Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.     

Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case

In this work, we are concerned with the regularities of the solutions to the Boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials, r ^−(p−1) with p > 2. We give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional Sobolev norms. As an application, we get the global entropy dissipation estimate which is a little stronger than that described by Alexandre et al. (Arch Rational Mech Anal 152(4):327–355, 2000). As another application, we prove the smoothing effects for the strong solutions constructed by Desvillettes and Mouhot (Arch Rational Mech Anal 193(2):227–253, 2009) of the spatially homogeneous Boltzmann equation with “true” hard potential and “true” moderately soft potential.     

A Beale–Kato–Majda Criterion for Three Dimensional Compressible Viscous Heat-Conductive Flows

We prove a blow-up criterion in terms of the upper bound of (ρ, ρ ⁻¹, θ) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof is an a priori estimate for a quantity independently introduced in Haspot (Regularity of weak solutions of the compressible isentropic Navier–Stokes equation, arXiv:1001.1581, 2010) and Sun et al. (J Math Pure Appl 95:36–47, 2011), whose divergence can be viewed as the effective viscous flux.     

The Erpenbeck High Frequency Instability Theorem for Zeldovitch–von Neumann–D?ring Detonations

The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–D?ring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function V(t,e){V(\tau,\epsilon)} whose zeros in ${\mathfrak{R}\tau > 0}${\mathfrak{R}\tau > 0} correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber e{\epsilon} , even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of V(t,e){V(\tau,\epsilon)} , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as e?￥{\epsilon\to\infty} of solutions to a linear system of ODEs in x, depending on e{\epsilon} and a complex frequency τ as parameters, with turning points x _* on the half-line [0,∞).     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity

Variational formulations are constructed for rate-independent problems in small-deformation single-crystal strain-gradient plasticity. The framework, based on that of Gurtin (J Mech Phys Solids 50: 5–32, 2002), makes use of the flow rule expressed in terms of the dissipation function. Provision is made for energetic and dissipative microstresses. Both recoverable and non-recoverable defect energies are incorporated into the variational framework. The recoverable energies include those that depend smoothly on the slip gradients, the Burgers tensor, or on the dislocation densities (Gurtin et al. J Mech Phys Solids 55:1853–1878, 2007), as well as an energy proposed by Ohno and Okumura (J Mech Phys Solids 55:1879–1898, 2007), which leads to excellent agreement with experimental results, and which is positively homogeneous and therefore not differentiable at zero slip gradient. Furthermore, the variational formulation accommodates a non-recoverable energy due to Ohno et al. (Int J Mod Phys B 22:5937–5942, 2008), which is also positively homogeneous, and a function of the accumulated dislocation density. Conditions for the existence and uniqueness of solutions are established for the various examples of defect energy, with or without the presence of hardening or slip resistance.     

A constrained theory for single crystal shape memory wires with application to restrained recovery

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 337:143–147, 2003) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399–436, 2000). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299–310, 2004; Tsoi in 50:3535–3544, 2002; S̆ittner et al. in Mater Sci Eng A 286:298–311, 2000; vokoun in Smart Mater Struct 12:680–685, 2003; Zheng and Cui in Intermetallics 12:1305–1309, 2004; Zheng et al. in J Mater Sci Technol 20(4):390–394, 2004). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction.     

Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation and its Application

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima–Shizuta stability condition formulated in Umeda et al. (Jpn J Appl Math 1:435–457, 1984) and Shizuta and Kawashima (Hokkaido Math J 14:249–275, 1985) and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (for example, the Timoshenko system and the Euler–Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (see Duan in J Hyperbolic Differ Equ 8:375–413, 2011; Ide et al. in Math Models Meth Appl Sci 18:647–667, 2008; Ide and Kawashima in Math Models Meth Appl Sci 18:1001–1025, 2008; Ueda et al. in SIAM J Math Anal 2012; Ueda and Kawashima in Methods Appl Anal 2012). Therefore our purpose in this paper is to formulate a new structural condition which includes the Kawashima–Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.     

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).     

Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model

Linear viscoelastic (LVE) measurements as well as non-linear elongation measurements have been performed on stoichiometrically imbalanced polymeric networks to gain insight into the structural influence on the rheological response (Jensen et al., Rheol Acta 49(1):1–13, 2010). In particular, we seek knowledge about the effect of dangling ends and soluble structures. To interpret our recent experimental results, we exploit a molecular model that can predict LVE data and non-linear stress–strain data. The slip-link model has proven to be a robust tool for both LVE and non-linear stress–strain predictions for linear chains (Khaliullin and Schieber, Phys Rev Lett 100(18):188302–188304, 2008, Macromolecules 42(19):7504–7517, 2009; Schieber, J Chem Phys 118(11):5162–5166, 2003), and it is thus used to analyze the experimental results. Initially, we consider a stoichiometrically balanced network, i.e., all strands in the ensemble are attached to the network in both ends. Next we add dangling strands to the network representing the stoichiometric imbalance, or imperfections during curing. By considering monodisperse network strands without dangling ends, we find that the relative low-frequency plateau, G₀/G_N⁰G_0/G_N^0, decreases linearly with the average number of entanglements. The decrease from G_N⁰G_N^0 to G ₀ is a result of monomer fluctuations between entanglements, which is similar to “longitudinal modes” in tube theory. It is found that the slope of G′ is dependent on the fraction of network strands and the structural distribution of the network. The power-law behavior of G ^″ is not yet captured quantitatively by the model, but our results suggest that it is a result of polydisperse dangling and soluble structures.     

