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 Archive for Rational Mechanics and Analysis   [SJR: 3.472]   [H-I: 57]   [3 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527    Published by Springer-Verlag  [2291 journals]
• Study of Island Formation in Epitaxially Strained Films on Unbounded
Domains
• Abstract: Abstract We consider a variational model related to the formation of islands in heteroepitaxial growth on unbounded domains. We first derive the scaling regimes of the minimal energy in terms of the volume of the film and the amplitude of the crystallographic misfit. For small volumes, non-existence of minimizers is then proven. This corresponds to the experimentally observed wetting effect. On the other hand, we show the existence of minimizers for large volumes. We finally study the asymptotic behavior of the optimal shapes.
PubDate: 2015-10-01

• The Optimal Decay Estimates on the Framework of Besov Spaces for Generally
Dissipative Systems
• Abstract: Abstract We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic–parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood–Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the $${L^{p}(\mathbb{R}^{n})}$$ embedding and the improved Gagliardo–Nirenberg inequality, the optimal $${L^{p}(\mathbb{R}^{n})-L^{2}(\mathbb{R}^{n})(1\leqq p < 2)}$$ decay rates and $${L^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})(1\leqq p < 2\leqq q\leqq \infty)}$$ decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained.
PubDate: 2015-10-01

• Bounded Minimisers of Double Phase Variational Integrals
• Abstract: Abstract Bounded minimisers of the functional $$w \mapsto \int ( Dw ^p+a(x) Dw ^q)\,{\rm d}x,$$ where $${0 \leqq a(\cdot) \in C^{0, \alpha}}$$ and $${1 < p < q}$$ , are $${C^{1, \beta}}$$ -regular provided the sharp bound $${q \leqq p + \alpha}$$ holds.
PubDate: 2015-10-01

• Global Existence of Renormalized Solutions to Entropy-Dissipating
Reaction–Diffusion Systems
• Abstract: Abstract In the present work we introduce the notion of a renormalized solution for reaction–diffusion systems with entropy-dissipating reactions. We establish the global existence of renormalized solutions. In the case of integrable reaction terms our notion of a renormalized solution reduces to the usual notion of a weak solution. Our existence result in particular covers all reaction–diffusion systems involving a single reversible reaction with mass-action kinetics and (possibly species-dependent) Fick-law diffusion; more generally, it covers the case of systems of reversible reactions with mass-action kinetics which satisfy the detailed balance condition. For such equations the existence of any kind of solution in general was an open problem, thereby motivating the study of renormalized solutions.
PubDate: 2015-10-01

• The Ginzburg–Landau Functional with Vanishing Magnetic Field
• Abstract: Abstract We study the infimum of the Ginzburg–Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed to vanish along a smooth curve. We obtain energy asymptotics which are valid when the Ginzburg–Landau parameter is large and the strength of the external field is below the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X.B. Pan–K.H. Kwek.
PubDate: 2015-10-01

• A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations
on Bounded Domains
• Abstract: Abstract We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ to the nonlinear fractional diffusion equation, $${\partial_t u + \mathcal{L} (u^m)=0}$$ , posed in a bounded domain, $${x\in\Omega\subset \mathbb{R}^N}$$ , with m > 1 for t > 0. As $${\mathcal{L}}$$ we use one of the most common definitions of the fractional Laplacian $${(-\Delta)^s}$$ , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.
PubDate: 2015-10-01

• On the Semi-geostrophic System in Physical Space with General Initial Data
• Abstract: Abstract In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space; as consequences we get conservation of Hamiltonian energy and some weak time-regularity of solutions.
PubDate: 2015-10-01

• Randomization and the Gross–Pitaevskii Hierarchy
• Abstract: Abstract We study the Gross–Pitaevskii hierarchy on the spatial domain $${\mathbb{T}^3}$$ . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $${\alpha > \frac{3}{4}}$$ . It was shown in our previous joint work with Gressman (J Funct Anal 266(7):4705–4764, 2014) that the range $${\alpha > 1}$$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1):169–185, 2008), where the spacetime estimate is known to hold whenever $${\alpha \geq 1}$$ . The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross–Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross–Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.
PubDate: 2015-10-01

• Construction of Almost-Sharp Fronts for the Surface Quasi-Geostrophic
Equation
• Abstract: Abstract In this paper we construct families of real analytic solutions of the Surface Quasi-Geostrophic equation (SQG) that are locally constant outside a thin neighborhood of a curve of arbitrarily small thickness. Despite the fact that only local existence results are known for SQG, and that our initial conditions have a arbitrarily large gradient we show that solutions exist for a time independent of the thickness of the neighborhood.
PubDate: 2015-10-01

• Strictly Physical Global Weak Solutions of a Navier–Stokes Q -tensor
System with Singular Potential
• Abstract: Abstract We study the existence, regularity and so-called ‘strict physicality’ of global weak solutions of a Beris–Edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials. An important contribution to the dynamics comes from a singular potential introduced by John Ball and Apala Majumdar which replaces the commonly employed Landau-de Gennes bulk potential. This is built into our model to ensure that a natural physical constraint on the eigenvalues of the Q-tensor order parameter is respected by the dynamics of this system. Moreover, by a maximum principle argument, we are able to construct global strong solutions in dimension two.
PubDate: 2015-10-01

• Stable Determination of a Scattered Wave from its Far-Field Pattern: The
High Frequency Asymptotics
• Abstract: Abstract We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order.
PubDate: 2015-10-01

