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 Archive for Rational Mechanics and Analysis    [5 followers]  Follow        Hybrid journal (It can contain Open Access articles)      ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527      Published by Springer-Verlag  [2210 journals]   [SJR: 3.43]   [H-I: 51]
• Homogenization of the Hele-Shaw Problem in Periodic Spatiotemporal Media
• Abstract: Abstract We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.
PubDate: 2014-12-16

• A Stable Self-Similar Singularity of Evaporating Drops: Ellipsoidal
Collapse to a Point
• Abstract: Abstract We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called D 2 law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converges to a self-similar ellipsoid whose center, orientation, and semi-axes are determined by the initial shape.
PubDate: 2014-12-16

• Crystal Dislocations with Different Orientations and Collisions
• Abstract: Abstract We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superposition of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. More precisely, we consider the evolutionary equation $$(v_\varepsilon)_t=\frac{1}{\varepsilon}\left( \mathcal{I}v_\varepsilon-\frac{1}{\varepsilon^{2s}}W'(v_\varepsilon)+\sigma(t, x)\right),$$ where $${v_\varepsilon=v_\varepsilon(t, x)}$$ is the atom dislocation function at time t >  0 at the point $${x \in \mathbb{R}, {\mathcal{I}_{s}}}$$ is an integro-differential operator of order $${2s \in (0, 2), W}$$ is a periodic potential, $${\sigma}$$ is an external stress and $${\varepsilon > 0}$$ is a small parameter that takes into account the small periodicity scale of the crystal. We suppose that $${v_\varepsilon(0, x)}$$ is the superposition of N−K transition layers in the positive direction and K in the negative one (with $${K \in\{0,\dots,N\}}$$ ); more precisely, we fix points $${x_1^0 < \dots < x_N^0}$$ and we take $$v_\varepsilon(0, x)= \frac{\varepsilon^{2s}}{W''(0)}\sigma(0, x)+\sum_{i=1}^N u\left(\zeta_i\frac{x-x_i^0}{\varepsilon}\right).$$ Here $${\zeta_i}$$ is either −1 or 1, depending on the orientation of the transition layer u, which in turn solves the stationary equation $${\mathcal{I}_{s} u=W'(u)}$$ . We show that our problem possesses a unique solution and that, as $${\varepsilon \to 0^+}$$ , it approaches the sum of Heaviside functions H with different orientations centered at points x i (t), namely $$\sum_{i=1}^N H(\zeta_i(x-x_i(t))).$$ The point x i evolves in time from $${x_i^0}$$ , being subject to the external stress and a singular potential, which may be either attractive or repulsive, according to the different orientation of the transitions; more precisely, the speed $${\dot x_i}$$ is proportional to $$\sum_{j\neq i}\zeta_i\zeta_j\frac{x_i-x_j}{2s x_i-x_j ^{1+2s}}-\zeta_i\sigma(t, x_i).$$ The evolution of such a dynamical system may lead to collisions in finite time. We give a detailed description of such collisions when N = 2, 3 and we show that the solution itself keeps track of such collisions; indeed, at the collision time T c the two opposite dislocations have the tendency to annihilate each other and make the dislocation vanish, but only outside the collision point
PubDate: 2014-12-16

• Existence and Stability for a Non-Local Isoperimetric Model of Charged
Liquid Drops
• Abstract: Abstract We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits local minimizers with respect to L 1 perturbations preserving the volume. However, we prove that the ball is stable under small C 1,1 perturbations when the charge is small enough.
PubDate: 2014-12-12

• On the Dirichlet Problem of Mixed Type for Lower Hybrid Waves in
Axisymmetric Cold Plasmas
• Abstract: Abstract For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms.
PubDate: 2014-12-11

• Optimal Shape and Location of Sensors for Parabolic Equations with Random
Initial Data
• Abstract: Abstract In this article, we consider parabolic equations on a bounded open connected subset $${\Omega}$$ of $${\mathbb{R}^n}$$ . We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of $${\Omega}$$ of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in Privat et al. (J Eur Math Soc, 2015) for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations.
PubDate: 2014-12-10

• Green’s Functions of Wave Equations in $$\mathbb{R}^n_+\times \mathbb{R}_+$$ R + n × R +
• Abstract: Abstract We study the d’Alembert equation with a boundary. We introduce the notions of Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations. This allows us to obtain the complete and simple formula of the Green’s functions for the wave equation with the presence of various boundary conditions. We are able to determine whether a Rayleigh surface wave is active or virtual, and study the lacunas of the wave equation in three dimensional with the presence of a boundary in the case of a virtual Rayleigh surface wave.
PubDate: 2014-12-09

