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.     

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory

The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u=u(X/T). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884–921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solution were studied using asymptotic expansions. Here we show that the asymptotic expansions correspond to true eigenvalue–eigenfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma.     

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.     

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.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

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.     

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.     

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u _ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.     

On the Impossibility, in Some Cases, of the Leray�CHopf Condition for Energy Estimates

Current proofs of time independent energy bounds for solutions of the time dependent Navier–Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of κ less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1–82, 1993), Hopf (Math Ann 117:764–775, 1941) and Finn (Acta Math 105:197–244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of κ, has been shown by Takeshita (Pac J Math 157:151–158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

On the Camassa–Holm and K–dV Hierarchies

It is known that a transform of Liouville type allows one to pass from an equation of the Korteweg–de Vries (K–dV) hierarchy to a corresponding equation of the Camassa–Holm (CH) hierarchy (Beals et al., Adv Math 154:229–257, 2000; McKean, Commun Pure Appl Math 56(7):998–1015, 2003). We give a systematic development of the correspondence between these hierarchies by using the coefficients of asymptotic expansions of certain Green’s functions. We illustrate our procedure with some examples.     

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

Stability of Viscous Profiles: Proofs Via Dichotomies

In this paper we give a self-contained approach to a nonlinear stability result, as t → ∞, for a viscous profile corresponding to a strong shock of a system of conservation laws. The initial perturbation is assumed to be small and to have zero mass. As t→ ∞, the solution with perturbed initial data is shown to approach the viscous profile in maximum norm.A complete proof of the stability result is given under slightly weaker assumptions than those in [Comm. Pure Appl. Math. LI (1998) 1397]; our assumptions, techniques, and results also differ from those in [Indiana Univ. Math. J. 47 (1998) 741]. To derive resolvent estimates for a linearized problem, we use the theory of exponential dichotomies for ODEs extensively. A main tool provided by this theory is a quantitative L ₁ perturbation theorem for dichotomies, which yields the delicate resolvent estimates for s near zero.When showing that the resolvent estimates imply nonlinear stability, we essentially follow the arguments in [Comm. Pure Appl. Math. LI (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

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.     

On the asymptotic behavior of solutions of second-order differential equations with nonlinearity close to an exponential

We obtain asymptotic representations for one class of solutions of a second-order differential equation with nonlinearity close to an exponential. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 541–553, October–December, 2008.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.