From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

Fourier Law,Phase Transitions and the Stationary Stefan Problem

We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. We construct stationary solutions with non-zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied: we show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface.     

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.     

Equilibrium Problems and Limit Analysis of Masonry Beams

We consider planar straight and curved masonry beams with the constitutive equation from Orlandi (Ph.D. thesis, 1999) and Zani (Eur. J. Mech. A, Solids 23:467–484, 2004). After stating some results about the solution to the boundary value problem, the limit analysis for this kind of bodies is outlined, based on energetic considerations (Lucchesi et al. in Q. Appl. Math. 68:713–746, 2010). The static and kinematic theorems of limit analysis, which usually are justified in a heuristic way (Heyman in The Masonry Arch, 1982; Kooharian in Proc. - Am. Concr. Inst. 89:317–328, 1953), are examined from this point of view. It is seen that the kinematic theorem does not always hold but can be proved under some hypotheses that are frequently met in applications.     

Degenerate Regularization of Forward–Backward Parabolic Equations: The Regularized Problem

We study a quasilinear parabolic equation of forward–backward type in one space dimension, under assumptions on the nonlinearity which hold for a number of important mathematical models (for example, the one-dimensional Perona–Malik equation), using a degenerate pseudoparabolic regularization proposed in Barenblatt et al. (SIAM J Math Anal 24:1414–1439, 1993), which takes time delay effects into account. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. We also study qualitative properties of such solutions, in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the Lebesgue measure. In this respect, the existence of a family of viscous entropy inequalities plays an important role.     

High-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation

The dynamics and stability of the high-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation have been studied using a simulation model equipped with governing equations of continuity, motion, energy, and crystallinity, along with the Phan-Thien–Tanner constitutive equation. Despite the fact that a simple one-phase model was incorporated into the governing equations to describe the spinline crystallinity, as opposed to the best-known two-phase model [Doufas et al. J Non-Newton Fluid Mech, 92:27–66, 2000a]; [Kohler et al. J Macromol Sci Phys, 44:185–202, 2005] that treats amorphous and crystalline phases separately in computing the spinline stress, the simulation has successfully portrayed the typical nonlinear characteristic of the high-speed spinning process called neck-like spinline deformation. It has been found that the criterion for the neck-like deformation to occur on the spinline is for the extensional viscosity to decrease on the spinline, so that the spinning is stabilized by the formation of the spinline neck-like deformation. The accompanying linear stability analysis explains this stabilizing effect of the spinline neck-like deformation, corroborating a recent experimental finding [Takarada et al. Int Polym Process, 19:380–387, 2004].This paper was presented at the 2nd Annual European Rheology Conference 2005 on April 21–23, 2005, in Grenoble, France.     

The MSF model: relation of nonlinear parameters to molecular structure of long-chain branched polymer melts

The elongational viscosity data of model PS combs (Hepperle J, Einfluss der Molekularen Struktur auf Rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen. Doctoral Thesis, University Erlangen-Nürnberg, 2003) are reconsidered by including the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37:3934–3942, 2004] in the Molecular Stress Function model [Wagner et al., J Rheol 47(3):779–793, 2003, Wagner et al., J Rheol 49:1317–1327, 2005d]. Two nonlinear model parameters are needed to describe elongational flow, β and . The parameterβ determines the slope of the elongational viscosity after the inception of strain hardening. It is directly related to the molecular structure of the polymer and represents the ratio of the molar mass of the (branched) polymer to the molar mass of the backbone alone. β follows from the hypothesis of Wagner et al. [J Rheol 47(3):779–793, 2003] that side chains are compressed onto the backbone. We consider also the case that side chains are oriented by deformation, but not stretched, and found little difference in the model predictions. The parameter represents the maximum strain energy stored in the polymeric system and determines the steady-state value of the viscosity in extensional flows. The relation of this energy parameter to the molecular structure is discussed. Good correlations between the energy parameter and different coil contraction ratios, as determined either experimentally or calculated theoretically by considering the topology of the macromolecule, are found. The smaller the relative size of the polymer coil, the larger is the energy parameter and the more strain energy can be stored in the polymeric system. Presented at the 3rd Annual European Rheology Conference, AERC2006, Crete, Greece.     

Convergence of Ginzburg–Landau Functionals in Three-Dimensional Superconductivity

In this paper we consider the asymptotic behavior of the Ginzburg–Landau model for superconductivity in three dimensions, in various energy regimes. Through an analysis via Γ-convergence, we rigorously derive a reduced model for the vortex density and deduce a curvature equation for the vortex lines. In the companion paper (Baldo et al. Commun. Math. Phys. 2012, to appear) we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H _c1, and the critical angular velocity of rotating Bose–Einstein condensates.     

Estimation of the largest Lyapunov exponent from?the?perturbation vector and its derivative dot product

Nowadays, the largest Lyapunov exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. The new method of the LLE estimation is presented in this paper. The method (LLEDP) applies the perturbation vector and its derivative dot product to calculate Lyapunov exponent in the direction of disturbance. Value of this exponent is the LLE. The theoretical improvement was introduced. Results of the numerical simulations were shown and compared with the Stefański method (Stefański in Chaos Solitons Fractals 11(15):2443–2451, 2000; Stefański and Kapitaniak in Chaos Solitons Fractals 15:233–244, 2003; Stefański et al. in Chaos Solitons Fractals 23:1651–1659, 2005; Stefański in J. Theor. Appl. Mech. 46:665–678, 2008). Investigations of the Duffing oscillators with external excitation and three Duffing oscillators coupled in ring scheme without external excitation were made with use of the presented method. Fast time LLE estimation results convergence was shown.     

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E _∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.