Showing papers in "Archive for Rational Mechanics and Analysis in 1996"

Nonholonomic Mechanical Systems with Symmetry

Abstract: This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

Conformal curvature flows: From phase transitions to active vision

Abstract: In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

Abstract: We consider as in Part I a family of linearly elastic shells of thickness 2ɛ, all having the same middle surfaceS=ϕ(ϖ)⊂R3, whereω⊂R2 is a bounded and connected open set with a Lipschitz-continuous boundary, andϕ∈l3 (ϖ;R3). The shells are clamped on a portion of their lateral face, whose middle line isϕ(γ0), whereγ0 is any portion of∂ω withlength γ0>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ0, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where\(\gamma _{\alpha \beta }\)(η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder andϕ(γ0) is contained in a generatrix ofS.

Well-posedness of characteristic symmetric hyperbolic systems

Abstract: We consider the initial-boundary-value problem for quasi-linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. We show the well-posedness in Hadamard's sense (i.e., existence, uniqueness and continuous dependence of solutions on the data) of regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary.

Critical points for multiple integrals of the calculus of variations

Abstract: In this paper we deal with the existence of critical points of functional defined on the Sobolev space W01,p(Ω), p>1, by $$J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}$$ where Ω is a bounded, open subset of ℝN. Even for very simple examples in ℝN the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.

Small-time existence for the three-dimensional navier-stokes equations for an incompressible fluid with a free surface

Abstract: In 1980 BEALE [2] studied the problem of describing the motion of a layer of heavy, viscous, incompressible fluid lying above an infinite rigid bottom and having a non-compact free surface with no surface tension. He proved its local solvability in an anisotropic Sobolev space for any initial data. Its global solvability was discussed by SYLV~SXER [20]. But a crucial point of her proof does not appear clear to me: the regularity of the free surface. With surface tension taken into account, BEALE [31 and BEALE & NISHIDA [4] respectively studied the large-time existence and regularity and the large-time behavior of the solution to this problem with initial data near equilibrium. TERAMOTO [24, 25] considered these problems for fluids lying above an inclined plane. For any initial data the same problem was solved locally in time by ALLAIN [11 in the two-dimensional case. The aim of this paper is to establish the analogous result in the three-dimensional case. A similar problem describing the motion of a finite isolated mass of incompressible viscous fluid was studied thoroughly by SOLONNIKOV. He proved local solvability in a H61der space in [12] and global solvability in the space W 2' 1, p > n (where n is the dimension of the domain) in [16] without surface tension and in [-13 15, 17 19,21] with surface tension. Let us formulate our problem. Given an initial domain f~ _ R 3 with x3 being the vertical component and an initial velocity vector field Vo in f~, we want to know the domain f~(t), t > 0, occupied by the fluid, which is bounded by the fixed bottom SB and the free surface Se(t), the velocity vector field v = v(x, t) = (vl, v2, v3) and the pressure p = p(x, t) so that

Stability of rarefaction waves in viscous media

Abstract: We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w 0 with small mass and localized as w 0(x)= $$\mathcal{O}(|x|^{ - 1} )$$ The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error. This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass $$\mathcal{O}$$ (log (t)). These “diffusion waves” have amplitude $$\mathcal{O}$$ (t -1/2logt) in linear degenerate transversal fields and $$\mathcal{O}$$ (t -1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

Abstract: We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.

Dynamics as a mechanism preventing the formation of finer and finer microstructure

Abstract: We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((ux)2 −1)2. What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]: •While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (unvn) of E[u,v] in the Sobolev space W1p(0, 1)×L2(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W1p(0,1)×L2(0,1) of all solutions (u(t),ut(t)) with low initial energy as time t → ∞).

Capillary Gravity Waves on the Free Surface of an Inviscid Fluid of Infinite Depth. Existence of Solitary Waves

Abstract: Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions This can be converted to an integro-differential equation with symbol −k2+4|k|−4(1+μ), where μ is a bifurcation parameter A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for |μ| small This normal form system has been studied thoroughly by several authors (Iooss &Kirchgassner [8],Iooss &Peroueme [10],Dias &Iooss [5]) It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgassner [11] By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|

Hamiltonian dynamics of an elastica and the stability of solitary waves

Abstract: A method is presented for deriving unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems. The method is applied to the theory of planar, finite-amplitude motions of inextensible and unshearable elastic rods. The constraints of inextensibility and unshearability become integrals of motion in the Hamiltonian formulation.

Bloch-Wave Homogenization for a Spectral Problem in Fluid-Solid Structures

Abstract: This paper is concerned with the study of the vibrations of a coupled fluid-solid periodic structure. As the period goes to zero, an asymptotic analysis of the spectrum (i.e., the set of eigenfrequencies) is performed with the help of a new method, the so-called Bloch-wave homogenization method (which is a blend of two-scale convergence and Bloch-wave decomposition). The limit spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and the boundary-layer spectrum. The two first parts are completely characterized: The homogenized and the Bloch spectra are purely essential, and have a band structure. The boundary-layer spectrum is shown to be empty in the special case of periodic boundary condition.

Boundary-Layer Behavior in the Fluid-Dynamic Limit for a Nonlinear Model Boltzmann Equation

Abstract: In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL ∞-norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.

