Authors:Jean-Michel Coron; Hoai-Minh Nguyen Pages: 993 - 1023 Abstract: Abstract Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1119-y Issue No:Vol. 225, No. 3 (2017)

Authors:Ana Cristina Barroso; José Matias; Marco Morandotti; David R. Owen Pages: 1025 - 1072 Abstract: Abstract Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1120-5 Issue No:Vol. 225, No. 3 (2017)

Authors:Cheng Yu Pages: 1073 - 1087 Abstract: Abstract In this paper, we prove the energy conservation for the weak solutions of the compressible Navier–Stokes equations for any time t > 0, under certain conditions. The results hold for the renormalized solutions of the equations with constant viscosities, as well as the weak solutions of the equations with degenerate viscosity. Our conditions do not depend on the dimensions. The energy may be conserved on the vacuum for the compressible Navier–Stokes equations with constant viscosities. Our results are the first ones on energy conservation for the weak solutions of the compressible Navier–Stokes equations. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1121-4 Issue No:Vol. 225, No. 3 (2017)

Authors:N. V. Krylov; E. Priola Pages: 1089 - 1126 Abstract: Abstract We show, among other things, how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on the time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other methods are available at this time and it is a very challenging problem to find a purely analytical approach to proving such results. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1122-3 Issue No:Vol. 225, No. 3 (2017)

Authors:Gang Bao; Hai Zhang Pages: 1127 - 1160 Abstract: Abstract Let g be a Riemannian metric for \({\mathbb{R}^d}\) ( \({d \geqq 3}\) ) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain M. The lens rigidity problem is concerned with recovering the metric g inside M from the corresponding lens relation on the boundary \({\partial M}\) . In this paper, the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non-trapping metric satisfying the “strong fold-regular” condition. A metric g is called strong fold-regular if for each point \({x\in M}\) , there exists a set of geodesics passing through x whose conormal bundle covers \({T^*_{x}M}\) . Moreover, these geodesics contain either no conjugate points or only fold conjugate points with a non-degeneracy condition. Examples of strong fold-regular metrics are constructed. Our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points. The approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow, which is a weighted geodesic X-ray transform problem for symmetric 2-tensor fields. A key ingredient is to show that the kernel of the X-ray transform on symmetric solenoidal 2-tensor fields is of finite dimension. It remains open whether the kernel space is trivial or not. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1123-2 Issue No:Vol. 225, No. 3 (2017)

Authors:Paolo Antonelli; Stefano Spirito Pages: 1161 - 1199 Abstract: Abstract In this paper we consider the Quantum Navier–Stokes system both in two and in three space dimensions and prove the global existence of finite energy weak solutions for large initial data. In particular, the notion of weak solutions is the standard one. This means that the vacuum region is included in the weak formulation. In particular, no extra terms like damping or cold pressure are added to the system in order to define the velocity field in the vacuum region. The main contribution of this paper is the construction of a regular approximating system consistent with the effective velocity transformation needed to get the necessary a priori estimates. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1124-1 Issue No:Vol. 225, No. 3 (2017)

Authors:Dustin Lazarovici; Peter Pickl Pages: 1201 - 1231 Abstract: Abstract We present a probabilistic proof of the mean field limit and propagation of chaos N-particle systems in three dimensions with positive (Coulomb) or negative (Newton) 1/r potentials scaling like 1/N and an N-dependent cut-off which scales like \({N^{-1/3+ \epsilon}}\) . In particular, for typical initial data, we show convergence of the empirical distributions to solutions of the Vlasov–Poisson system with either repulsive electrical or attractive gravitational interactions. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1125-0 Issue No:Vol. 225, No. 3 (2017)

Authors:Guy Bouchitté; Christophe Bourel; Didier Felbacq Pages: 1233 - 1277 Abstract: Abstract It is now well established that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a (possibly) negative frequency-dependent effective permeability. However this result based on a two-dimensional approach holds merely in the case of linearly polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this paper we consider a dielectric structure placed in a bounded domain of \({\mathbb{R}^3}\) and perform a full three dimensional asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We give evidence of a vectorial spectral problem on the periodic cell which determines micro-resonances and encodes the oscillating behavior of the magnetic field from which artificial magnetism arises. At a macroscopic level we deduce an effective permeability tensor that we can make explicit as a function of the frequency. As far as sign-changing permeability is sought after, we may foresee that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. Part of these results have been announced in Bouchitté et al. (C R Math Acad Sci Paris 347(9–10):571–576, 2009). PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1132-1 Issue No:Vol. 225, No. 3 (2017)

Authors:Mihaela Ifrim; Daniel Tataru Pages: 1279 - 1346 Abstract: Abstract This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position–velocity potential holomorphic coordinates, and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions. PubDate: 2017-09-01 DOI: 10.1007/s00205-017-1126-z Issue No:Vol. 225, No. 3 (2017)

