Authors:Heinrich Freistühler; Matthias Kotschote Pages: 1 - 20 Abstract: Abstract Various versions of the Navier–Stokes–Allen–Cahn (NSAC), the Navier–Stokes–Cahn–Hilliard (NSCH), and the Navier–Stokes–Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. One main purpose of this paper consists in (re-)deriving NSAC, NSCH and NSK from first principles, in the spirit of rational mechanics, for fluids of very general constitutive laws. For NSAC, this deduction confirms and extends a proposal of Blesgen. Regarding NSCH, it continues work of Lowengrub and Truskinovsky and provides the apparently first justified formulation in the non-isothermal case. For NSK, it yields a most natural correction to the formulation by Dunn and Serrin. The paper uniformly recovers as examples various classes of fluids, distinguished according to whether none, one, or both of the phases are compressible, and according to the nature of their co-existence. The latter is captured not only by the mixing energy, but also by a ‘mixing rule’—a constitutive law that characterizes the type of the mixing. A second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature-independent specific volumes, NSAC reduces literally to NSK. This finding may be considered as an independent justification of NSK. An analogous fact is shown for NSCH, which under the same assumption reduces to a new non-local version of NSK. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1065-0 Issue No:Vol. 224, No. 1 (2017)

Authors:Jonathan Eckhardt Pages: 21 - 52 Abstract: Abstract We establish the inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. In particular, it is employed to prove the existence of weak solutions for the corresponding Cauchy problem. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1066-z Issue No:Vol. 224, No. 1 (2017)

Authors:Raphaël Danchin; Jiang Xu Pages: 53 - 90 Abstract: Abstract The global existence issue for the isentropic compressible Navier–Stokes equations in the critical regularity framework was addressed in Danchin (Invent Math 141(3):579–614, 2000) more than 15 years ago. However, whether (optimal) time-decay rates could be shown in critical spaces has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of Danchin (Invent Math 141(3):579–614, 2000) but also in the general L p critical framework of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010), Chen et al. (Commun Pure Appl Math 63(9):1173–1224, 2010), Haspot (Arch Ration Mech Anal 202(2):427–460, 2011): we show that under a mild additional decay assumption that is satisfied if, for example, the low frequencies of the initial data are in \({L^{p/2}(\mathbb{R}^{d})}\) , the L p norm (the slightly stronger \({\dot B^0_{p,1}}\) norm in fact) of the critical global solutions decays like \({t^{-d(\frac 1p-\frac14)}}\) for \({t\to+\infty,}\) exactly as firstly observed by Matsumura and Nishida in (Proc Jpn Acad Ser A 55:337–342, 1979) in the case p = 2 and d = 3, for solutions with high Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1067-y Issue No:Vol. 224, No. 1 (2017)

Authors:Thomas Chen; Younghun Hong; Nataša Pavlović Pages: 91 - 123 Abstract: Abstract In this paper, we study the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at zero temperature, in dimensions d = 2, 3, and prove global well-posedness of the corresponding Cauchy problem. Our work extends some of the recent important results obtained by Lewin and Sabin in [33,34], who addressed this problem for more regular pair interactions. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1068-x Issue No:Vol. 224, No. 1 (2017)

Authors:Z. Bradshaw; Z. Grujić Pages: 125 - 133 Abstract: Abstract Two regularity criteria are established to highlight which Littlewood–Paley frequencies play an essential role in possible singularity formation in a Leray–Hopf weak solution to the Navier–Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya–Prodi–Serrin-type regularity criteria restricted to a finite window of frequencies, the lower bound of which diverges to \({+\infty}\) as t approaches an initial singular time. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1069-9 Issue No:Vol. 224, No. 1 (2017)

Authors:Robert L. Jerrard; Christian Seis Pages: 135 - 172 Abstract: Abstract In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in \({\mathbb{R}^{3}}\) , we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Poisson structures with stability estimates recently developed in Jerrard and Smets (J Eur Math Soc (JEMS) 17(6):1487–1515, 2015). PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1070-3 Issue No:Vol. 224, No. 1 (2017)

