Showing papers in "Archive for Rational Mechanics and Analysis in 2008"

Global Solutions for Incompressible Viscoelastic Fluids

Abstract: We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near- equilibrium initial data. The results hold in both two- and three-dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals, and mixture problems.

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

Abstract: We consider the nonlinear elliptic system $$\left \{ \begin{aligned} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\, \mathbb B,\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm{in}\, \mathbb B,\\ &u,v > 0 \quad \rm{in}\, \mathbb B,\quad u=v=0 \quad \rm{on}\, \partial \mathbb B, \end{aligned} \right.$$ where \(N\leqq 3\) and \(\mathbb B \subset \mathbb {R}^N\) is the unit ball. We show that, for every \(\beta \leqq -1\) and \(k \in \mathbb N\), the above problem admits a radially symmetric solution (uβ, vβ) such that uβ − vβ changes sign precisely k times in the radial variable. Furthermore, as \(\beta \to -\infty\), after passing to a subsequence, uβ → w+ and vβ → w− uniformly in \(\mathbb B\), where w = w+− w− has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − w + w3 = 0 in \(\mathbb B\), w = 0 on \(\partial \mathbb B\). Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.

Exponential Decay for Soft Potentials near Maxwellian

Abstract: We consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches zero at the rate of \(e^{-\lambda t^{p}}\) for some λ > 0 and 0 < p < 1. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical result Caflisch of [2] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by [5, 14].

A Model for the Formation and Evolution of Traffic Jams

Abstract: In this paper, we establish and analyze a traffic flow model which describes the formation and dynamics of traffic jams. It consists of a pressureless gas dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. From this analysis, we deduce the particular dynamical behavior of clusters (or traffic jams), defined as intervals where the density limit is reached. An existence result for a generic class of initial data is proved by means of an approximation of the solution by a sequence of clusters. Finally, numerical simulations are produced.

Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture

Abstract: Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform. He then conjectured that the converse is true, that is, that if the field inside an inclusion is uniform for all uniform loadings, then the inclusion is of elliptic or ellipsoidal shape. We call this the weak Eshelby conjecture. In this paper we prove this conjecture in three dimensions. In two dimensions, a stronger conjecture, which we call the strong Eshelby conjecture, has been proved: if the field inside an inclusion is uniform for a single uniform loading, then the inclusion is of elliptic shape. We give an alternative proof of Eshelby’s conjecture in two dimensions using a hodographic transformation. As a consequence of the weak Eshelby’s conjecture, we prove in two and three dimensions a conjecture of Polya and Szego on the isoperimetric inequalities for the polarization tensors (PTs). The Polya–Szego conjecture asserts that the inclusion whose electrical PT has the minimal trace takes the shape of a disk or a ball.

Serrin-Type Overdetermined Problems: an Alternative Proof

Abstract: We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely the Hessian equations. In the case of the Poisson equation, our proof is alternative to the proofs proposed by Serrin (moving planes) and by Weinberger. Moreover, our proof makes no direct use of the maximum principle while it sheds light on a relation between the Serrin problem and the isoperimetric inequality.

A Hardy inequality in twisted waveguides

Abstract: We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.

Traveling Waves in Discrete Periodic Media for Bistable Dynamics

Abstract: This paper is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for a system of ordinary differential equations exhibiting bistable dynamics. The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics. The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media.

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Abstract: Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number \(M > \sqrt{2}\) and instability when \(M \sqrt{2}\) . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.

Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

Abstract: A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction This prediction is rigorously confirmed for the full water-wave problem in the present paper The theory is variational in nature A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations

Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

Abstract: In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system.

A Vanishing Viscosity Approach to Quasistatic Evolution in Plasticity with Softening

Abstract: We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analyse a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.

Nonlinear Instability in Gravitational Euler–Poisson Systems for $$\gamma=\frac{6}{5}$$

Abstract: The dynamics of gaseous stars can be described by the Euler–Poisson system. Inspired by Rein’s stability result for \(\gamma > \frac{4}{3}\), we prove the nonlinear instability of steady states for the adiabatic exponent \(\gamma=\frac{6}{5}\) under spherically symmetric and isentropic motion.

Exact Solitary Water Waves with Vorticity

Abstract: The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.

Crack Initiation in Brittle Materials

Abstract: In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time ti > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional \(\epsilon( u, \Gamma)\,:=\int_{\Omega} f(x, abla v)\,{\rm d}x+k{\mathcal{H}}^{1} (\Gamma),\) where \(\Omega \subseteq {\mathbb{R}}^{2}\) , k > 0 and f is a suitable Caratheodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with \({\mathcal{H}}^{1} (\Gamma)\) small enough are such that \(\epsilon(u,\emptyset) < \epsilon(u_{\Gamma},\Gamma)\) .

