We give a rigorous proof for the linear stability of the Skyrmion. In addition, we provide new proofs for the existence of the Skyrmion and the GGMT bound.
Due to the complexity of the field equation, there are not many rigorous results on dynamical aspects of the Skyrme model. In @cite_17 , small data global well-posedness and scattering is proved and @cite_25 establishes large-data global well-posedness. There is also some recent activity on the related but simpler Adkins-Nappi model, see e.g. @cite_13 @cite_20 @cite_18 . From a numerical point of view, the linear stability of the Skyrmion is addressed in @cite_6 and @cite_7 studies the nonlinear stability. As far as the method of proof is concerned, we note that our approach is in parts inspired by @cite_24 .
@cite_18 @cite_7 @cite_6 @cite_24 @cite_13 @cite_25 @cite_20 @cite_17
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Abstract We investigate the stability of our recently constructed self-gravitating skyrmions in the framework of small time-dependent perturbations. It is shown that the frequency spectrum for radial perturbations of the coupled Einstein-Skyrme equations can be reduced to the energy spectrum of a p-wave Schrodinger equation with a bounded effective potential, which is determined by the equilibrium solution. Bound states of this Schrodinger equation correspond to exponentially growing modes. It turns out that there are no such modes, as long as the relevant dimensionless coupling constant ( κ ) is less than a critical value, κ c . For larger values of κ there is exactly one unstable mode. Therefore, the Skyrme “stars” become unstable (in the sense of Liapunov) for κ > κ c , in spite of the fact that their topological winding number is conserved. The linear stability for κ κ c (with respect to radial modes) is, of course, only a necessary condition for (non-linear) stability. Similar results are expected to hold for our new black hole solutions of the Einstein-Skyrme system. We consider the dynamics of hedgehog solutions in the @math -dimensional Skyrme model which is an energy-supercritical problem. We introduce a new strategy to prove global wellposedness for arbitrarily large initial data. We study a generalization of energy super-critical wave maps due to Adkins and Nappi that can also be viewed as a simplified version of the Skyrme model. These are maps from 1 + 3 dimensional Minkowski space that take values in the 3-sphere, and it follows that every finite energy Adkins–Nappi wave map has a fixed topological degree which is an integer. Here we initiate the study of the large data dynamics for Adkins–Nappi wave maps by proving that there is no type II blow-up in the class of maps with topological degree zero. In particular, any degree zero map whose critical norm stays bounded must be global-in-time and scatter to zero as ( t ). Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point... Linear perturbations of black hole models by a variety of fields are considered. Perturbing fields include the zero rest mass scalar field in the case of Reissner-Nordstrom, and gravitational, electromagnetic and zero rest mass scalar perturbation in the case of the Kerr model. The analysis deals with the Ψ 0 components (in the Newman-Penrose (1962) formalism) of non-zero spin fields. The symmetry properties of the models are used to derive the crucial condition th at the field be singular on the inner horizon. This condition is independent of the field propagation equation. Initial data are then given in terms of incoming radiation from f - is shown that there exist wellbehaved initial data sets for which the resultant fields are singular on the inner horizon. It is emphasized that this instability result is dependent only on the global symmetries and causal structure of the models considered, and is independent of the precise nature of the perturbing field. This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation ( g =0 ) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon ( H ^ + ). We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon ( H ^ + ). A hierarchy of conservations laws on degenerate horizons is also derived. This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein–Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L±, satisfy the gap property, at least over the radial functions. This means that the interval (0, 1] does not contain any eigenvalues of L± and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the centre-stable manifold associated with these states. This paper therefore provides the final instalment in the proof of this scattering property for the cubic Klein–Gordon equation in the radial case, see the recent theory of Nakanishi and the third author. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations. We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc. Abstract Global radiation measured on fixed-tilt, south-facing planes (40° and vertical) and a 2-axis tracker at NREL’s Solar Radiation Research Lab. in Golden, CO is compared to predictions from ten transposition models, in combination with either optimal or suboptimal input data of horizontal irradiance. Suboptimal inputs are typically used in everyday engineering calculations, for which the necessary data are usually unavailable for the site under scrutiny, and must be estimated in some way. The performance of the transposition models is first evaluated for ideal conditions when optimal data are used. In this specific case, it is found that the Gueymard and Perez models provide the best estimates of global tilted irradiance under clear skies in particular. The performance of four direct diffuse separation models is also evaluated. Their predictions of direct and diffuse radiation appear biased in most cases, with a model-dependent magnitude. Finally, the performance of the resulting combinations of separation and transposition models is analyzed in a variety of situations. When only global irradiance is known, the accuracy of the tilted irradiance predictions degrades significantly, and is mainly conditioned by the local performance of the direct diffuse separation method. For the south-facing vertical surface, inaccuracies in the ground reflection calculations becomes another key factor and significantly increase the prediction error. The Reindl transposition algorithm appears to perform best in this case. When using suboptimal input data for the prediction of plane-of-array irradiance on a moderately tilted plane (40°S) or a 2-axis tracking plane, the Hay, Reindl and Skartveit models are less penalized than others and tend to perform better. It is concluded that further research should be conducted to improve the overall process of predicting irradiance on tilted planes in realistic situations where no local high-quality irradiance or albedo measurements are available. In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method. The stability of the inner Reissner-Nordstroem geometry is studied with test massless integer-spin fields. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate most of the results. Monochromatic waves which are of small amplitude and ingoing near the outer horizon develop infinite energy densities near the inner Cauchy horizon (as measured by a freely falling observer). Previous work has shown that certain derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon (r = r sub - ) if energy densities are to remain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past null infinity (I sup - ), cannot be localized near r sub + , the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the black hole. Price, Sibgatullin, and Alekseev have shown that such moments are radiated away as ''tails'' which travel outward and are rescattered inward yielding a wave field with a time dependence t sup -p , p > 0. This decay in time is sufficiently slow that themore » tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with the time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. have further shown that even perturbations which are localized as they cross the outer horizon produce singularities at the inner horizon. It is shown that this singularity arises when the incoming radiation is first scattered just inside the outer horizon« less It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner–Nordstrom data for the Einstein–Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably; in fact, it cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner–Nordstrom, there is no spacelike component of either the future or the past singularity. We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation ( g =0 ) on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ0 crossing the future event horizon ( H ^ + ) . We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon ( H ^ + ) . The fundamental new aspect of this problem is the degeneracy of the redshift on ( H ^ + ) . Several new analytical features of degenerate horizons are also presented. We prove sharp pointwise t−3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t−4, and t−6, respectively. We proceed by decomposition into angular momentum l and summation of the decay estimates on the Regge-Wheeler equation for fixed l. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in l is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials. We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+?j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I?I?Rd,f?TdI?I?Rd,f?Td, p,q?Rnp,q?Rn, t?T1t?T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I?I?RdI?I?Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in and A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price’s law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.’s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price’s law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou’s C0-formulation, the conjecture is proven to be false. This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime.