Authors:Antonio Gaudiello; Olivier Guibé; François Murat Pages: 1 - 64 Abstract: We consider a domain which has the form of a brush in 3D or the form of a comb in 2D, i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one tooth to another and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to \({\varepsilon}\) , and the asymptotic volume fraction of the teeth (as \({\varepsilon}\) tends to zero) is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth. In this domain we study the asymptotic behavior (as \({\varepsilon}\) tends to zero) of the solution of a second order elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is satisfied on the whole of the boundary. First, we revisit the problem where the source term belongs to L 2. This is a classical problem, but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new. Then, we study the case where the source term belongs to L 1. Working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1079-2 Issue No:Vol. 225, No. 1 (2017)

Authors:Paweł Goldstein; Piotr Hajłasz Pages: 65 - 88 Abstract: We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit n-dimensional cube onto itself, whose Jacobian is equal to −1 almost everywhere. Moreover, we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1085-4 Issue No:Vol. 225, No. 1 (2017)

Authors:Henryk Gerlach; Philipp Reiter; Heiko von der Mosel Pages: 89 - 139 Abstract: To describe the behavior of knotted loops of springy wire with an elementary mathematical model we minimize the integral of squared curvature, \({E = \int \varkappa^2}\) , together with a small multiple of ropelength \({\mathcal{R}}\) = length/thickness in order to penalize selfintersection. Our main objective is to characterize all limit configurations of energy minimizers of the total energy \({E_{\vartheta} \equiv E + \vartheta \mathcal{R}}\) as \({\vartheta}\) tends to zero. For short, these limit configurations will be referred to as elastic knots. The elastic unknot turns out to be the once covered circle with squared curvature energy \({(2\pi)^2}\) . For all (non-trivial) knot classes for which the natural lower bound \({(4\pi)^2}\) on E is sharp, the respective elastic knot is the doubly covered circle. We also derive a new characterization of two-bridge torus knots in terms of E, proving that the only knot classes for which the lower bound \({(4\pi)^2}\) on E is sharp are the \({(2,b)}\) -torus knots for odd b with \({ b \ge 3}\) (containing the trefoil knot class). In particular, the elastic trefoil knot is the doubly covered circle. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1100-9 Issue No:Vol. 225, No. 1 (2017)

Authors:Thomas C. Sideris Pages: 141 - 176 Abstract: The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in \({{\rm GL}^+(3, \mathbb{R})}\) . The evolution of the fluid domain is described by a family of ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank r = 1, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along 3−r of its principal axes. In the compressible case, the asymptotic limit has rank r = 3, and asymptotic completeness holds, when the adiabatic index \({\gamma}\) satisfies \({4/3 < \gamma < 2}\) . The number of possible degeneracies, 3−r, increases with the value of the adiabatic index \({\gamma}\) . In the incompressible case, affine motion reduces to geodesic flow in \({{\rm SL}(3, \mathbb{R})}\) with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1106-3 Issue No:Vol. 225, No. 1 (2017)

Authors:Christian Heinemann; Christiane Kraus; Elisabetta Rocca; Riccarda Rossi Pages: 177 - 247 Abstract: In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature concerning phase separation and damage processes in elastic media, in our model we encompass thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More particularly, we prove the existence of “entropic weak solutions”, resorting to a solvability concept first introduced in Feireisl (Comput Math Appl 53:461–490, 2007) in the framework of Fourier–Navier–Stokes systems and then recently employed in Feireisl et al. (Math Methods Appl Sci 32:1345–1369, 2009) and Rocca and Rossi (Math Models Methods Appl Sci 24:1265–1341, 2014) for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1102-7 Issue No:Vol. 225, No. 1 (2017)

Authors:Jonathan J. Bevan Pages: 249 - 285 Abstract: We prove the local Hölder continuity of strong local minimizers of the stored energy functional $$E(u)=\int_{\Omega}\lambda \nabla u ^{2}+h({\rm det} \nabla u)\,{\rm d}x$$ subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as \({s\to 0+}\) , linearly as \({s \to +\infty}\) , and satisfies \({h(s)=+\infty}\) if \({s \leqq 0}\) . These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term \({\int_{\Omega} h({\rm det} \nabla u)\,{\rm d}x}\) can have by analysing the regularity of local minimizers in the class of ‘shear maps’. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1104-5 Issue No:Vol. 225, No. 1 (2017)

Authors:Yasunori Maekawa Pages: 287 - 374 Abstract: We study the stability of some exact stationary solutions to the two-dimensional Navier–Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1105-4 Issue No:Vol. 225, No. 1 (2017)

Authors:Renjun Duan; Feimin Huang; Yong Wang; Tong Yang Pages: 375 - 424 Abstract: The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new \({L^\infty_xL^1_{v}\cap L^\infty_{x,v}}\) approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted \({L^\infty}\) norm under some smallness condition on the \({L^1_xL^\infty_v}\) norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the \({L^\infty_{x,v}}\) norm with explicit rates of convergence are also studied. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1107-2 Issue No:Vol. 225, No. 1 (2017)

Authors:Manuel Friedrich Pages: 425 - 467 Abstract: We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via \({\Gamma}\) -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1108-1 Issue No:Vol. 225, No. 1 (2017)

Authors:Rowan Killip; Tadahiro Oh; Oana Pocovnicu; Monica Vişan Pages: 469 - 548 Abstract: We consider the cubic–quintic nonlinear Schrödinger equation: $$i\partial_t u = -\Delta u - u ^2u + u ^4u.$$ In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region \({{\mathcal{R}}}\) of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology. PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1109-0 Issue No:Vol. 225, No. 1 (2017)

Authors:Dongho Chae; Jörg Wolf Pages: 549 - 572 Abstract: We prove Liouville type theorems for the self-similar solutions to the Navier–Stokes equations. One of our results generalizes the previous ones by Nečas–Ru̇žička–Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier–Stokes equations with the profile belonging to \({L^{p, \infty} (\mathbb{R}^3)}\) with \({p > \frac{3}{2}}\) . PubDate: 2017-07-01 DOI: 10.1007/s00205-017-1110-7 Issue No:Vol. 225, No. 1 (2017)

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

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

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

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

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

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

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

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

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