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Abstract: A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

The Toda System and Clustering Interfaces in the Allen Cahn equation

Abstract: We consider the Allen–Cahn equation \({\varepsilon^{2}\Delta u + (1-u^2)u = 0}\) in a bounded, smooth domain Ω in \({\mathbb{R}^2}\) , under zero Neumann boundary conditions, where \({\varepsilon > 0}\) is a small parameter. Let Γ0 be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ0, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are \({O(\varepsilon|\log\varepsilon|)}\) and which collapse onto Γ0 as \({\varepsilon\to 0}\) . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.

The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System

Abstract: We study the gravitational Vlasov Poisson system \({f_t + v \cdot abla_{x} f - E \cdot abla_{v} f = 0}\) = 0 where \({E(x) = abla_{x}\phi(x)}\), \({\Delta_{x}\phi = \rho(x)}\), \({\rho(x) = \int_{\mathbb{R}^{N}} f(x, v)\,{\rm d}x\,{\rm d}v}\), in dimension N = 3, 4. In dimension N = 3 where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies, in particular, the orbital stability in the energy space of the spherically symmetric polytropes and improves the nonlinear stability results obtained for this class in [11, 16, 19]. In dimension N = 4 where the problem is L1 critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy. We then derive using an explicit pseudo-conformal symmetry the existence of critical mass finite time blow-up solutions, and prove more generally a mass concentration phenomenon for finite time blow up solutions. This is the first result of description of a singularity formation in a Vlasov setting. The global structure of the problem is reminiscent of the one for the focusing nonlinear Schrodinger equation iut = −Δu−|u|p−1u in the energy space \({H^1(\mathbb{R}^N)}\) .

Spatial Analyticity of the Solutions to the Subcritical Dissipative Quasi-geostrophic Equations

Abstract: We employ a new bilinear estimate to show that solutions to the subcritical dissipative quasi-geostrophic equations with initial data in the scaling-invariant Lebesgue space are analytic in space variables. Some decay in time estimates for space–time derivatives are also obtained.

Lagrangean Structure and Propagation of Singularities in Multidimensional Compressible Flow

Abstract: We study the propagation of singularities in solutions of the Navier–Stokes equations of compressible, barotropic fluid flow in two and three space dimensions. The solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded, initial energies are small, and initial velocities are in certain fractional Sobolev spaces. We show that, if the initial density is bounded below away from zero in an open set V, then each point of V determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi-Holder homeomorphism of V onto its image at each positive time. This “Lagrangean structure” is then applied to show that, if the initial density has a limit at a point of such a set V from a given side of a continuous hypersurface in V, then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface, which is also a continuous manifold. If the limits from both sides exist, then the Rankine–Hugoniot conditions hold in a strict pointwise sense, showing that the jump in the divergence of the velocity is proportional to the jump in the pressure. This leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density, from which it follows that discontinuities persist for all time, convecting along fluid particle paths, and in the case that the pressure is strictly increasing in density, having strengths which decay exponentially in time.

A Continuum Theory of Deformable, Semiconducting Ferroelectrics

Abstract: Ferroelectric solids, especially ferroelectric perovskites, are widely used as sensors, actuators, filters, memory devices, and optical components. While these have traditionally been treated as insulators, they are in reality wide-band-gap semiconductors. This semiconducting behavior affects the microstructures or domain patterns of the ferroelectric material and the interaction of ferroelectrics with electrodes, and is affected significantly by defects and dopants. In this paper, we develop a continuum theory of deformable, semiconducting ferroelectrics. A key idea is to introduce space charges and dopant density as field (state) variables in addition to polarization and deformation. We demonstrate the theory by studying oxygen vacancies in barium titanate. We find the formation of depletion layers, regions of depleted electrons, and a large electric field at the ferroelectric–electrode boundary. We also find the formation of a charge double layer and a large electric field across 90° domain walls but not across 180° domain walls. We show that these internal electric fields can give rise to a redistribution or forced diffusion of oxygen vacancies, which provides a mechanism for aging of ferroelectric materials.

