Showing papers in "Quarterly of Applied Mathematics in 2009"

The Euler-Poincaré theory of metamorphosis

Abstract: In the pattern matching approach to imaging science, the process of ``metamorphosis'' is template matching with dynamical templates. Here, we recast the metamorphosis equations of into the Euler-Poincare variational framework of and show that the metamorphosis equations contain the equations for a perfect complex fluid \cite{Ho2002}. This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.

Multiplicative finite difference methods

Abstract: Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.

A nonlocal quenching problem arising in a micro-electro mechanical system

Abstract: In this paper, we study a nonlocal parabolic problem arising in the study of a micro-electro mechanical system. The nonlocal nonlinearity involved is related to an integral over the spatial domain. We first give the structure of stationary solutions. Then we derive the convergence of a global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.

Bounded solutions for the Boltzmann equation

Abstract: In either a periodic box T d or R d (1 ≤ d ≤ 3), we establish a unified L ∞ estimate for solutions near Maxwellians for the Boltzmann equation, in terms of natural mass, momentum, energy conservation and the entropy inequality.

Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations

Abstract: Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schrodinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.

Stability of random-switching systems of differential equations

Abstract: This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.

On the stability of Mindlin–Timoshenko plates

Abstract: We consider a Mindlin-Timoshenko model with frictional dissipations acting on the equations for the rotation angles. We prove that this system is not exponentially stable independent of any relations between the constants of the system, which is different from the analogous one-dimensional case. Moreover, we show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data.

On the particle paths in solitary water waves

Abstract: We provide the qualitative flow pattern beneath a solitary water wave by describing the individual particle trajectories.

Modelling supply networks with partial differential equations

Abstract: A continuum-discrete model for supply networks is introduced. The model consists of a system of conservation laws: a conservation law for the goods density and an evolution equation for the processing rate. The network is formed by subchains and nodes at which, motivated by real cases, two routing algorithms are considered: the first maximizes fluxes taking into account the goods' final destinations, while the second maximizes fluxes without constraints. We analyze waves produced at nodes and equilibria for both algorithms, relating the latter to production rates in real supply networks. In particular, we show how the model can reproduce the well-known Bullwhip effect.

Kinematics of a shock wave of arbitrary strength in a non-ideal gas

Abstract: Singular surface theory is used to study the evolutionary behaviour of an unsteady three-dimensional motion of a shock wave of arbitrary strength propagating through a non-ideal gas. The dynamical coupling between the shock front and the induced discontinuities behind it is investigated by considering an infinite system of transport equations governing the strength of a shock wave and the induced discontinuities behind it. This infinite system, when subjected to a truncation approximation, efficiently describes the shock motion. Disturbances propagating on the shock and the onset of shock-shocks are briefly discussed. For a two-dimensional shock motion, our transport equations bear a structural resemblance to those of geometrical shock dynamics. Attention is drawn to the connection between the transport equation obtained by using the truncation rule and the one obtained by using the characteristic rule. The effects of van der Waals' excluded volume and wavefront geometry on the evolutionary behaviour of shocks are discussed.

Thermo-visco-elasticity at small strains with ¹-data

Abstract: Existence of a very weak solution to the d-dimensional thermo-viscoelasticity system for Kelvin-Voigt-type material at small strains involving (possibly nonlinear) monotone viscosity of a p-Laplacian type and temperature-dependent heat capacity of an (ω−1)-polynomial growth is proved by a successive passage to a limit in a suitably regularized Galerkin approximation and sophisticated a priori estimates for the temperature gradient performed for the coupled system. A global solution for arbitrarily large data having an L-structure is obtained under the conditions p ≥ 2, ω ≥ 1, and p > 1 + d/(2ω).

Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains

Abstract: First, we introduce the concept of pullback omega-limit compactness for multivalued processes, as an extension of the similar concept in the autonomous and nonautonomous framework. Next, we present the necessary and sufficient, conditions (pullback dissipativeness and pullback omega-limit compactness) for the existence of a nonempty local bounded kernel (kernel sections are all compact, invariant and pullback attracting) of an infinite dimensional multi-valued process. In addition, we prove a result ensuring the existence of a uniform attractor and the uniform forward attraction of the inflated kernel sections of a family of multi-valued processes under the general assumptions of point dissipativeness and uniform omega-limit compactness. Finally, we illustrate the abstract theory with a nonlinear reaction-diffusion model in an unbounded domain.

