at non-zero pressure. Finally, comparisons are made among the approximate values of μ(λ) and μ'(λ), determined from successively higher order series expansions (i.e. with the coefficients based on the zero pressure, higher order moduli), and the “exact” values evaluated in the current state.

Optimal feedback control of a class of stochastic systems permitting jumps in the diffusion processes

In this paper we consider certain problems of optimal feedback control of systems governed by a general Ito stochastic differential equation that allows for jumps in tho diffusion process. It will bo shown that these stochastic optimal feedback eontrol problems can be converted into problems of optimal control of deterministic systems described by a parabolic integro-partial-difforential equation. Both tho fixed terminal time and the Markov terminal time problems are converted to optimal control problems of distributed parameter systems with Cauchy condition. However, tho stochastic optimal feedback control problem with Markov terminal time can also bo reduced to the optimal eontrol problem of deterministic systems described by a partial integro-difforcntial equation of parabolic type with the first boundary condition. Further it is interesting to note that tho objective funetionals for all tho reduced problems have precisely the same form.

Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes

Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes

The sodium chloride primary pressure gauge

The failure of a central force model for sodium chloride is discussed. It is noted that it does not closely satisfy the Cauchy conditions at low temperatures, and that it fails the central force requirement of the Love condition. It is also noted that the model as used in the usual way, with B0 as the zero-pressure value of the bulk modulus and the lattice parameter as input, fails to predict the pressure derivative of the bulk modulus. It is therefore concluded that there is no scientific justification for basing a pressure scale on a central force equation of state. The available shock data for sodium chloride and its analysis is examined, and two reasons why the Hugoniot transformation pressure is likely to be less than 231 kbar are discussed. These reasons are (1) the presence of overpressure (the additional pressure beyond the equilibrium transformation pressure needed to cause the transformation under dynamic conditions) and (2) the lack of data points between 212 and 231 kbar. The important (but unjustified) theoretical assumptions made in converting Hugoniot to isothermal data is discussed; it is noted that serious error can enter for very large pressures (P ? B0) for a given material and that at such high pressures the isothermal data should thus be considered only semiquantitative even if the Hugoniot data itself is accurate. An alternate method of estimating the isothermal transformation pressure from the Hugoniot transformation pressure is used. This method is based on the temperature derivative of the transformation pressure. On this basis it is concluded that an upper bound for the isothermal transformation of NaCl (to a CsCl-type structure) at room temperature is 257 kbar; it is noted that the actual value may be considerably less than this.