Authors:Cristian E. Gutiérrez; Ahmad Sabra Pages: 341 - 399 Abstract: We show the existence of a lens, when its lower face is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves a given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper face of the lens satisfies a pde of Monge-Ampère type. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1196-y Issue No:Vol. 228, No. 2 (2018)

Authors:Alessio Figalli; Connor Mooney Pages: 401 - 429 Abstract: A developable cone (“d-cone”) is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance \({\varepsilon}\) . Starting from a nonlinear model depending on the thickness h > 0 of the sheet, we prove a \({\Gamma}\) -convergence result as \({h \rightarrow 0}\) to a fourth-order obstacle problem for curves in \({\mathbb{S}^2}\) . We then describe the exact shape of minimizers of the limit problem when \({\varepsilon}\) is small. In particular, we rigorously justify previous results in the physics literature. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1195-z Issue No:Vol. 228, No. 2 (2018)

Authors:Aifang Qu; Wei Xiang Pages: 431 - 476 Abstract: In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1197-x Issue No:Vol. 228, No. 2 (2018)

Authors:L. Caffarelli; D. De Silva; O. Savin Pages: 477 - 493 Abstract: We prove Lipschitz continuity of solutions to a class of rather general two-phase anisotropic free boundary problems in 2D and we classify global solutions. As a consequence, we obtain \({C^{2,1}}\) regularity of solutions to the Bellman equation in 2D. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1198-9 Issue No:Vol. 228, No. 2 (2018)

Authors:Dominic Breit; Sebastian Schwarzacher Pages: 495 - 562 Abstract: We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies \({\gamma > \frac{12}{7}}\) ( \({\gamma >1 }\) in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and Růžičkaka (Arch Ration Mech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1199-8 Issue No:Vol. 228, No. 2 (2018)

Authors:Luis C. García-Naranjo; James Montaldi Pages: 563 - 602 Abstract: We consider nonholonomic systems with symmetry possessing a certain type of first integral which is linear in the velocities. We develop a systematic method for modifying the standard nonholonomic almost Poisson structure that describes the dynamics so that these integrals become Casimir functions after reduction. This explains a number of recent results on Hamiltonization of nonholonomic systems, and has consequences for the study of relative equilibria in such systems. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1200-6 Issue No:Vol. 228, No. 2 (2018)

Authors:Mitsuo Higaki; Yasunori Maekawa; Yuu Nakahara Pages: 603 - 651 Abstract: We study the two-dimensional stationary Navier–Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of the obstacle and the given exterior force are sufficiently small. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1201-5 Issue No:Vol. 228, No. 2 (2018)

Authors:Johannes Elschner; Guanghui Hu Pages: 653 - 690 Abstract: Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be any number in \({(0,2\pi)\backslash\{\pi\}}\) . We prove that such an obstacle scatters any incoming wave non-trivially (that is, the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1202-4 Issue No:Vol. 228, No. 2 (2018)

Authors:Alessandro Morando; Yuri Trakhinin; Paola Trebeschi Pages: 691 - 742 Abstract: We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional planar flows provided that the Rayleigh–Taylor sign condition \({[\partial p/\partial N] <0 }\) on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis (Morando et al. in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh–Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser iteration. PubDate: 2018-05-01 DOI: 10.1007/s00205-017-1203-3 Issue No:Vol. 228, No. 2 (2018)

Authors:Tam Do; Alexander Kiselev; Lenya Ryzhik; Changhui Tan Pages: 1 - 37 Abstract: We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian \({(-\partial _{xx})^{\alpha/2}, \alpha \in (0, 1)}\) . The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all \({\alpha \in (0, 1)}\) . To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1184-2 Issue No:Vol. 228, No. 1 (2018)

Authors:Hailiang Li; Yi Wang; Tong Yang; Mingying Zhong Pages: 39 - 127 Abstract: The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov–Poisson–Boltzmann (VPB) system. To this end, motivated by the micro–macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133–179, 2004) and Liu et al. (Physica D 188:178–192, 2004), we first set up a new micro–macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665–725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1185-1 Issue No:Vol. 228, No. 1 (2018)

Authors:Roberto Castelli; Marcio Gameiro; Jean-Philippe Lessard Pages: 129 - 157 Abstract: In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton–Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted ℓ1 Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1186-0 Issue No:Vol. 228, No. 1 (2018)