On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a Stellarator geometry

Abstract: We prove the existence and some qualitative properties of the solution to a two-dimensional free-boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. The nonlinear elliptic partial differential equation on the plasma region was obtained from the three-dimensional magnetohydrodynamic system by Hender & Carreras in 1984 by using averaging arguments and a suitable system of coordinates (Boozer's vacuum coordinates). The free boundary represents the separation between the plasma and vacuum regions, and the model is described by an inverse-type problem (some nonlinear terms of the equation are unknown). Using the zero net current condition for the Stellarator configurations, we reformulate the problem with the help of the notion of relative rearrangement, leading to a new problem involving nonlocal terms in the equation. We use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution.

Relaxation of bulk and interfacial energies

Abstract: In this paper we obtain an integral representation for the relaxation inBV(Ω; ℝp) of the functional $$u \mapsto \int\limits_\Omega {f(x. abla u(x))dx + \int\limits_{\sum _{(u)} } {\varphi (x,[u](x),v(x))dH_{N - 1} (x)} }$$ with respect to theBV weak * convergence.

On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

Abstract: We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrodinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t→∞, of the solutions which are close enough to a stationary asymptotically stable solution.

Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

Abstract: It is proved that under a physically realistic truncation of the collision kernel, the Boltzmann equation in the one-dimensional slab [0,1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. This solution satisfies the boundary conditions almost everywhere, and has, at worst, exponentially growing total energy.

Classification and unfoldings of 1:2 resonant Hopf Bifurcation

Abstract: In this paper, we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1:2 ratio. This corresponds to a Hopf-Hopf mode interaction with 1:2 resonance, as occurs in the context of dissipative mechanical systems. Using an approach based on Liapunov-Schmidt reduction and singularity theory, we give a framework in which to study these problems and their perturbations in two cases: no distinguished parameter, and one distinguished (bifurcation) parameter. We give a complete classification of the generic cases and their unfoldings.

Global solvability of the maxwell-bloch equations from nonlinear optics

Abstract: The nonlinear self-focusing of optical beams is a well known and abundantly documented phenomenon M, Sh]. The mechanism is simple. If the speed of propagation of electromagnetic waves decreases as a function of the intensity, then ray tracing suggests that a planar wavefront with intensity which is large at the center and decreases away from the center will propagate so that the center lags behind the edges. This initial curvature translates into focussing after a nite period of time. Interest in this phenomenon has been recently renewed with the advent of ultrashort and ultraintense pulses Ro]. For ultraintense pulses self-focusing is undesirable as it often leads to the extinction of the beam and to the destruction of very costly optical devices. In order to better understand focussing, with one goal being avoidance, good models are needed for the regions of space time where laser beams focus. The usual analysis of self-focusing uses the nonlinear Schrr odinger equation (NLS). This arises from more fundamental eld equations by the slowly varying envelope approximation. One seeks an approximate solution in the form of a slowly varying eld envelope times a rapidly oscillating term with linear phase function. A medium with an

A Cauchy integral approach to Hele-Shaw problems with a free boundary: The case of zero surface tension

Abstract: In this paper, we study a nonlinear and nonlocal free-boundary dynamics — the Hele-Shaw problem without surface tension when the fluid domain is either bounded or unbounded. The key idea is to use a global quantity, the Cauchy integral of the free boundary, to capture the motion of the boundary. This Cauchy integral is shown to be linear in time. The free boundary at a fixed time is then recovered from its Cauchy integral at that time. The main tool in our analysis isCherednichenko's theorem concerning the inverse properties of the Cauchy integrals. As products of our approach, we establish the short-time existence and uniqueness of classical solutions for analytic initial boundaries. We also show the non-existence of classical solutions for all smooth but non-analytic initial boundaries when there is a sink at either a finite point or at infinity. When the fluid domain is bounded, all solutions except the circular one break down before all the fluid is sucked out from the sink. Regularity results are also obtained when there is a source at a finite point or at infinity.

Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems

Abstract: A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astala's theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.

Propagation of cracks in elastic media

Abstract: Throughout this paper the normal n denotes the interior normal direction; i.e., n = ( f (x1); 1)= p 1 + jf (x1)j when x approaches from above (denoted by ) and n = (f (x1); 1)= p 1 + jf (x1)j from below (denoted by ). In a recent paper [2] the present authors studied the asymptotic behavior of any solution of (1.2){(1.4) in a neighborhood of the tip X(t). Taking for simplicity t = 0 and assuming that

Self-gravitating relativistic fluids: The formation of a free phase boundary in the phase transition from soft to hard

Abstract: In the 1990s Christodoulou introduced an idealized fluid model intended to capture some of the features of the gravitational collapse of a massive star to form a neutron star or a black hole. This was the two-phase model introduced in ‘Self-gravitating relativistic fluids: a two phase model’ (Demeterios, Arch Ration Mech Anal 130:343–400, 1995). The present work deals with the formation of a free phase boundary in the phase transition from hard to soft in this model. In this case the phase boundary has corners at the null points; the points which separate the timelike and spacelike components of the interface between the two phases. We prove the existence and uniqueness of a free phase boundary. Also the local form of the shock near the null point is established.

Bifurcations of travelling waves in the thermo-diffusive model for flame propagation

Abstract: The main topic of this paper is the study of steady-state bifurcations occurring in the two-dimensional thermo-diffusive model in the framework of large activation energies.