Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 2012"

How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”

Abstract: Navier–Cauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how—by means of the Principle of Virtual Powers—it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green–Saint-Venant tensor and all its N − 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.

Multiple solutions to a magnetic nonlinear Choquard equation

Abstract: We consider the stationary nonlinear magnetic Choquard equation $$(- {\rm i} abla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$$ where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 − (α/N) < p < (2N − α)/(N−2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition $$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$$ where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Abstract: We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Malek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).

Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces

Abstract: We investigate the Cauchy problem of a multidimensional chemotaxis model with initial data in critical Besov spaces. The global existence and uniqueness of the strong solution is shown for initial data close to a constant equilibrium state.

Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions

Abstract: We study the eigenpairs of a model Schr¨ odinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit ! 0 and we prove a full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.

Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

Abstract: As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that starts on sonic curves and ends at transonic shocks. Both the vacuum and the sonic sets cause parabolic degeneracy that results in substantial difficulties such as singularities of solutions and uniform a priori estimates. Main ingredients in this study are various characteristic decompositions for the pseudo-steady Euler equations in order to obtain necessary a priori estimates. Furthermore, we are able to verify the uniform Holder continuity of solutions with exponent 1/2 for the gas expansion problem and up to 2/7 for the semi-hyperbolic problem.

MHD oblique stagnation-point flow of a Newtonian fluid

Abstract: The steady two-dimensional oblique stagnation-point flow of an electrically conducting Newtonian fluid in the presence of a uniform external electromagnetic field (E0, H0) is analysed, and some physical situations are examined. In particular, if E0 vanishes, H0 lies in the plane of the flow, with a direction not parallel to the boundary, and the induced magnetic field is neglected, it is proved that the oblique stagnation-point flow exists if and only if the external magnetic field is parallel to the dividing streamline. In all cases it is shown that the governing nonlinear partial differential equations admit similarity solutions, and the resulting ordinary differential problems are solved numerically. Finally, the behaviour of the flow near the boundary is analysed; this depends on the Hartmann number if H0 is parallel to the dividing streamline.

On the development and generalizations of Allen–Cahn and Stefan equations within a thermodynamic framework

Abstract: Starting from a simplified framework of the theory of interacting continua in which the mass balance equations are considered for each constituent but the balance of linear momentum and the balance of energy are considered for the mixture as a whole, we provide a thermodynamic basis for models that include the Allen–Cahn and Stefan equations as particular cases. We neglect the mass flux due to diffusion associated with the components of the mixture but permit the possibility of mass conversion of the phases. As a consequence of the analysis, we are able to show that the reaction (source) term in the mass balance equation leads to the Laplace operator that appears in the Allen–Cahn model and that this term is not related to a diffusive process. This study is complementary to that by Heida et al. (Zeitschrift fur Angewandte Mathematik und Physik (ZAMP) 63, 145–169, 2012), where we neglected mass conversion of the species but considered mass diffusion effects and derived the constitutive equations for diffusive mass flux (the framework suitable for capturing other interface phenomena such as capillarity and for generalizing the Cahn–Hilliard and Lowengrub–Truskinovsky models).

Analysis and simulations of a nonlinear elastic dynamic beam

Abstract: A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions.

Riemann problem for the isentropic relativistic Chaplygin Euler equations

Abstract: This paper studies the Riemann problem of the isentropic relativistic Euler equations for a Chaplygin gas. The solutions exactly include five kinds. The first four consist of different contact discontinuities while the rest involves delta-shock waves. Under suitable generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.

Global dynamics of delayed reaction–diffusion equations in unbounded domains

Abstract: We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflow $${\Phi}$$ on a bounded and positively invariant set Y in C + = C([−1, 0], X +) that attracts every solution of the equation, where X + is the set of all bounded and uniformly continuous functions from $${\mathbb{R}}$$ to [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

Abstract: In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial u}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial u}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain Ω. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.

Entropy stable schemes for initial-boundary-value conservation laws

Abstract: We consider initial-boundary-value problems for systems of conservation laws and design entropy stable finite difference schemes to approximate them. The schemes are shown to be entropy stable for a large class of systems that are equipped with a symmetric splitting, derived from the entropy formulation. Numerical examples for the Euler equations of gas dynamics are presented to illustrate the robust performance of the proposed method.

Numerical modelling of rarefied gas flow through a slit at arbitrary pressure ratio based on the kinetic equation

Abstract: A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kinetic model equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those based on the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditions of applicability of the linearized theory are discussed.

Hinged and supported plates with corners

Abstract: We consider the Kirchhoff–Love model for the supported plate, that is, the fourth-order differential equation Δ2 u = f with appropriate boundary conditions. Due to the expectation that a downwardly directed force f will imply that the plate, which is supported at its boundary, touches that support everywhere, one commonly identifies those boundary conditions with the ones for the so-called hinged plate: u = 0 = Δu − (1 − σ ) κ u n . Structural engineers however are usually aware that rectangular roofs tend to bend upwards near the corners, and this would mean that u = 0 is not appropriate. We will confirm this behavior and show the difference of the supported and the hinged plates in case of domains with corners.

On the dissipative effect of a magnetic field in a Mindlin-Timoshenko plate model

Abstract: In this paper we are concerned with a linear model for the magnetoelastic interactions in a two-dimensional electrically conducting Mindlin-Timoshenko plate. The magnetic field that permeates the plate consists of a non-stationary part and a uniform (constant) part. When the uniform magnetic field is aligned with the mid-plane of the plate, a strongly interactive system emerges with direct coupling between the elastic field and the magnetic field occurring in all the equations of the system. The unique solvability of the model is established within the framework of semigroup theory. Spectral analysis methods are used to show strong asymptotic stability and determine the polynomial decay rate of weak solutions.

