The Model Plasma Problem: Existence of a Solution Branch

We are interested in the model plasma problem –u = u⁺in ,u = –d on , _au⁺ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.     

Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition

Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R², there exists a curve suchthat u(x)=—K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=²+ ?(—³) + 2u that have a transition layer behaviour (from=—1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

Asymptotic Solution of Hyperbolic Systems with Constant Coefficients

An integral representation of the exact solution of the initialvalue problem for the hyperbolic equation of the form is derived. Here A^o, A^v, B, and Care constant m x m matrices, u(t, X; ) is an m-component columnvector, and is a positive parameter. Various conditions areimposed on the coefficient matrices that permit the applicationof the method of stationary phase in several variables to theintegral representation of the exact solution. The leading termof the asymptotic expansion as of the exact solution is obtainedfor several types of initial data and source functions whichdepend on the parameter .     

The Structure of the Solution Set for Periodic Oscillations in a Suspension Bridge Model

Numerical results are reported for the computation of periodicsolution paths for a suspension bridge model represented bythe equation. u_n + EIu_xxxx + u_t, + Ku⁺ = W(^x) + sin x sin µt. with hinged-end boundary conditions, as the forcing amplitude and frequency µ are varied. The term Ku⁺ models the factthat there is restoring force due to the cables only when theyare being stretched. It is found that an S-shaped curve is obtainedwhen the displacement amplitudes are plotted against the forcingamplitudes for some frequency regimes. As a two-parameter problem,it appears that the solution set resembles a cusp-like surfacewith the singular point near linear resonance. While the effectof strengthening the cable (i.e. increasing K) will enhancethe occurrence of the multiple solutions, the effect of increasingthe damping coefficient gives the opposite effect.     

An analogue of Berry's conjecture for the phase in fractal obstacle scattering

In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRⁿ with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant C_n, dependingonlyon n and such that where     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption

The autonomous differential equations for the temperature andreactant consumption in a first-order well-stirred exothermicreaction are considered. An examination of the phase-plane solutionsallows the qualitative behaviour of the Semenov number as afunction of maximum temperature rise ^* to be established. Inthe limit of infinite adiabatic temperature rise (B) and zeroactivation energy parameter ( = 0), the relationship between and stationary temperature _s is known to be e_¹ = _s. Criticalityarises at the maximum of (_s) and leads to the critical Semenovvalues (_s)_cr = 1, _cr = e^–1. For sufficiently large B,it is shown that the (^*) curve has a bifurcation at ^* = 1, withthe upper branch monotonically increasing and the lower branchmonotonically decreasing for ^* > 1. In the limit B thesebecome respectively the straight line = e^–1, _s 1 andthe unstable branch of = _se^–1, _s 1 and the unstablebranch of = _s e^–. Criticality for finite B is definedas occurring at the bifurcation, namely ^*_cr = 1, with _cr(B)the value of at this point. Values of these Semenoy numbersare obtainable from the numerical calculations of Boddingtonet al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The newcriterion is applied to an approximate phase-plane solution.The corresponding critical parameter is found to be _cr = e^–1[1+B^–(2–e^–1)+O(B^–1)].     

Numerical Evaluation of Fourier Integrals

A method is developed for evaluating Fourier integrals of theform A() = ¹_–1f(x) e^fax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJ_r+(), r= 0, 1, 2,.... A partialsum A_N() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of A_N() for all , and alsothe truncation error A()–A_N() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.     

The Decay of Eddy-currents in Thin Sheets and Related Water-wave Problems

The decay of the eddy-currents that are induced in a thin, uniform,imperfectly-conducting sheet by switching off the source ofan external magnetic field is investigated. For the two-dimensionalproblem of an infinite strip the (non-dimensional) decay constants_n and eddy-current distributions i_n(x) are the eigenvalues andeigenfunctions of the integral equation with the constraint. For the circular disc the corresponding equation is where and K and E are complete elliptic integrals. For both problemsthe initial eddy-currents have inverse-square-root singularitiesat the edges but during their decay the eddy currents are finiteat the edges and the normal magnetic fields have logarithmicsingularities there. Numerical results are given for variousinitial-value problems. The eddy current problems are closely related to water-waveproblems in which there is a strip-shaped or circular aperturein a horizontal rigid dock. If _n and _n are the decay constantsand magnetic scalar potentials for the strip and _n and _n theangular frequencies and velocity potentials for the normal modesin the strip-shaped aperture, then _n =_n² and _n and _n are thereal and imaginary parts respectively of a holomorphic function.The velocities in the normal modes are deduced from the solutionof the eddy-current problem and are found to agree with resultsgiven in Miles (1972). For circular geometries the eigenvaluesand eigenfunctions of the axisymmetric eddy-current problemare the same as those of the water-wave problem that has angularvariation eⁱ; where (, , z) are cylindrical polar co-ordinateslocated at the centre of the basin.     

Experimental Observation of the Large-Amplitude Solutions of Duffing's and Related Equations

Experiments with a nonlinear electronic model show that certainsimple features of the solutions of where f(u) is an odd monotonic function of u for example u³,repeat in a regular pattern as either is decreased or U isincreased. For fixed U, the position of these features is periodicin 1/ and, when f(u) has the form u|u|^k–1 a quantitativerelation between the period in 1/ and U can be found. The occurrenceof large-amplitude chaotic solutions is found to depend notonly on the nonlinearity of f(u) for large U but also on itsbehaviour near u = 0. For the Duffing equation, which can bereduced to the range of parameters accessible to experiment is 0<1 and0<F5000.     