Authors:Blair Davey Pages: 159 - 196 Abstract: The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows. Specifically, we give new proofs of an L 2 Carleman estimate for the heat operator, and the monotonicity formulas for the frequency function associated to the heat operator, the two-phase free boundary problem, the flow of harmonic maps, and the mean curvature flow. The proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory. In particular, each parabolic theorem is proved by taking a high-dimensional limit of the related elliptic result. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1187-z Issue No:Vol. 228, No. 1 (2018)

Authors:Claude Bardos; Edriss S. Titi Pages: 197 - 207 Abstract: The goal of this note is to show that, in a bounded domain \({\Omega \subset \mathbb{R}^n}\) , with \({\partial \Omega\in C^2}\) , any weak solution \({(u(x,t),p(x,t))}\) , of the Euler equations of ideal incompressible fluid in \({\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t}\) , with the impermeability boundary condition \({u\cdot \vec n =0}\) on \({\partial\Omega\times(0,T)}\) , is of constant energy on the interval (0,T), provided the velocity field \({u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))}\) , with \({\alpha > \frac13.}\) PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1189-x Issue No:Vol. 228, No. 1 (2018)

Authors:Gareth E. Roberts Pages: 209 - 236 Abstract: Morse theoretical ideas are applied to the study of relative equilibria in the planar n-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1190-4 Issue No:Vol. 228, No. 1 (2018)

Authors:Diego Chamorro; Pierre-Gilles Lemarié-Rieusset; Kawther Mayoufi Pages: 237 - 277 Abstract: We study the role of the pressure in the partial regularity theory for weak solutions of the Navier–Stokes equations. By introducing the notion of dissipative solutions, due to Duchon and Robert (Nonlinearity 13:249–255, 2000), we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach sheels new light on the role of the pressure in this theory in connection to Serrin’s local regularity criterion. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1191-3 Issue No:Vol. 228, No. 1 (2018)

Authors:Alessio Brancolini; Carolin Rossmanith; Benedikt Wirth Pages: 279 - 308 Abstract: We consider two different variational models of transport networks: the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford–Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives. As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove a lower bound for the network cost that matches a known upper bound (in terms of how the cost scales in the model parameters) which helps better understand optimal networks and their minimal costs. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1192-2 Issue No:Vol. 228, No. 1 (2018)

Authors:Onur Alper Pages: 309 - 339 Abstract: In [2], Hardt, Lin and the author proved that the defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Hölder continuous curves with finitely many crossings. In this article, we show that each Hölder continuous curve in the defect set is of finite length. Hence, locally, the defect set is rectifiable. For the most part, the proof closely follows the work of De Lellis et al. (Rectifiability and upper minkowski bounds for singularities of harmonic q-valued maps, arXiv:1612.01813, 2016) on harmonic \({\mathcal{Q}}\) -valued maps. The blow-up analysis in Alper et al. (Calc Var Partial Differ Equ 56(5):128, 2017) allows us to simplify the covering arguments in [11] and locally estimate the length of line defects in a geometric fashion. PubDate: 2018-04-01 DOI: 10.1007/s00205-017-1193-1 Issue No:Vol. 228, No. 1 (2018)

Authors:Simon Markfelder; Christian Klingenberg Pages: 967 - 994 Abstract: In this paper we consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states, where one state lies in the lower and the other state in the upper half plane. The aim is to investigate whether there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. For some initial states this question has been answered by Feireisl and Kreml (J Hyperbolic Differ Equ 12(3):489–499, 2015), and also Chen and Chen (J Hyperbolic Differ Equ 4(1):105–122, 2007), where there exists a unique entropy solution. For other initial states Chiodaroli and Kreml (Arch Ration Mech Anal 214(3):1019–1049, 2014) and Chiodaroli et al. (Commun Pure Appl Math 68(7):1157–1190, 2015), showed that there are infinitely many entropy solutions. For still other initial states the question on uniqueness remained open and this will be the content of this paper. This paper can be seen as a completion of the aforementioned papers by showing that the solution is non-unique in all cases (except if the solution is smooth). PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1179-z Issue No:Vol. 227, No. 3 (2018)

Authors:Li Li; YanYan Li; Xukai Yan Pages: 1091 - 1163 Abstract: We classify all (−1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier–Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of (−1)-homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of solutions for any point on the other part of the boundary. We also establish asymptotic expansions for every (−1)-homogeneous axisymmetric solutions in a neighborhood of the singular point on the unit sphere. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1181-5 Issue No:Vol. 227, No. 3 (2018)