Thermal grooving by surface diffusion: mullins revisited and extended to multiple grooves

Abstract: We extend the original Mullins theory of surface grooving due to a single interface to multiple interacting grooves formed due to closely spaced flat interfaces. First, we show that Mullins’ analysis for one groove can be simplified by using Fourier cosine transforms instead of Laplace transforms. Second, we solve the corresponding problem for an infinite periodic row of grooves. For both of these problems, symmetry considerations ensure that the interface conditions reduce to boundary conditions. Third, we solve the problem for two interacting grooves. Continuity requirements at the groove roots require sliding at the interfaces or tilting of the groove roots. We adopt the latter model. We find that the groove roots tilt until the surface curvature of the semi-infinite profiles is eliminated.

Fractional radial diffusion in an infinite medium with a cylindrical cavity

Abstract: The time-fractional diffusion equation is employed to study the radial diffusion in an unbounded body containing a cylindrical cavity. The Caputo fractional derivative is used. The solution is obtained by application of Laplace and Weber integral transforms. Several examples of problems with Dirichlet and Neumann boundary conditions are presented. Numerical results are illustrated graphically.

Critical Rayleigh number in Rayleigh-Bénard convection

Abstract: The Rayleigh-Benard convection is a classical problem in fluid dynamics. In the presence of rigid boundary condition, we identify the critical Rayleigh number R* a by a reduced variational problem. We prove nonlinear asymptotic stability for motionless steady states for R a R* a . The dynamic of such instability is determined by the leading growing mode(s) for the corresponding linearized system within the time interval of instability.

Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system

Abstract: We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general context, and we include arguments which show that the spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain a black hole in the sense of suitable mathematical definitions of these concepts which are available in the literature.

A derivation of the Aw–Rascle traffic models from Fokker–Planck type kinetic models

Abstract: We show how the Aw–Rascle model, a hyperbolic system of PDEs modeling traffic flow, can be derived from a simplified Fokker–Planck type kinetic equation.

Cosine effect on shallow water equations and mathematical properties

Abstract: This paper presents a viscous Shallow Water type model with new Coriolis terms, and some limits according to the values of the Rossby and Froude numbers. We prove that the extension to the bidimensional case of the unidimensional results given by [J.―F. GERBEAU, B. PERTHAME. Discrete Continuous Dynamical Systems, (2001)] including the Coriolis force has to add new terms, omitted up to now, depending on the latitude cosine, when the viscosity is assumed to be of the order of the aspect ratio. We show that the expressions for the waves are modified, particularly at the equator, as well as the Quasi-Geostrophic and the Lake equations. To conclude, we also study the mathematical properties of these equations.

Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

Abstract: We study the existence and uniqueness of traveling wave solutions for a class of twocomponent reaction diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor model, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinburgh 2005; 135A:663–675), but also establishes the existence and uniqueness of traveling wave solutions to the reaction-diffusion system for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.

Sensitivity equations for a size-structured population model

Abstract: In this paper we consider the classical Sinko-Streifer size-structured population model and derive sensitivity partial differential equations for the sensitivities of solutions with respect to initial conditions, growth rate, mortality rate and fecundity rate. Sample numerical results to illustrate use of these equations are also presented.

Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays

Abstract: This paper studies an incompressible non-Newtonian fluid with delays in two-dimensional bounded domains. We first prove the existence and uniqueness of solutions. Then we establish the existence of pullback attractors {A cH (t)} t∈ℝ (has L 2 -regularity), {A cw (t)) t∈ℝ (has H 2 -regularity), and {A E 2 H (t)} t∈ℝ (has L 2 -regularity), {A E 2 W (t)} t∈ℝ (has H 2 -regularity) corresponding to two different processes associated to E 2 W the fluid, respectively. Meanwhile, we verify the regularity of the pullback attractors by proving A CH (t) = A cW (t), A E 2 H (t) = A E 2 W (t), ∀ t ∈ ℝ and A E 2 H (t) = J(A cH (t)) = J(A cW (t)) = A E 2 W (t), ∀ t ∈ℝ, where J is a linear operator. By the regularity we reveal the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data. This effect implies, in the case of delays, that the regularity of the fluid in its history state does not play an important role on the regularity of its eventual state. Finally, we give some remarks.