Finite-Element Approximation of a Plasma Equilibrium Problem

The plasma problem studied is: given R⁺ find (, d, u) R ?R ? H¹() such that Let ₁ < ₂ be the first two eigenvalues of the associatedlinear eigenvalue problem: find $$\left(\lambda ,\phi \right)\in\mathrm{R;}\times {\hbox{ H }}_{0}^{1}\left(\Omega \right)$$such that For ₀(0,₂) it is well known that there exists a unique solution(₀, d₀, u₀) to the above problem. We show that the standard continuous piecewise linear Galerkinfinite-element approximatinon $$\left({\lambda }_{0},{\hbox{d }}_{0}^{k},{u}_{0}^{h}\right)$$, for ₀(0,₂), converges atthe optimal rate in the H¹, L², and L norms as h, the mesh length,tends to 0. In addition, we show that dist (, ^h)Ch² ln 1/h,where $${\Gamma }^{\left(h\right)}=\left\{x\in \Omega :{u}_{0}^{\left(h\right)}\left(x\right)=0\right\}$$.Finally we consider a more practical approximation involvingnumerical integration.     

On a Functional Differential Equation

This paper considers some analytical and numerical aspects ofthe problem defined by an equation or systems of equations ofthe type (d/dt)y(t) = ay(t)+by(t), with a given initial conditiony(0) = 1. Series, integral representations and asymptotic expansions fory are obtained and discussed for various ranges of the parametersa, b and (> 0), and for all positive values of the argumentt. A perturbation solution is constructed for |1–| <<1, and confirmed by direct computation. For > 1 the solutionis not unique, and an analysis is included of the eigensolutionsfor which y(0) = 0. Two numerical methods are analysed and illustrated. The first,using finite differences, is applicable for < 1, and twotechniques are demonstrated for accelerating the convergenceof the finite-difference solution towards the true solution.The second, an adaptation of the Lanczos method, is applicablefor any > 0, though an error analysis is available onlyfor < 1. Numerical evidence suggests that for > 1 themethod still gives good approximations to some solution of theproblem.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

The Critical Conditions in a Reactive Slab with Slowly Varying Surface Temperature

Liouville's non-linear partial differential equation is consideredfor an infinite rectangular strip domain with a slowly varyingboundary condition. The equation describes a layer of chemicallyreactive material under conditions where the resistance to surfaceheat transfer is negligible and the ambient temperature variesslowly along the surface. Symmetrical heating by a zero orderexothermic reaction is assumed. If is a small dimensionlesstemperature difference between regions where the surface temperatureis effectively constant, a perturbation series solution in may be determined provided the Frank-Kamenetskii parameter satisfies _c(). It is shown that a plausible value for thecritical parameter is _c() = _c(0) e^–e,where _c(0) = 0.878.The corresponding critical temperature distribution is shownto have a dependence on different from that for subcriticalcases.     

Infinitely many turning points for an elliptic problem with a singular non-linearity

We consider the problem – u = |x|/(1 – u)^p inB, u = 0 on B, 0 < u < 1 in B, where 0, p 1 and Bis the unit ball in ^N (N 2). We show that there exists a _*> 0 such that for < _*, the minimizer is the only positiveradial solution. Furthermore, if , then the branch of positive radial solutions must undergoinfinitely many turning points as the maximums of the radialsolutions on the branch converge to 1. This solves ConjectureB in [N. Ghoussoub and Y. Gun, SIAM J. Math. Anal. 38 (2007)1423–1449]. The key ingredient is the use of monotonicityformula.     

Blow up of solutions of a generalized Boussinesq equation

Consider the Cauchy problem u_tt = (f(u))_xx + u_xxtt x R, t 0, u(x,0) = u₀(x), u_t(x,0) = u₁(x),7rcub; where f : R R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let > 0 be real, such that 2(l + 2)F(u) uf(u), (v₀, Pv₀)_l² + _- F(u₀)dx < 0 where P = 1 - d²/dx², F'(s), and v₀ is given by u₁(x,0) = (v₀(x))_x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.     

A Phase-Change Model with a Zone of Coexistence of Phases

A mathematical model for change of phase is presented, accountingfor the presence of regions in which liquid and solid coexist.The basic variables are temperature and solid fraction v. Westart from a relationship of the type =(v), supposed valid inthermodynamical equilibrium. Then for dynamical processes weintroduce a perturbation causing v to be less than its equilibriumvalue in any solidification process. This solid fraction deficiencyis governed by an ordinary differential equation containing_t, in the forcing term. The heat-balance equation is in turncoupled to the ordinary differential equation through the termv_t, ( is latent heat). Some existence and uniqueness resultsare proved and some monotonicity properties are described forpure melting or pure solidification processes.     

Precise Spectral Asymptotics for Logistic Equations of Population Dynamics

The semilinear elliptic eigenvalue problem with superlinearpure power nonlinearity is considered. This problem is treatedfrom the standpoint of L²-theory and the precise asymptoticformula for the eigenvalue parameter = () as is established,where is the L²-norm of the solution u associated with . 2000Mathematics Subject Classification 35P30 (primary), 35J60 (secondary).     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((₀, ₀), 0, P₀) in the solution-parameter space of ((₀, ₀),0, P₀) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((₀, ₀), 0, P₀) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P₀ and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((₀, ₀), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((₀, ₀),0).