Refined Jacobian Estimates and Gross-Pitaevsky Vortex Dynamics

Abstract: We study the dynamics of vortices in solutions of the Gross–Pitaevsky equation $${ i u_t = \Delta u + { 1 \over {\varepsilon^2}} u \left(1 - |u|^2 \right)}$$ in a bounded, simply connected domain $${\Omega\subset {\mathbb{R}^2}}$$ with natural boundary conditions on ∂Ω. Previous rigorous results have shown that for sequences of solutions $${u_\varepsilon}$$ with suitable well-prepared initial data, one can determine limiting vortex trajectories, and moreover that these trajectories satisfy the classical ODE for point vortices in an ideal incompressible fluid. We prove that the same motion law holds for a small, but fixed $${\varepsilon}$$ , and we give estimates of the rate of convergence and the time interval for which the result remains valid. The refined Jacobian estimates mentioned in the title relate the Jacobian J(u) of an arbitrary function $${u\in H^1(\Omega;\mathbb{C})}$$ to its Ginzburg–Landau energy. In the analysis of the Gross–Pitaevsky equation, they allow us to use the Jacobian to locate vortices with great precision, and they also provide a sort of dynamic stability of the set of multi-vortex configurations.

Incompressible Viscous Flows in Borderline Besov Spaces

Abstract: We establish two new estimates for a transport-diffusion equation. As an application we treat the problem of global persistence of the Besov regularity \(B_{p,1}^{\frac{2}{p}+1},\) with \(p \in ]2,+\infty]\) , for the two-dimensional Navier–Stokes equations with uniform bounds on the viscosity. We provide also an inviscid global result.

An Initial and Boundary Value Problem Modeling of Fish-like Swimming

Abstract: In this paper we consider an initial and boundary value problem that models the self-propelled motion of solids in a bidimensional viscous incompressible fluid. The self-propelling mechanism, consisting of appropriate deformations of the solids, is a simplified model of the propulsion mechanism of fish-like swimmers. The governing equations consist of the Navier–Stokes equations for the fluid, coupled to Newton’s laws for the solids. Since we consider the case in which the fluid–solid system fills a bounded domain we have to tackle a free boundary value problem. The main theoretical result in the paper asserts the global existence and uniqueness (up to possible contacts) of strong solutions of this problem. The second novel result of this work is the provision of a numerical method for the fluid–solid system. This method provides a simulation of the simultaneous displacement of several swimmers and is tested on several examples.

Solutions of a Nonlinear Dirac Equation with External Fields

Abstract: We study the stationary Dirac equation $$-ic\hbar{\sum^3_{k=1}}\alpha_k\partial_k u+mc^2\beta u+M(x) u= R_u(x,u),$$ where M(x) is a matrix potential describing the external field, and R(x, u) stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption. For ħ fixed our discussion includes the Coulomb potential as a special case, and for the semiclassical situation (ħ → 0), we handle the scalar fields. We obtain existence and multiplicity results of stationary solutions via critical point theory.

On the Passage from Atomic to Continuum Theory for Thin Films

Abstract: We give a rigorous derivation of a continuum theory from atomic models for thin films. This scheme has been proposed by Friesecke and James in [J. Mech. Phys. Solids 48, 1519–1540 (2000)]. The resulting continuum energy expression is obtained by integrating a stored energy density which not only depends on the deformation gradient, but also on ν-1 director fields when ν is the (fixed) number of atomic film layers.

On Positivity for the Biharmonic Operator under Steklov Boundary Conditions

Abstract: The positivity-preserving property for the inverse of the biharmonic operator under Steklov boundary conditions is studied. It is shown that this property is quite sensitive to the parameter involved in the boundary condition. Moreover, positivity of the Steklov boundary value problem is linked with positivity under boundary conditions of Navier and Dirichlet type.

Steady Periodic Hydroelastic Waves

Abstract: This is a study of steady periodic travelling waves on the surface of an infinitely deep irrotational ocean when the top streamline is in contact with a light frictionless membrane that strongly resists stretching and bending and the pressure in the air above is constant. It is shown that this is a free-boundary problem for the domain of a harmonic function (the stream function) which is zero on the boundary and at which its normal derivative is determined by the boundary geometry. With the wavelength fixed at 2π, we find travelling waves with arbitrarily large speeds for a significant class of membranes. Our approach is based on Zakharov’s Hamiltonian theory of water waves to which elastic effects at the surface have been added. However we avoid the Hamiltonian machinery by first defining a Lagrangian in terms of kinetic and potential energies using physical variables. A conformal transformation then yields an equivalent Lagrangian in which the unknown function is the wave height. Once critical points of that Lagrangian have been shown to correspond to the physical problem, the existence of hydroelastic waves for a class of membranes is established by maximization. Hardy spaces on the unit disc and the Hilbert transform on the unit circle play a role in the analysis.

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

Abstract: Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in Hs for data in Hs. We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.

Vanishing Viscosity Method for Transonic Flow

Abstract: A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant $$\gamma \in [1,3)$$ This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields Mathematical entropy pairs are constructed through the Loewner–Morawetz relation via entropy generators governed by a generalized Tricomi equation of mixed elliptic–hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation with uniformly bounded velocity angles, converges to an entropy solution of the inviscid transonic flow problem