Authors:Connor Mooney Pages: 1039 - 1055 Abstract: Abstract We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1052-5 Issue No:Vol. 223, No. 3 (2017)

Authors:Yan Guo; Lijia Han; Jingjun Zhang Pages: 1057 - 1121 Abstract: Abstract It is shown that smooth solutions with small amplitude to the one dimensional Euler–Poisson system for electrons persist forever with no shock formation. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1053-4 Issue No:Vol. 223, No. 3 (2017)

Authors:Alexander Lytchak; Stefan Wenger Pages: 1123 - 1182 Abstract: Abstract We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1054-3 Issue No:Vol. 223, No. 3 (2017)

Authors:Enzo Vitillaro Pages: 1183 - 1237 Abstract: Abstract The aim of this paper is to study the problem $$\left\{\begin{array}{ll} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \quad & {\rm in} \, (0,\infty)\times\Omega, \\ u=0 & {\rm on} \, (0,\infty)\times \Gamma_0, \\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t)=g(x,u)\quad & {\rm on} \, (0,\infty)\times \Gamma_1,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & {\rm in} \, \overline \Omega, \end{array}\right.$$ where \({\Omega}\) is a open bounded subset of \({{\mathbb R}^N}\) with C 1 boundary ( \({N \ge 2}\) ), \({\Gamma = \partial\Omega}\) , \({(\Gamma_{0},\Gamma_{1})}\) is a measurable partition of \({\Gamma}\) , \({\Delta_{\Gamma}}\) denotes the Laplace–Beltrami operator on \({\Gamma}\) , \({\nu}\) is the outward normal to \({\Omega}\) , and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear subcritical perturbations. In the paper a local Hadamard well-posedness result for initial data in the natural energy space associated to the problem is given. Moreover, when \({\Omega}\) is C 2 and \({\overline{\Gamma_{0}} \cap \overline{\Gamma_{1}} = \emptyset}\) , the regularity of solutions is studied. Next a blow-up theorem is given when P and Q are linear and f and g are superlinear sources. Finally a dynamical system is generated when the source parts of f and g are at most linear at infinity, or they are dominated by the damping terms. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1055-2 Issue No:Vol. 223, No. 3 (2017)

Authors:Grégoire Nadin; Luca Rossi Pages: 1239 - 1267 Abstract: Abstract This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When our hypothesis is no longer satisfied, such generalized transition fronts still exist for an interval of average speeds, with explicit bounds. Our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1056-1 Issue No:Vol. 223, No. 3 (2017)

Authors:Alberto Boscaggin; Walter Dambrosio; Susanna Terracini Pages: 1269 - 1306 Abstract: Abstract For the N-centre problem in the three dimensional space, $${\ddot{x}} = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in {\mathbb{R}}^3 {\setminus} \{c_1,\ldots,c_N\},$$ where \({N \geqq 2}\) , \({m_i > 0}\) and \({\alpha \in [1,2)}\) , we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min–max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1057-0 Issue No:Vol. 223, No. 3 (2017)

Authors:Christophe Lacave; Takéo Takahashi Pages: 1307 - 1335 Abstract: Abstract We consider a single disk moving under the influence of a two dimensional viscous fluid and we study the asymptotic as the size of the solid tends to zero. If the density of the solid is independent of \({\varepsilon}\) , the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal L p −L q decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, we will deduce the convergence to the solution of the Navier–Stokes equations in \({\mathbb{R}^{2}}\) . PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1058-z Issue No:Vol. 223, No. 3 (2017)

Authors:Sergio Conti; Matteo Focardi; Flaviana Iurlano Pages: 1337 - 1374 Abstract: Abstract We prove an integral representation result for functionals with growth conditions which give coercivity on the space \({SBD^p(\Omega)}\) , for \({\Omega\subset\mathbb{R}^{2}}\) , which is a bounded open Lipschitz set, and \({p\in(1,\infty)}\) . The space SBD p of functions whose distributional strain is the sum of an L p part and a bounded measure supported on a set of finite \({\mathcal{H}^{1}}\) -dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W 1,p functions. We also obtain a generalization of Korn’s inequality in the SBD p setting. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1059-y Issue No:Vol. 223, No. 3 (2017)

Authors:Eduard Feireisl; Piotr Gwiazda; Agnieszka Świerczewska-Gwiazda; Emil Wiedemann Pages: 1375 - 1395 Abstract: Abstract We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by Constantin et al. for the homogeneous incompressible Euler equations. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1060-5 Issue No:Vol. 223, No. 3 (2017)

Authors:Seung-Yeal Ha; Tommaso Ruggeri Pages: 1397 - 1425 Abstract: Abstract We present a thermodynamically consistent particle (TCP) model motivated by the theory of multi-temperature mixture of fluids in the case of spatially homogeneous processes. The proposed model incorporates the Cucker-Smale (C-S) type flocking model as its isothermal approximation. However, it is more complex than the C-S model, because the mutual interactions are not only “mechanical” but are also affected by the “temperature effect” as individual particles may exhibit distinct internal energies. We develop a framework for asymptotic weak and strong flocking in the context of the proposed model. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1062-3 Issue No:Vol. 223, No. 3 (2017)

Authors:Jan Giesselmann; Corrado Lattanzio; Athanasios E. Tzavaras Pages: 1427 - 1484 Abstract: Abstract We consider a Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and we derive a relative energy identity. When applied to specific energies, this yields relative energy identities for the Euler–Korteweg, the Euler–Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler–Korteweg system. For the Euler–Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier–Stokes–Korteweg system (NSK) with non-monotone pressure laws, and prove stability for the NSK system via a modified relative energy approach. We prove the continuous dependence of solutions on initial data and the convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, compensated by higher-order gradients. PubDate: 2017-03-01 DOI: 10.1007/s00205-016-1063-2 Issue No:Vol. 223, No. 3 (2017)