Exponential and polynomial decay for first order linear Volterra evolution equations

Abstract: We consider, in an abstract setting, an instance of the Coleman-Gurtin model for heat conduction with memory, that is, the Volterra integro-differential equation ∂tu(t) − β∆u(t) − ∫ t 0 k(s)∆u(t − s)ds = 0. We establish new results for the exponential and polynomial decay of solutions, by means of conditions on the convolution kernel which are weaker than the classical differential inequalities.

Recovery of multiple obstacles by probe method

Abstract: We consider an inverse scattering problem for multiple obstacles D = U N j=1 D j ⊂ R 3 with different types of boundary for D j . By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

Asymmetric and moving-frame approaches to navier-stokes equations

Abstract: In this paper, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent variables and a method of moving frame for solving the three dimensional Navier-Stokes equations. Seven families of non-steady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blow up at any point on a moving plane with a line deleted, which may be used to study turbulence. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in study of shock waves. Another family of solutions are partially cylindrical invariant, contain two parameter functions of $t$ and structurally depend on two arbitrary polynomials, which may be used to describe incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spacial variables.

Structural conditions for full MHD equations

Abstract: In this paper, we investigate the characteristic structure of the full equations of magnetohydrodynamics (MHD) and show that it satisfies the hypotheses of a general variable-multiplicity stability framework introduced by Metivier and Zumbrun, thereby extending to the general case various results obtained by Metivier and Zumbrun for the isentropic equations of MHD.

On spatial behavior in linear viscoelasticity

Abstract: Within the framework of linear viscoelasticity this paper deals with the study of spatial behavior of solutions describing harmonic vibrations in a right cylinder of finite extent. Some exponential decay estimates of Saint–Venant type, in terms of the distance from the excited end of the cylinder are obtained from a first-order differential inequality concerning an appropriate measure associated with the amplitude of the steady-state vibration. The dissipative mechanism guarantees the validity of the result for every value of the frequency of vibration and for the class of viscoelastic materials compatible with thermodynamics whose relaxation tensor is supposed to be symmetric and sufficiently regular. The case of a semi-infinite cylinder is also studied, and some alternatives of Phragmén–Lindelöf type are established. Introduction. The present paper is concerned with the study of the spatial behavior of solutions in a right cylinder made of an anisotropic and homogeneous viscoelastic solid. We consider a finite cylinder subject to boundary data varying harmonically in time on one end, while the other end and lateral surface are clamped. The history of the displacement up to time t = 0 is supposed to be known and the body force is assumed to be absent. Initial boundary value problems of this type have been treated by Flavin and Knops [1] in the framework of the linearly damped wave equation and the linearly elastic damped cylinder. They proved that in both cases the existence of damping gives rise ultimately to a steady-state oscillation, whose amplitude decays exponentially from the excited end provided the exciting frequency is less than a certain critical value. The latter case has Received May 16, 2008. 2000 Mathematics Subject Classification. Primary 74D05, 74G50; Secondary 74H45, 74E10.

On a contact problem in thermoelasticity with second sound

Abstract: We investigate the existence and stability of a thermoelastic contact problem with second sound. Previous results established the existence and stability of a solution of the corresponding classical system in the case of radial symmetry. However, recent works have shown that sometimes stability can be lost when the classical Fourier heat conduction is substituted by Cattaneo’s Law. We show that also in this case this substitution does indeed lead to a loss in regularity that proves to be a major problem prohibiting the transfer of the existence proof for the classical problem to the problem with second sound, leaving the existence of a solution an open question. We then prove that, if a viscoelastic term is added to the equations providing additional regularity, existence and exponential stability the second, as can be expected, only in the case of radial symmetry follow.

Readily computable Green’s and Neumann functions for symmetry-preserving triangles

Abstract: Neumann and Green's functions of the Laplacian operator on 30-60-90° and 45-45-90° triangles can be generated with appropriately placed multiple sources/sinks in a rectangular domain. Highly accurate and easily computable Neumann and Green's function formulas already exist for rectangles. The extension to equilateral triangles is illustrated. In applications, closed-form expressions can be constructed for the potential, the streamfunction, or the various spatial derivatives of these properties. The derivation of analytic line integrals of these functions allows the proper handling of singularities and facilitates extended applications to problems on domains with open boundaries. Using a boundary integral method, it is demonstrated how one can construct semi-analytical solutions to problems defined on domains that exhibit spatially-dependent properties (heterogeneous media) or possess irregular boundaries.