Authors:Alessio Figalli; Moon-Jin Kang; Javier Morales Pages: 869 - 896 Abstract: Abstract We consider the so-called spatially homogenous Kolmogorov–Vicsek model, a non-linear Fokker–Planck equation of self-driven stochastic particles with orientation interaction under the space-homogeneity. We prove the global existence and uniqueness of weak solutions to the equation. We also show that weak solutions exponentially converge to a steady state, which has the form of the Fisher-von Mises distribution. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1176-2 Issue No:Vol. 227, No. 3 (2018)

Authors:Lihui Chai; Carlos J. García-Cervera; Xu Yang Pages: 897 - 928 Abstract: Abstract The Schrödinger–Poisson–Landau–Lifshitz–Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. This system has been used in connection with the study of spin transfer and magnetization reversal in ferromagnetic materials. In this paper, we rigorously prove the existence of weak solutions to SPLLG and derive the Vlasov–Poisson–Landau–Lifshitz–Glibert system as the semiclassical limit. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1177-1 Issue No:Vol. 227, No. 3 (2018)

Authors:Ibrahim Ekren; H. Mete Soner Pages: 929 - 965 Abstract: The classical duality theory of Kantorovich (C R (Doklady) Acad Sci URSS (NS) 37:199–201, 1942) and Kellerer (Z Wahrsch Verw Gebiete 67(4):399–432, 1984) for classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice \({\mathcal{X}}\) with an order unit. The problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of \({\mathcal{X}}\) and the dual problem is defined on the bi-dual of \({\mathcal{X}}\) . These results are then applied to several extensions of the classical optimal transport. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1178-0 Issue No:Vol. 227, No. 3 (2018)

Authors:Simon Markfelder; Christian Klingenberg Pages: 967 - 994 Abstract: 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:Xiangdi Huang; Jing Li Pages: 995 - 1059 Abstract: Abstract For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1188-y Issue No:Vol. 227, No. 3 (2018)

Authors:Yuning Liu; Wei Wang Pages: 1061 - 1090 Abstract: Abstract We study the relationship between Onsager’s molecular theory, which involves the effects of nonlocal molecular interactions and the Oseen–Frank theory for nematic liquid crystals. Under the molecular setting, we prove the existence of global minimizers for the generalized Onsager’s free energy, subject to a nonlocal boundary condition which prescribes the second moment of the number density function near the boundary. Moreover, when the re-scaled interaction distance tends to zero, the global minimizers will converge to a uniaxial distribution predicted by a minimizing harmonic map. This is achieved through the investigations of the compactness property and the boundary behaviors of the corresponding second moments. A similar result is established for critical points of the free energy that fulfill a natural energy bound. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1180-6 Issue No:Vol. 227, No. 3 (2018)

Authors:Li Li; YanYan Li; Xukai Yan Pages: 1091 - 1163 Abstract: 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)

Authors:Richard M. Höfer; Juan J. L. Velázquez Pages: 1165 - 1221 Abstract: Abstract We study the convergence of the method of reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which exist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) method of reflections. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1182-4 Issue No:Vol. 227, No. 3 (2018)

Authors:Alberto Bressan; Marta Lewicka Pages: 1223 - 1266 Abstract: Abstract We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with \({\mathcal{C}^{2,\alpha}}\) boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion. PubDate: 2018-03-01 DOI: 10.1007/s00205-017-1183-3 Issue No:Vol. 227, No. 3 (2018)

Authors:Luca Minotti; Giuseppe Savaré Pages: 477 - 543 Abstract: Abstract We propose the new notion of Visco-Energetic solutions to rate-independent systems \({(X, \mathcal{E},}\) d) driven by a time dependent energy \({\mathcal{E}}\) and a dissipation quasi-distance d in a general metric-topological space X. As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation quasi-distance d is incremented by a viscous correction \({\delta}\) (for example proportional to the square of the distance d), which penalizes far distance jumps by inducing a localized version of the stability condition. We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable of covering a wide range of applications. The new refined Energy Balance condition compensates for the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1165-5 Issue No:Vol. 227, No. 2 (2018)

Authors:Yann Brenier; Xianglong Duan Pages: 545 - 565 Abstract: Abstract We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic, change of time. A typical example is the curve-shortening flow in \({\mathbb{R}^d}\) , which is a particular case of mean-curvature flow with a co-dimension higher than one (except in the case d = 2). Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtained from the Nambu–Goto action. Using the concept of “relative entropy” (or “modulated energy”), borrowed from the theory of hyperbolic systems of conservation laws, we introduce a notion of generalized solutions, that we call dissipative solutions, for the curve-shortening flow. For given initial conditions, the set of generalized solutions is convex and compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1166-4 Issue No:Vol. 227, No. 2 (2018)

Authors:Stefano Lisini; Edoardo Mainini; Antonio Segatti Pages: 567 - 606 Abstract: Abstract We consider a family of porous media equations with fractional pressure, recently studied by Caffarelli and Vázquez. We show the construction of a weak solution as the Wasserstein gradient flow of a square fractional Sobolev norm. The energy dissipation inequality, regularizing effect and decay estimates for the L p norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1168-2 Issue No:Vol. 227, No. 2 (2018)