Authors:Theodoros Katsaounis; Julien Olivier; Athanasios E. Tzavaras Pages: 173 - 208 Abstract: Abstract Shear localization occurs in various instances of material instability in solid mechanics and is typically associated with Hadamard-instability for an underlying model. While Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. The latter is a nonlinear effect and its analysis is the main objective of this article. We consider a model that captures the main mechanisms observed in high strain-rate deformation of metals, and describes shear motions of temperature dependent non-Newtonian fluids. For a special dependence of the viscosity on the temperature, we carry out a linearized stability analysis around a base state of uniform shearing solutions, and quantitatively assess the effects of the various mechanisms affecting the problem: thermal softening, momentum diffusion and thermal diffusion. Then, we turn to the nonlinear model, and construct localized states—in the form of similarity solutions—that emerge as coherent structures in the localization process. This justifies a scenario for localization that is proposed on the basis of asymptotic analysis in Katsaounis and Tzavaras (SIAM J Appl Math 69:1618–1643, 2009). PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1071-2 Issue No:Vol. 224, No. 1 (2017)

Authors:Wei Luo; Zhaoyang Yin Pages: 209 - 231 Abstract: Abstract In this paper we mainly study the finite extensible nonlinear elastic (FENE) dumbbell model with dimension \({d \geqq 2}\) in the whole space. We first prove that there is only the trivial solution for the steady-state FENE model under some integrable condition. The obtained results generalize and cover the classical results for the stationary Navier–Stokes equations. We then obtain that the L 2 decay rate of the velocity of the co-rotation FENE model is \({(1+t)^{-\frac{d}{4}}}\) when \({d \geqq 3}\) , and \({\ln^{-k}{(e+t)}, k\in \mathbb{N}^{+}}\) when d = 2. This result improves considerably the recent result of Schonbek (SIAM J Math Anal 41:564–587, 2009). Moreover, we investigate the L 2 decay of solutions to the general FENE model. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1072-1 Issue No:Vol. 224, No. 1 (2017)

Authors:Grégoire Allaire; Jeffrey Rauch Pages: 233 - 268 Abstract: Abstract This article proves most of the assertion in §116 of Maxwell’s treatise on electromagnetism. The results go under the name Earnshaw’s Theorem and assert the absence of stable equilibrium configurations of conductors and dielectrics in an external electrostatic field. PubDate: 2017-04-01 DOI: 10.1007/s00205-016-1073-0 Issue No:Vol. 224, No. 1 (2017)

Authors:Giovanni Cupini; Francesco Leonetti; Elvira Mascolo Pages: 269 - 289 Abstract: Abstract We give a regularity result for local minimizers \({u}:{\Omega \subset {\mathbb{R}}^3 \to {\mathbb{R}}^3}\) of a special class of polyconvex functionals. Under some structure assumptions on the energy density, we prove that local minimizers u are locally bounded. For each component \({u^{\alpha}}\) of u, we first prove a Caccioppoli’s inequality and then apply De Giorgi’s iteration method to get the boundedness of \({u^{\alpha}}\) . Our result can be applied to the polyconvex integral $$\int_\Omega \left( \sum\limits_{\alpha = 1}^{3} D u^\alpha ^{p} + \operatorname{adj}_2 Du ^q + \operatorname{det} Du ^{r}\right) {\rm d}x$$ with suitable \({p,q,r > 1}\) . PubDate: 2017-04-01 DOI: 10.1007/s00205-017-1074-7 Issue No:Vol. 224, No. 1 (2017)

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: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: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:T. Hudson Abstract: Abstract We formulate and study a stochastic model for the thermally-driven motion of interacting straight screw dislocations in a cylindrical domain with a convex polygonal cross-section. Motion is modelled as a Markov jump process, where waiting times for transitions from state to state are assumed to be exponentially distributed with rates expressed in terms of the potential energy barrier between the states. Assuming the energy of the system is described by a discrete lattice model, a precise asymptotic description of the energy barriers between states is obtained. Through scaling of the various physical constants, two dimensionless parameters are identified which govern the behaviour of the resulting stochastic evolution. In an asymptotic regime where these parameters remain fixed, the process is found to satisfy a Large Deviations Principle. A sufficiently explicit description of the corresponding rate functional is obtained such that the most probable path of the dislocation configuration may be described as the solution of Discrete Dislocation Dynamics with an explicit anisotropic mobility which depends on the underlying lattice structure. PubDate: 2017-02-04 DOI: 10.1007/s00205-017-1076-5