• Metric Description of Singular Defects in Isotropic Materials
• Abstract: Abstract Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.
PubDate: 2014-12-09

• Vanishing Shear Viscosity and Boundary Layer for the Navier–Stokes
Equations with Cylindrical Symmetry
• Abstract: Abstract Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier–Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L 1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components.
PubDate: 2014-12-09

• Fick’s Law in a Random Lattice Lorentz Gas
• Abstract: Abstract We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick’s law when connected to particles reservoirs. We consider a box on a d + 1-dimensional lattice and when $${d\geqq7}$$ , we show that under a diffusive rescaling of space and time, the probability of finding a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in a high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.
PubDate: 2014-12-09

• Γ -convergence Approximation of Fracture and Cavitation in Nonlinear
Elasticity
• Abstract: Abstract Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.
PubDate: 2014-12-05

• Abstract: Abstract We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves of finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave is, generically, a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.
PubDate: 2014-12-03

• A Derivation of the Magnetohydrodynamic System from
Navier–Stokes–Maxwell Systems
• Abstract: Abstract We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier–Stokes–Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case).
PubDate: 2014-12-02

• Well-Posedness and Singular Limit of a Semilinear Hyperbolic Relaxation
System with a Two-Scale Discontinuous Relaxation Rate
• Abstract: Abstract Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L 1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.
PubDate: 2014-12-01

• Hyperelastic Internal Balance by Multiplicative Decomposition of the
• Abstract: Abstract The multiplicative decomposition of the deformation gradient $${{\bf F} = {{\hat{\bf F}}}{\bf F}^*}$$ is often used in finite deformation continuum mechanics as a basis for treating mechanical effects including plasticity, biological growth, material swelling, and notions of material morphogenesis. Evolution rules for the particular effect from this list are then posed for F*. The tensor $${{{\hat{\bf F}}}}$$ is then invoked to describe a subsequent elastic accommodation, and a hyperelastic framework is put in place for its determination using an elastic energy density function, say $${W({\hat{\bf F}})}$$ , as a constitutive specification. Here we explore the theory that emerges if both F* and $${{\hat{\bf F}}}$$ are governed by hyperelastic criteria; thus we consider energy densities $${W({{\hat{\bf F}}}, {\bf F}^*)}$$ . The decomposition of F is itself determined by energy minimization, and the variation associated with the multiplicative decomposition gives a tensor relation that is interpreted as an internal balance requirement. Our initial development purposefully proceeds with minimal presumptions on the kinematic interpretation of the factors in the deformation gradient decomposition. Connections are then made to treatments that ascribe particular kinematic properties to the decomposition factors—the theory of structured deformations is especially significant in this regard. Such theories have broad utility in describing certain substructural reconfigurations in solids. To demonstrate in the context of the present variational treatment we consider a boundary value problem that involves an imposed twist. If the twist is small then the minimizer is classically smooth. At larger values of twist the energy minimizer exhibits a non-smooth deformation that localizes slip at a singular surface.
PubDate: 2014-12-01

• Dispersion Equation for Water Waves with Vorticity and Stokes Waves on
Flows with Counter-Currents
• Abstract: Abstract The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.
PubDate: 2014-12-01

• Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal
Fokker–Planck Equations with Dynamical Control
• Abstract: Abstract The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker–Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.
PubDate: 2014-12-01

• Newtonian Limit of Maxwell Fluid Flows
• Abstract: Abstract In this paper, we revise Maxwell’s constitutive relation and formulate a system of first-order partial differential equations with two parameters for compressible viscoelastic fluid flows. The system is shown to possess a nice conservation–dissipation (relaxation) structure and therefore is symmetrizable hyperbolic. Moreover, for smooth flows we rigorously verify that the revised Maxwell’s constitutive relations are compatible with Newton’s law of viscosity.
PubDate: 2014-12-01

• Quantitative Stochastic Homogenization of Elliptic Equations in
Nondivergence Form
• Abstract: Abstract We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge–Ampère equation.
PubDate: 2014-12-01

• Existence, Uniqueness and Lipschitz Dependence for
Patlak–Keller–Segel and Navier–Stokes in
$${\mathbb{R}^2}$$ R 2 with Measure-Valued Initial Data
• Abstract: Abstract We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic–elliptic Patlak–Keller–Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption $${\max_{x \in \mathbb{R}^2} \mu (\{x\}) < 8 \pi}$$ . This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier–Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.
PubDate: 2014-12-01

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