Authors:F. Q. Nazar; C. Ortner Pages: 817 - 870 Abstract: Abstract We establish a pointwise stability estimate for the Thomas–Fermi–von Weiz-säcker (TFW) model, which demonstrates that a local perturbation of a nuclear arrangement results also in a local response in the electron density and electrostatic potential. The proof adapts the arguments for existence and uniqueness of solutions to the TFW equations in the thermodynamic limit by Catto et al. (The mathematical theory of thermodynamic limits: Thomas–Fermi type models. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, New York, 1998). To demonstrate the utility of this combined locality and stability result we derive several consequences, including an exponential convergence rate for the thermodynamic limit, partition of total energy into exponentially localised site energies (and consequently, exponential locality of forces), and generalised and strengthened results on the charge neutrality of local defects. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1075-6 Issue No:Vol. 224, No. 3 (2017)

Authors:Jean-Yves Chemin; Ping Zhang; Zhifei Zhang Pages: 871 - 905 Abstract: Abstract Let us consider initial data \({v_0}\) for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to \({L^{\frac 32}\cap L^2}\) . We prove that if the solution associated with \({v_0}\) blows up at a finite time \({T^\star}\) , then for any p in \({]4,\infty[}\) , and any unit vector e of \({\mathbb{R}^3}\) , the L p norm in time with value in \({\dot{H}^{\frac 12+\frac 2 p }}\) of \({(v e)_{\mathbb{R}^3}}\) blows up at \({T^\star}\) . PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1089-0 Issue No:Vol. 224, No. 3 (2017)

Authors:Julian Braun Pages: 907 - 953 Abstract: Abstract We prove the long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the Cauchy–Born rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic Gårding inequality. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1091-6 Issue No:Vol. 224, No. 3 (2017)

Authors:Adam M. Oberman; Yuanlong Ruan Pages: 955 - 984 Abstract: Abstract A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn–Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1092-5 Issue No:Vol. 224, No. 3 (2017)

Authors:Heiner Olbermann Pages: 985 - 1019 Abstract: Abstract We consider a single disclination in a thin elastic sheet of thickness h. We prove ansatz-free lower bounds for the free elastic energy in three different settings: first, for a geometrically fully non-linear plate model; second, for three-dimensional nonlinear elasticity; and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of h. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1093-4 Issue No:Vol. 224, No. 3 (2017)

Authors:Vu Hoang; Maria Radosz Pages: 1021 - 1036 Abstract: Abstract Córdoba et al. (Ann Math 162(3):1377–1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the Córdoba–Córdoba–Fontelos equation, and prove that either a cusp- or needle-like singularity forms in finite time. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1094-3 Issue No:Vol. 224, No. 3 (2017)

Authors:Cecilia Cavaterra; Elisabetta Rocca; Hao Wu Pages: 1037 - 1086 Abstract: Abstract In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen–Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg–Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1095-2 Issue No:Vol. 224, No. 3 (2017)

Authors:Guido De Philippis; Filip Rindler Pages: 1087 - 1125 Abstract: Abstract This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer–Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The “local” proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti’s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1096-1 Issue No:Vol. 224, No. 3 (2017)

Authors:Tobias Ramming; Gerhard Rein Pages: 1127 - 1159 Abstract: Abstract We consider spherically symmetric steady states of the Vlasov–Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley–Dickens, and related models this mass–radius relation takes the form of a spiral. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1098-z Issue No:Vol. 224, No. 3 (2017)

Authors:Jacob Bedrossian; Michele Coti Zelati Pages: 1161 - 1204 Abstract: Abstract We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic \({\mathbb{T}^2}\) setting and the case of a bounded channel \({\mathbb{T} \times [0,1]}\) with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity \({\nu\to 0}\) ), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in \({H^1}\) in the limit \({\nu \to 0}\) , we show that viscous invariant measures still converge to a unique inviscid measure. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1099-y Issue No:Vol. 224, No. 3 (2017)

Authors:Jun Geng; Zhongwei Shen; Liang Song Pages: 1205 - 1236 Abstract: Abstract This paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with L 2 boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in Shen (Anal PDE, arXiv:1505.00694v2). PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1103-6 Issue No:Vol. 224, No. 3 (2017)

Authors:Kleber Carrapatoso; Laurent Desvillettes; Lingbing He Pages: 381 - 420 Abstract: Abstract This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation D(f) by weighted relative Fisher information of f with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani’s conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at-most linearly growing in-time rate. As an application of these estimates, we show the convergence of any (H- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher L 1-moment. More precisely, if the initial data have some (large enough) polynomial L 1-moment, then we obtain an algebraic decay. If the initial data have a stretched exponential L 1-moment, then we recover a stretched exponential decay. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1078-3 Issue No:Vol. 224, No. 2 (2017)

Authors:Sameer Iyer Pages: 421 - 469 Abstract: Abstract This paper concerns the validity of the Prandtl boundary layer theory for steady, incompressible Navier-Stokes flows over a rotating disk. We prove that the Navier-Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid limit. In so doing, we develop a new set of function spaces and prove several embedding theorems which capture the interaction between the Prandtl scaling and the geometry of our domain. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1080-9 Issue No:Vol. 224, No. 2 (2017)

Authors:Sara Daneri; László Székelyhidi Pages: 471 - 514 Abstract: Abstract In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. We prove that the set of Hölder \({1/5 - \varepsilon}\) wild initial data is dense in \({L^{2}}\) , where we call an initial datum wild if it admits infinitely many admissible Hölder \({1/5 - \varepsilon}\) weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows in order to show that a general form of the h-principle applies to Hölder-continuous weak solutions of the Euler equations. Our result indicates that in a deterministic theory of three dimensional turbulence the Reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1081-8 Issue No:Vol. 224, No. 2 (2017)

Authors:Chengchun Hao Pages: 515 - 553 Abstract: Abstract For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The L 2-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1082-7 Issue No:Vol. 224, No. 2 (2017)

Authors:Chao Wang; Yuxi Wang; Zhifei Zhang Pages: 555 - 595 Abstract: Abstract In this paper, we consider the zero-viscosity limit of the Navier–Stokes equations in a half space with non-slip boundary condition. Based on the vorticity formulation and the use of conormal derivative, we develop an energy method to justify the zero-viscosity limit for the analytic data. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1083-6 Issue No:Vol. 224, No. 2 (2017)

Authors:Habib Ammari; Pierre Millien; Matias Ruiz; Hai Zhang Pages: 597 - 658 Abstract: Abstract Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is fourfold: (1) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (2) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle; (3) to derive optimal bounds on the enhancement factors; (4) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1084-5 Issue No:Vol. 224, No. 2 (2017)

Authors:Lorenzo Bertini; Paolo Buttà; Adriano Pisante Pages: 659 - 707 Abstract: Abstract Consider the Allen–Cahn equation on the d-dimensional torus, d = 2, 3, in the sharp interface limit. As is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen–Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviations upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1086-3 Issue No:Vol. 224, No. 2 (2017)

Authors:Marco Barchiesi; Duvan Henao; Carlos Mora-Corral Pages: 743 - 816 Abstract: Abstract We define a class of deformations in \({W^{1,p}(\Omega,\mathbb{R}^n)}\) , \({p > n-1}\) , with a positive Jacobian, that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality Det = det (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in \({W^{1,p}}\) , and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity. PubDate: 2017-05-01 DOI: 10.1007/s00205-017-1088-1 Issue No:Vol. 224, No. 2 (2017)