Authors:Antonio Gaudiello; Olivier Guibé; François Murat Pages: 1 - 64 Abstract: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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)

Authors:F. Q. Nazar; C. Ortner Pages: 817 - 870 Abstract: Abstract We establish a pointwise stability estimate for the Thomas–Fermi–von Weiz-säcker (TFW) model, which demonstrates that a local perturbation of a nuclear arrangement results also in a local response in the electron density and electrostatic potential. The proof adapts the arguments for existence and uniqueness of solutions to the TFW equations in the thermodynamic limit by Catto et al. (The mathematical theory of thermodynamic limits: Thomas–Fermi type models. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, New York, 1998). To demonstrate the utility of this combined locality and stability result we derive several consequences, including an exponential convergence rate for the thermodynamic limit, partition of total energy into exponentially localised site energies (and consequently, exponential locality of forces), and generalised and strengthened results on the charge neutrality of local defects. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1075-6 Issue No:Vol. 224, No. 3 (2017)

Authors:Jean-Yves Chemin; Ping Zhang; Zhifei Zhang Pages: 871 - 905 Abstract: Abstract Let us consider initial data \({v_0}\) for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to \({L^{\frac 32}\cap L^2}\) . We prove that if the solution associated with \({v_0}\) blows up at a finite time \({T^\star}\) , then for any p in \({]4,\infty[}\) , and any unit vector e of \({\mathbb{R}^3}\) , the L p norm in time with value in \({\dot{H}^{\frac 12+\frac 2 p }}\) of \({(v e)_{\mathbb{R}^3}}\) blows up at \({T^\star}\) . PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1089-0 Issue No:Vol. 224, No. 3 (2017)

Authors:Adam M. Oberman; Yuanlong Ruan Pages: 955 - 984 Abstract: Abstract A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn–Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1092-5 Issue No:Vol. 224, No. 3 (2017)

Authors:Heiner Olbermann Pages: 985 - 1019 Abstract: Abstract We consider a single disclination in a thin elastic sheet of thickness h. We prove ansatz-free lower bounds for the free elastic energy in three different settings: first, for a geometrically fully non-linear plate model; second, for three-dimensional nonlinear elasticity; and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of h. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1093-4 Issue No:Vol. 224, No. 3 (2017)

Authors:Vu Hoang; Maria Radosz Pages: 1021 - 1036 Abstract: Abstract Córdoba et al. (Ann Math 162(3):1377–1389, 2005) introduced a nonlocal active scalar equation as a one-dimensional analogue of the surface-quasigeostrophic equation. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the Córdoba–Córdoba–Fontelos equation, and prove that either a cusp- or needle-like singularity forms in finite time. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1094-3 Issue No:Vol. 224, No. 3 (2017)

Authors:Cecilia Cavaterra; Elisabetta Rocca; Hao Wu Pages: 1037 - 1086 Abstract: Abstract In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen–Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg–Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1095-2 Issue No:Vol. 224, No. 3 (2017)

Authors:Tobias Ramming; Gerhard Rein Pages: 1127 - 1159 Abstract: Abstract We consider spherically symmetric steady states of the Vlasov–Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley–Dickens, and related models this mass–radius relation takes the form of a spiral. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1098-z Issue No:Vol. 224, No. 3 (2017)

Authors:Jacob Bedrossian; Michele Coti Zelati Pages: 1161 - 1204 Abstract: Abstract We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic \({\mathbb{T}^2}\) setting and the case of a bounded channel \({\mathbb{T} \times [0,1]}\) with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity \({\nu\to 0}\) ), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in \({H^1}\) in the limit \({\nu \to 0}\) , we show that viscous invariant measures still converge to a unique inviscid measure. PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1099-y Issue No:Vol. 224, No. 3 (2017)

Authors:Jun Geng; Zhongwei Shen; Liang Song Pages: 1205 - 1236 Abstract: Abstract This paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with L 2 boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in Shen (Anal PDE, arXiv:1505.00694v2). PubDate: 2017-06-01 DOI: 10.1007/s00205-017-1103-6 Issue No:Vol. 224, No. 3 (2017)