• Face-Centered Cubic Crystallization of Atomistic Configurations
• Abstract: Abstract We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance as it presents a significant milestone towards understanding, on the atomistic level, phenomena such as melting or plastic behavior. We characterize a set of rotation- and translation-invariant two- and three-body potentials V 2, V 3 such that the energy minimum of $$\frac{1}{\#Y}E(Y) = \frac{1}{\# Y} \left(2\sum_{\{y,y'\} \subset Y}V_2(y, y') + 6\sum_{\{y,y',y''\} \subset Y} V_3(y,y',y'')\right)$$ over all $${Y \subset \mathbb{R}^3}$$ , #Y = n, converges to the energy per particle in the face-centered cubic (fcc) lattice as n tends to infinity. The proof involves a careful analysis of the symmetry properties of the fcc lattice.
PubDate: 2015-10-01

• Rigorous Derivation of Nonlinear Scalar Conservation Laws from
Follow-the-Leader Type Models via Many Particle Limit
• Abstract: Abstract We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1–Wasserstein topology (respectively in $${\mathbf{L^{1}_{loc}}}$$ ) to the unique Kružkov entropy solution of the conservation law. The initial data are taken in $${\mathbf{L}^\infty}$$ , nonnegative, and with compact support, hence we are able to handle densities with a vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, for example in the Lighthill-Whitham-Richards model for traffic flow) with a possible degenerate slope near the vacuum state. The proof of the result is based on discrete $${\mathbf{BV}}$$ estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect $${\mathbf{L}^\infty \mapsto \mathbf{BV}}$$ for nonlinear scalar conservation laws is intrinsic to the discrete model.
PubDate: 2015-09-01

• Regularity for a Local–Nonlocal Transmission Problem
• Abstract: Abstract We formulate and study an elliptic transmission-like problem combining local and nonlocal elements. Let $${\mathbb{R}^n}$$ be separated into two components by a smooth hypersurface Γ. On one side of Γ, a function satisfies a local second-order elliptic equation. On the other, it satisfies a nonlocal one of lower order. In addition, a nonlocal transmission condition is imposed across the interface Γ, which has a natural variational interpretation. We deduce the existence of solutions to the corresponding Dirichlet problem, and show that under mild assumptions they are Hölder continuous, following the method of De Giorgi. The principal difficulty stems from the lack of scale invariance near Γ, which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter. We then turn to the question of optimal regularity and qualitative properties of solutions, and show that (in the case of constant coefficients and flat Γ) they satisfy a kind of transmission condition, where the ratio of “fractional conormal derivatives” on the two sides of the interface is fixed. A perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface. Throughout, we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution, which has an interpretation in the context of quasigeostrophic dynamics.
PubDate: 2015-09-01

• Hopf Bifurcation from Fronts in the Cahn–Hilliard Equation
• Abstract: Abstract We study Hopf bifurcation from traveling-front solutions in the Cahn–Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of physical phenomena, including pattern formation in chemical deposition and precipitation processes. Technically, we study bifurcation in the presence of an essential spectrum. We contribute a simple and direct functional analytic method and determine bifurcation coefficients explicitly. Our approach uses exponential weights to recover Fredholm properties and spectral flow ideas to compute Fredholm indices. Simple mass conservation helps compensate for negative indices. We also construct an explicit, prototypical example, prove the existence of a bifurcating front, and determine the direction of bifurcation.
PubDate: 2015-09-01

• Unique Conservative Solutions to a Variational Wave Equation
• Abstract: Abstract Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $${u_{tt} - c (u) (c(u)u_{x}) x = 0}$$ . Given a solution u(t, x), even if the wave speed c(u) is only Hör continuous in the t – x plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X, Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $${u(0, \cdot) \in H^{1}(I\!R), u_{t} (0, \cdot) \in L^{2}(I\!R).}$$
PubDate: 2015-09-01

• DiPerna–Lions Flow for Relativistic Particles in an Electromagnetic
Field
• Abstract: Abstract We show the existence and uniqueness of a DiPerna–Lions flow for relativistic particles subject to a Lorentz force in an electromagnetic field. The electric and magnetic fields solve the linear Maxwell system in the vacuum but for singular initial conditions which are only in the physical energy space. As the corresponding force field is only in L 2, we have to perform a careful analysis of the cancellations over a trajectory.
PubDate: 2015-09-01

• Multi-Value Microstructural Descriptors for Complex Materials: Analysis of
Ground States
• Abstract: Abstract We propose and analyze a multi-scale and multi-field description of complex materials in which we consider every material element as a system composed by $${Q \in \mathbb{N}}$$ indistinct substructures. We pay attention to the equilibrium configurations of such bodies. We consider first rigid bodies with microstructure represented by means of Q-valued maps from the reference place to the manifold of microstructural shapes. In that case, just microenergetics appears. Then, we enlarge the stage considering large strains and energies of Ginzburg–Landau type with respect to the microstructural descriptor fields. In both cases we provide conditions for semicontinuity of the relevant energies and the existence of ground states.
PubDate: 2015-09-01

• Negative Eigenvalues of Two-Dimensional Schrödinger Operators
• Abstract: Abstract We prove a certain upper bound for the number of negative eigenvalues of the Schrödinger operator H = −Δ − V in $${\mathbb{R}^{2}.}$$
PubDate: 2015-09-01

• Existence of Compactly Supported Global Minimisers for the Interaction
Energy
• Abstract: Abstract The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove the existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions.
PubDate: 2015-09-01

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