Authors:Yury Grabovsky Pages: 607 - 636 Abstract: Abstract Examples of non-quasiconvex functions that are rank-one convex are rare. In this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials, developed to identify G-closed sets of positive codimensions. The algebraic methods are used to construct a set of materials of positive codimension that is closed under lamination but is not G-closed. The well-known link between G-closed sets and quasiconvex functions and sets closed under lamination and rank-one convex functions is then used to construct a family of rotationally invariant, nonnegative, and 2-homogeneous rank-one convex functions, that are not quasiconvex. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1169-1 Issue No:Vol. 227, No. 2 (2018)

Authors:Ronghua Pan; Yi Zhou; Yi Zhu Pages: 637 - 662 Abstract: Abstract In this paper, we study the global existence of classical solutions to the three dimensional incompressible viscous magneto-hydrodynamical system without magnetic diffusion on periodic boxes, that is, with periodic boundary conditions. We work in Eulerian coordinates and employ a time-weighted energy estimate to prove the global existence result, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have some symmetries. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1170-8 Issue No:Vol. 227, No. 2 (2018)

Authors:Benny Avelin; Tuomo Kuusi; Giuseppe Mingione Pages: 663 - 714 Abstract: Abstract We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderón and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that if $$-{\rm div} \, ( Du ^{p-2}Du)=\mu \quad \mbox{in} \ \Omega\subset\mathbb{R}^n,$$ with \({\mu}\) being a Borel measure with locally finite mass on the open subset \({\Omega\subset \mathbb{R}^n}\) and \({p > 2-1/n}\) , then $$ Du ^{p-2}Du \in W^{\sigma, 1}_{\rm{loc}}(\Omega)\quad \mbox{for \, every} \ \sigma \in (0,1).$$ The case \({\sigma=1}\) is obviously forbidden already in the classical linear case of the Poisson equation \({-\triangle u=\mu}\) . PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1171-7 Issue No:Vol. 227, No. 2 (2018)

Authors:Xiuqing Chen; Esther S. Daus; Ansgar Jüngel Pages: 715 - 747 Abstract: Abstract The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager’s principle in thermodynamics. Under detailed balance (and without reaction) the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1172-6 Issue No:Vol. 227, No. 2 (2018)

Authors:Marco Barchiesi Pages: 749 - 766 Abstract: Abstract We present a simple example of a toughening mechanism in the homogenization of composites with soft inclusions, produced by a crack deflection at microscopic level. We show that the mechanism is connected to the irreversibility of the crack process; because of that it cannot be detected through the standard homogenization tool of the \({\Gamma}\) -convergence. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1173-5 Issue No:Vol. 227, No. 2 (2018)

Authors:Yan Guo; Ian Tice Pages: 767 - 854 Abstract: Abstract In an effort to study the stability of contact lines in fluids, we consider the dynamics of an incompressible viscous Stokes fluid evolving in a two-dimensional open-top vessel under the influence of gravity. This is a free boundary problem: the interface between the fluid in the vessel and the air above (modeled by a trivial fluid) is free to move and experiences capillary forces. The three-phase interface where the fluid, air, and solid vessel wall meet is known as a contact point, and the angle formed between the free interface and the vessel is called the contact angle. We consider a model of this problem that allows for fully dynamic contact points and angles. We develop a scheme of a priori estimates for the model, which then allow us to show that for initial data sufficiently close to equilibrium, the model admits global solutions that decay to equilibrium exponentially quickly. PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1174-4 Issue No:Vol. 227, No. 2 (2018)

Authors:Leo T. Butler Pages: 855 - 867 Abstract: Abstract Let \({H(q,p) = \frac{1}{2}{\parallel p \parallel}^2 + V(q)}\) be an n-degree of freedom C r mechanical Hamiltonian on \({T^{*}{\bf T}^n}\) where \({r > 2n+2}\) . When the metric \({{\parallel \cdot \parallel}}\) is flat, the Nosé-thermostated system associated to H is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nosé’s, too. These results complement results of Bulter (Nonlinearity 11(29):3454–3463, 2016), Legoll et al. (Arch Ration Mech Anal 184(3):449–463, 2007, Nonlinearity 22(7):1673–1694, 2009). PubDate: 2018-02-01 DOI: 10.1007/s00205-017-1175-3 Issue No:Vol. 227, No. 2 (2018)

Authors:William C. Troy Pages: 367 - 385 Abstract: Abstract We prove the existence of a new class of solutions, called shadow kinks, of the Painleve II equation \({{\frac {{\rm d}^{2} w}{{\rm d}z^{2}}}=2w^{3} +zw+\alpha,}\) where \({\alpha < 0}\) is a constant. Shadow kinks are sign changing solutions which satisfy \({ w(z) \sim -{\sqrt {-z/2}}\ {\rm as}\ z \to - \infty}\) and \({w(z) \sim -{\frac {\alpha}{z}} \ {\rm as}\ z \to \infty.}\) These solutions play a critical role in the prediction of a new class of topological defects, one dimensional shadow kinks and two dimensional shadow vortices, in light-matter interaction experiments on nematic liquid crystals. These new defects are physically important since it has recently been shown (Wang et al. in Nat Mater 15:106–112, 2016) that topological defects are a “template for molecular self-assembly” in liquid crystals. Connections with the modified KdV equation are also discussed. PubDate: 2018-01-01 DOI: 10.1007/s00205-017-1162-8 Issue No:Vol. 227, No. 1 (2018)