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  • Articles  (503)
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  • Articles  (503)
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  • 11
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    Blowup for biharmonic Schrödinger equation with critical nonlinearity (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: We consider the minimizers for the biharmonic nonlinear Schrödinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$ with the mass constraint \(\int |u|^2=1\) . We focus on the special power \(q=2(1+4/d)\) , which makes the nonlinear term \(\int |u|^q\) scales similarly to the biharmonic term \(\int |\Delta u|^2\) . Our main results are the existence and blowup behavior of the minimizers when a tends to a critical value \(a^*\) , which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.
    Print ISSN: 0044-2275
    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 12
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    Elastic layer under axisymmetric indentation and surface energy effects (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: In this paper, a continuum-based approach is adopted to investigate the contact problem of an elastic layer with finite thickness and rigid base subjected to axisymmetric indentation with the consideration of surface energy effects. A complete Gurtin–Murdoch surface elasticity is employed to consider the influence of surface stresses. The indentation problem of a rigid frictionless punch with arbitrary axisymmetric profiles is formulated by employing the displacement Green’s functions, derived with the aid of Hankel integral transform technique. The problem is solved by assuming the contact pressure distribution in terms of a linear combination of admissible functions and undetermined coefficients. Those coefficients are then obtained by employing a collocation technique and an efficient numerical quadrature scheme. The accuracy of proposed solution technique is verified by comparing with existing solutions for rigid indentation on an elastic half-space. Selected numerical results for the indenters with flat-ended cylindrical and paraboloidal punch profiles are presented to portray the influence of surface energy effects on elastic fields of the finite layer. It is found that the presence of surface stresses renders the layer stiffer, and the size-dependent behavior of elastic fields is observed in the present solutions. In addition, the surface energy effects become more pronounced with smaller contact area; thus, the influence of surface energy cannot be ignored in the analysis of indentation problem especially when the indenter size is very small such as in the case of nanoindentation.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 13
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    Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which arises in modeling the motion of such a fluid down an inclined plane, after a coordinate change. We consider the problem both with and without surface tension for horizontally periodic flows. This problem gives rise to shear-flow equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of the equilibria in certain parameter regimes. We prove that there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time \(t=0\) give rise to global-in-time solutions that return to equilibrium exponentially in the case with surface tension and almost exponentially in the case without surface tension. We also establish a vanishing surface tension limit, which connects the solutions with and without surface tension.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 14
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    Global dynamics of a nonlocal delayed reaction–diffusion equation on a half plane (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: We consider a delayed reaction–diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 15
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    Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper deals with a reaction–diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green’s identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 16
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    Expressions to Rayleigh circumferential phase velocity and dispersion relation for a cylindrical surface under mechanical pressure (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper describes a substantiated mathematical theory for Rayleigh waves propagated on some types of metal cylinders. More specifically, it presents not only a new way to express the dispersion relation of Rayleigh waves propagated on the cylindrical surface, but also how it can be used to construct a mathematical equation showing that the applied static mechanical pressure affects the shear modulus of the metal cylinder. All steps, required to conclude the process, consider the equation of motion as a function of radial and circumferential coordinates only, while the axial component can be overlooked without causing any problems. Some numerical experiments are done to illustrate the changes in the Rayleigh circumferential phase velocity in a metal cylindrical section due to static mechanical pressure around its external surface.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 17
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    Torsion of a Cosserat elastic bar with square cross section: theory and experiment (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: An approximate analytical solution for the displacement and microrotation vector fields is derived for pure torsion of a prismatic bar with square cross section comprised of homogeneous, isotropic linear Cosserat elastic material. This is accomplished by analytical simplification coupled with use of the principle of minimum potential energy together with polynomial representations for the desired field components. Explicit approximate expressions are derived for cross section warp and for applied torque versus angle of twist of the bar. These show that torsional rigidity exceeds the classical elasticity value, the difference being larger for slender bars, and that cross section warp is less than the classical amount. Experimental measurements on two sets of 3D printed square cross section polymeric bars, each set having a different microstructure and four different cross section sizes, revealed size effects not captured by classical elasticity but consistent with the present analysis for physically sensible values of the Cosserat moduli. The warp can allow inference of Cosserat elastic constants independently of any sensitivity the material may have to dilatation gradients; warp also facilitates inference of Cosserat constants that are difficult to obtain via size effects.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 18
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    On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper concerns the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum on \(\varOmega \subset \mathbb {R}^3\) . The domain \(\varOmega \subset \mathbb {R}^3\) is a general connected smooth one, either bounded or unbounded. In particular, the initial density can have compact support when \(\varOmega \) is unbounded. First, we obtain the local existence and uniqueness of strong solution to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations without any compatibility condition assumed on the initial data. Then, we also prove the continuous dependence of strong solution on the initial data under an additional compatibility condition.
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    Topics: Mathematics , Physics
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  • 19
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    Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in \(\mathbb {R}^3\) . In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 20
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    Existence and smoothness for a class of d -dimensional models in elasticity theory of small deformations (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: We consider a model for deformations of a homogeneous isotropic body, whose shear modulus remains constant, but its bulk modulus can be a highly nonlinear function. We show that for a general class of such models, in an arbitrary space dimension, the respective PDE problem has a unique solution. Moreover, this solution enjoys interior smoothness. This is the first full regularity result for elasticity problems that covers the most natural space dimension 3 and that captures the behaviour of real-life elastic materials (considered in small deformations), primarily certain beta-phase titanium alloys.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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