Authors:Tarek M. Elgindi Pages: 573 - 599 Abstract: Abstract We study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation (IPM). We show that solutions which are near certain stable stationary solutions must converge as t → ∞ to a stationary solution of the IPM equation. It turns out that linearizing the IPM equation about certain stable stationary solutions gives a non-local partial damping mechanism. On the torus, the linearized problem has a very large set of stationary (undamped) modes. This makes the problem of long-time behavior more difficult since there is the possibility of a cascading non-linear growth along the stationary modes of the linearized problem. We solve this by, more or less, doing a second linearization around the undamped modes, exploiting a special non-linear structure, and showing that the stationary modes can be controlled. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1090-7 Issue No:Vol. 225, No. 2 (2017)

Authors:Jean-Marie Barbaroux; Dirk Hundertmark; Tobias Ried; Semjon Vugalter Pages: 601 - 661 Abstract: Abstract It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties to the fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties to the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in \({L^1_2(\mathbb{R}^d)\cap L {\rm log} L(\mathbb{R}^d)}\) , i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1101-8 Issue No:Vol. 225, No. 2 (2017)

Authors:Meng Wang; Wendong Wang; Zhifei Zhang Pages: 663 - 683 Abstract: Abstract In this paper, we consider the solutions of the relaxed Q-tensor flow in \({\mathbb{R}^3}\) with small parameter \({\epsilon}\) . We show that the limiting map is the so-called harmonic map flow. As a consequence, we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in [18, 23], where the Ginzburg–Landau approximation approach was used. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1111-6 Issue No:Vol. 225, No. 2 (2017)

Authors:Peter W. Bates; Giorgio Fusco; Panayotis Smyrnelis Pages: 685 - 715 Abstract: We present a systematic study of entire symmetric solutions \({u : \mathbb{R}^n \rightarrow\mathbb{R}^m}\) of the vector Allen–Cahn equation $$\Delta u - W_u(u) = 0 \quad\text{for all}\quad x \in \mathbb{R}^n,$$ where \({W:\mathbb{R}^m \rightarrow \mathbb{R}}\) is smooth, symmetric, nonnegative with a finite number of zeros, and where \({ W_u= (\partial W / \partial u_1,\dots,\partial W / \partial u_m)^{\top}}\) . We introduce a general notion of equivariance with respect to a homomorphism \({f:G\rightarrow\Gamma}\) ( \({G, \Gamma}\) reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen–Cahn equation and on a pointwise estimate for vector minimizers. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1112-5 Issue No:Vol. 225, No. 2 (2017)

Authors:Lisa Beck; Miroslav Bulíček; Josef Málek; Endre Süli Pages: 717 - 769 Abstract: Abstract We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic radial structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously to the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1113-4 Issue No:Vol. 225, No. 2 (2017)

Authors:Yu Deng; Alexandru D. Ionescu; Benoit Pausader Pages: 771 - 871 Abstract: Abstract A basic model for describing plasma dynamics is given by the Euler–Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the “one-fluid” Euler–Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background. In 2 dimensions our global solutions have relatively slow (strictly less than 1/t) pointwise decay and the system has a large (codimension 1) set of quadratic time resonances. The issue in such a situation is to solve the “division problem”. To control the solutions we use a combination of improved energy estimates in the Fourier space, an L 2 bound on an oscillatory integral operator, and Fourier analysis of the Duhamel formula. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1114-3 Issue No:Vol. 225, No. 2 (2017)

Authors:Juhi Jang; Tetu Makino Pages: 873 - 900 Abstract: Abstract We construct stationary axisymmetric solutions of the Euler–Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent \({\gamma}\) belongs to \({(\frac65,\frac32]}\) . The problem is formulated as a nonlinear integral equation, and is solved by an iteration technique. By this method, not only do we get the existence, but we also clarify properties of the solutions such as the physical vacuum condition and the oblateness of the star surface. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1115-2 Issue No:Vol. 225, No. 2 (2017)

Authors:Guowei Yu Pages: 901 - 935 Abstract: Abstract In the N-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian N-body problem with equal masses, N ≧ 3, there are at least 2 N-3 + 2[(N-3)/2] different main simple choreographies. This confirms a conjecture given by Chenciner et al. (Geometry, mechanics, and dynamics. Springer, New York, pp 287–308, 2002). All the simple choreoagraphies we prove belong to the linear chain family. PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1116-1 Issue No:Vol. 225, No. 2 (2017)

Authors:Scott Armstrong; Jessica Lin Pages: 937 - 991 Abstract: Abstract We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto (Ann Probab 39(3):779–856, 2011; Ann Appl Probab 22(1):1–28, 2012) for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green’s functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a C 1,1 regularity theory down to microscopic scale, which is of independent interest and is inspired by the C 0,1 theory introduced in the divergence form case by the first author and Smart (Ann Sci Éc Norm Supér (4) 49(2):423–481, 2016). PubDate: 2017-08-01 DOI: 10.1007/s00205-017-1118-z Issue No:Vol. 225, No. 2 (2017)

Authors:Ken Abe Abstract: Abstract We prove global well-posedness of the two-dimensional exterior Navier–Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers. PubDate: 2017-07-31 DOI: 10.1007/s00205-017-1157-5

Authors:Christophe Gomez; Olivier Pinaud Abstract: Abstract This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence. PubDate: 2017-07-28 DOI: 10.1007/s00205-017-1150-z