Authors:Giacomo Canevari Pages: 591 - 676 Abstract: Abstract We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set \({\mathscr{S}_{\rm line}}\) of finite length, such that minimizers converge to a locally harmonic map away from \({\mathscr{S}_{\rm line}}\) . Moreover, \({\mathscr{S}_{\rm line}}\) restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1040-9 Issue No:Vol. 223, No. 2 (2017)

Authors:Patrick W. Dondl; Antoine Lemenant; Stephan Wojtowytsch Pages: 693 - 736 Abstract: Abstract This article is concerned with the problem of minimising the Willmore energy in the class of connected surfaces with prescribed area which are confined to a small container. We propose a phase field approximation based on De Giorgi’s diffuse Willmore functional to this variational problem. Our main contribution is a penalisation term which ensures connectedness in the sharp interface limit. The penalisation of disconnectedness is based on a geodesic distance chosen to be small between two points that lie on the same connected component of the transition layer of the phase field. We prove that in two dimensions, sequences of phase fields with uniformly bounded diffuse Willmore energy and diffuse area converge uniformly to the zeros of a double-well potential away from the support of a limiting measure. In three dimensions, we show that they converge \({\mathcal{H}^1}\) -almost everywhere on curves. This enables us to show \({\Gamma}\) -convergence to a sharp interface problem that only allows for connected structures. The results also imply Hausdorff convergence of the level sets in two dimensions and a similar result in three dimensions. Furthermore, we present numerical evidence of the effectiveness of our model. The implementation relies on a coupling of Dijkstra’s algorithm in order to compute the topological penalty to a finite element approach for the Willmore term. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1043-6 Issue No:Vol. 223, No. 2 (2017)

Authors:Mouhamed Moustapha Fall; Ignace Aristide Minlend; Tobias Weth Pages: 737 - 759 Abstract: Abstract We study the existence of nontrivial unbounded domains \({\Omega}\) in \({{\mathbb R}^{N}}\) such that the overdetermined problem $${-\Delta u = 1 \quad {\rm in} \, \Omega}, \quad u = 0, \quad \partial_{\nu} u = {\rm const} \quad {\rm on} \partial \Omega$$ admits a solution u. By this, we complement Serrin’s classification result from 1971, which yields that every bounded domain admitting a solution of the above problem is a ball in \({{\mathbb R}^{N}}\) . The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1044-5 Issue No:Vol. 223, No. 2 (2017)

Authors:Hui Chen; Daoyuan Fang; Ting Zhang Pages: 817 - 843 Abstract: Abstract In this paper, we investigate the global well-posedness for the three dimensional inhomogeneous incompressible Navier–Stokes system with axisymmetric initial data. We obtain the global existence and uniqueness of the axisymmetric solution provided that $$\left\ \frac{a_{0}}{r}\right\ _{\infty} {\rm and}\ u_{0}^{\theta}\ _{3} {\rm are sufficiently small}.$$ Furthermore, if \({u_0 \in L^{1}}\) and \({ru^{\theta}_{0}\in L^{1} \cap L^{2}}\) , we have the decay estimate $$\begin{aligned} \ u^{\theta}(t)\ _{2}^{2} + \langle t \rangle \ \nabla(u^{\theta}e_{\theta})(t)\ _{2}^{2} + t\langle t \rangle(\ u_{t}^{\theta}(t)\ _{2}^{2} + \ \Delta(u^{\theta}e_{\theta})(t)\ _{2}^{2}) \leqq C \langle t\rangle^{-\frac{5}{2}}, \\ \quad \forall t > 0. \end{aligned}$$ PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1046-3 Issue No:Vol. 223, No. 2 (2017)

Authors:Hind Al Baba; Chérif Amrouche; Miguel Escobedo Pages: 881 - 940 Abstract: Abstract In this article we consider the Stokes problem with Navier-type boundary conditions on a domain \({\Omega}\) , not necessarily simply connected. Since, under these conditions, the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypotheses on the initial data u 0 and external force f. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers, we deduce maximal \({L^{p}-L^{q}}\) regularity for the inhomogeneous Stokes problem. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1048-1 Issue No:Vol. 223, No. 2 (2017)

Authors:Roberto Castelli Pages: 941 - 975 Abstract: Abstract This work concerns the planar \({N}\) -center problem with homogeneous potential of degree \({-\alpha}\) ( \({\alpha\in[1,2)}\) ). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive \({h}\) the existence of a classical periodic solution with energy \({h}\) for the \({N}\) -center problem with \({\alpha\in (1,2)}\) is proven. In case \({\alpha=1}\) a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1049-0 Issue No:Vol. 223, No. 2 (2017)

Authors:Andrea Braides; Marco Cicalese Pages: 977 - 1017 Abstract: Abstract We introduce a class of n-dimensional (possibly inhomogeneous) spin-like lattice systems presenting modulated phases with possibly different textures. Such systems can be parameterized according to the number of ground states, and can be described by a phase-transition energy which we compute by means of variational techniques. Degeneracies due to frustration are also discussed. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1050-7 Issue No:Vol. 223, No. 2 (2017)

Authors:Karl-Mikael Perfekt; Mihai Putinar Pages: 1019 - 1033 Abstract: Abstract Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors–Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors–Beurling transform and the Neumann–Poincaré operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann–Poincaré operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field. PubDate: 2017-02-01 DOI: 10.1007/s00205-016-1051-6 Issue No:Vol. 223, No. 2 (2017)