Infinitely many solutions for a differential inclusion problem in {\mathbb{R}^N} involving p(x)-Laplacian and oscillatory terms

Abstract: In this paper, we consider the differential inclusion in $${\mathbb{R}^N}$$ involving the p(x)-Laplacian of the type $${\begin{array}{lll}-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u(x)),\;\;{\rm in}\;\;\mathbb{R}^N,\quad\quad\quad\quad\quad\quad ({\rm P})\end{array}}$$ where $${p: \mathbb{R}^N \to {\mathbb{R}}}$$ is Lipschitz continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many solutions of (P). We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces.

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Abstract: The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method The material behaviour is modelled by using a general Kelvin–Voigt constitutive law

General decay for a von Karman equation of memory type with acoustic boundary conditions

Abstract: A Karman equation of memory type with acoustic boundary conditions is considered This work is devoted to investigate the influence of kernel function g and prove general decay rates of solutions when g does not necessarily decay exponentially

Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity

Abstract: We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L 2 initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139–166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L 2 initial data.

Revisiting total, matric, and osmotic suction in partially saturated geomaterials

Abstract: An accurate quantification of negative pore pressure (commonly referred to as ‘suction’) in the pore network is necessary for modeling the mechanical response of unsaturated geomaterials. Traditional definitions and formulations of total, matric, and osmotic suction suggest incorrect pore fluid pressures under certain conditions. In this paper, the notion of suction is revisited by deriving an expression for pore fluid pressure in a simple osmotic, capillary tube using the framework of mixture theory in conjunction with the fundamental laws of thermodynamics. Based on the derived expression for the tube, expressions are derived for total, matric, and osmotic suction for partially saturated geomaterials. Particular attention is given to osmotic suction since confusion regarding its mechanisms has apparently contributed to its misapplication in geomechanics. The new expressions derived herein adequately explain behavior that is incorrectly explained by the traditional formulations and unifies two approaches to modeling osmotic suction previously considered to be in contradiction.

A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

Abstract: In this paper, a quasi-periodically forced nonlinear beam equation $${u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}$$ with hinged boundary conditions is considered, where μ > 0, $${\varepsilon}$$ is a small positive parameter, $${\phi}$$ is a real analytic quasi-periodic function in t with a frequency vector ω = (ω 1,ω 2 . . . , ω m ), and the nonlinearity h is a real analytic odd function of the form $${h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} eq0, \bar{r} \in {\mathbb {N}}.}$$ The above equation admits a quasi-periodic solution.

Shallow water equations: viscous solutions and inviscid limit

Abstract: We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C2 test-functions, are confined in a compact set in H−1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.

Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

Abstract: We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation $$\left\{\begin{array}{ll}-{div}(| abla u|^{p-2} abla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.$$ with singular radial potentials V,Q and bounded nonlinearity f. The approaches used here are based on a compact embedding from \({W_r^{1,p}(\mathbb{R}^N; V)}\) into \({L^1(\mathbb{R}^N; Q)}\) and minimax methods. A uniqueness result is given for f ≡ 1.

Covariantization of nonlinear elasticity

Abstract: In this paper we make a connection between covariant elasticity based on covariance of energy balance and Lagrangian field theory of elasticity with two background metrics. We use Kuchař’s idea of reparametrization of field theories and make elasticity generally covariant by introducing a "covariance field", which is a time-independent spatial diffeomorphism. We define a modified action for parameterized elasticity and show that the Doyle-Ericksen formula and spatial homogeneity of the Lagrangian density are among its Euler–Lagrange equations.

Boundary stabilization of memory-type thermoelasticity with second sound

Abstract: In this paper, we consider an n-dimensional thermoelastic system of second sound with a viscoelastic damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

Stability analysis of a thermo-elastic system of type II with boundary viscoelastic damping

Abstract: The stability of a kind of one-dimensional thermo-elastic system of type II is considered. This system consists of two strongly coupled wave equations. Suppose that there exists an viscoelastic damping at one end of the 1-d domain. If without coupling, this damping always makes one of these two wave equations (the corresponding pure elastic system) achieve exponential stability and the other wave system (heat equation of type II) be conservative. Whether the coupling can pass the damping effect from the dissipative elastic system to the conservative heat system of type II is discussed. However, by a detailed spectral analysis, it is proved that this thermo-elastic system is at most asymptotically stable but not exponentially stable. A numerical simulation is given to support these results obtained in this paper.

On Love-type waves in a finitely deformed magnetoelastic layered half-space

Abstract: In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.

Self-diffusion in remodeling and growth

Abstract: Self-diffusion, or the flux of mass of a single species within itself, is viewed as an independent phenomenon amenable to treatment by the introduction of an auxiliary field of diffusion velocities. The theory is shown to be heuristically derivable as a limiting case of that of an ordinary binary mixture.

Stokes flow between eccentric rotating spheres with slip regime

Abstract: The steady axisymmetric flow problem of a viscous fluid contained between two eccentric spheres that rotate about an axis joining their centers with different angular velocities is considered. A linear slip of Basset-type boundary condition at both surfaces of the spherical particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in the spherical coordinate systems based on the inner solid particle and the spherical container. The boundary conditions on the particle’s surface and spherical container are satisfied by a collocation technique. Numerical results for the coupling coefficient acting on the particle are obtained with good convergence for various values of the ratio of particle-to-container radii, the relative distance between the centers of the particle and container, the slip coefficients and the relative angular velocity. In the limiting cases, the numerical values of the coupling coefficient for the solid sphere in concentric position with the container and when the particle is near the inner surface of the container are obtained, and the results are in good agreement with the available values in the literature. The variation of the coupling coefficient with respect the parameters considered are tabulated and displayed graphically.