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Archive for Rational Mechanics and Analysis    [7 followers]  Follow
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ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527
• On the Rigorous Derivation of the 2D Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-Body Dynamics
• Abstract: Abstract We consider the 3D quantum many-body dynamics describing a dilute Bose gas with strong confinement in one direction. We study the corresponding Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, which contains a diverging coefficient as the strength of the confining potential tends to ∞. We find that this diverging coefficient is counterbalanced by the limiting structure of the density matrices and we establish the convergence of the BBGKY hierarchy. Moreover, we prove that the limit is fully described by a 2D cubic nonlinear Schrödinger equation (NLS) and we obtain the exact 3D to 2D coupling constant.
PubDate: 2013-12-01

• A Free Boundary Problem Arising from Segregation of Populations with High Competition
• Abstract: Abstract In this work, we show how to obtain a free boundary problem as the limit of a fully nonlinear elliptic system of equations that models population segregation of the Gause–Lotka–Volterra type. We study the regularity of the solutions. In particular, we prove Lipschitz regularity across the free boundary. The problem is motivated by the work done by Caffarelli, Karakhanyan and Fang-Hua Lin for the linear case.
PubDate: 2013-12-01

• Well-Posedness of the Ericksen–Leslie System
• Abstract: Abstract We prove the local well-posedness of the Ericksen–Leslie system, and the global well-posedness for small initial data under a physical constraint condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg–Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.
PubDate: 2013-12-01

• Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations
• Abstract: Abstract Initial value problems for quasilinear parabolic equations having Radon measures as initial data have been widely investigated, looking for solutions which for positive times take values in some function space. In contrast, it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the Radon measures of the initial data. We call such solutions measure-valued, in contrast to function-valued solutionspreviously considered in the literature. We first show that there is a natural notion of measure-valued solution of problem (P) below, in spite of its nonlinear character. A major consequence of our definition is that, if the space dimension is greater than one, the concentrated part of the solution with respect to the Newtonian capacity is constant in time. Subsequently, we prove that there exists exactly one solution of the problem, such that the diffuse part with respect to the Newtonian capacity of the singular part of the solution (with respect to the Lebesgue measure) is concentrated for almost every positive time on the set where “the regular part (with respect to the Lebesgue measure) is large”. Moreover, using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time. Finally, the regularity problem is addressed, as we give conditions (depending on the space dimension, the initial data and the rate of convergence at infinity of the nonlinearity ψ) to ensure that the measure-valued solution of problem (P) is, in fact, function-valued.
PubDate: 2013-12-01

• Equivalent Theories of Liquid Crystal Dynamics
• Abstract: Abstract There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen–Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the micropolar theory includes the director theory. In this paper we show that this is the case by using symmetry reduction techniques and introducing a new system that is equivalent to the Ericksen–Leslie equations and may include disclination dynamics. The resulting equations of motion are verified to be completely equivalent, although one of the two different reductions offers the possibility of accounting for orientational defects. After applying these two approaches to the ordered micropolar theory of Lhuiller and Rey, all the results are eventually extended to flowing complex fluids, such as nematic liquid crystals.
PubDate: 2013-12-01

• Boundary Regularity for Solutions to the Linearized Monge–Ampère Equations
• Abstract: Abstract We obtain boundary Hölder gradient estimates and regularity for solutions to the linearized Monge–Ampère equations under natural assumptions on the domain, Monge–Ampère measures and boundary data. Our results are affine invariant analogues of the boundary Hölder gradient estimates of Krylov.
PubDate: 2013-12-01

• Stability of the Self-similar Dynamics of a Vortex Filament
• Abstract: Abstract In this paper we continue our investigation of self-similar solutions of the vortex filament equation, also known as the binormal flow or the localized induction equation. Our main result is the stability of the self-similar dynamics of small perturbations of a given self-similar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic one-dimensional Schrödinger equation, connected to the binormal flow by Hasimoto’s transform.
PubDate: 2013-12-01

• Nonlinear Elliptic–Parabolic Problems
• Abstract: Abstract We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).
PubDate: 2013-12-01

• On the Euler–Poincaré Equation with Non-Zero Dispersion
• Abstract: Abstract We consider the Euler–Poincaré equation on ${\mathbb{R}^d, \, d \geqq 2}$ R d , d ≧ 2 . For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu (Commun Math Phys 314:671–687, 2012). Our analysis exhibits some new concentration mechanisms and hidden monotonicity formulas associated with the Euler–Poincaré flow. In particular we show an abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.
PubDate: 2013-12-01

• Embedded Surfaces of Arbitrary Genus Minimizing the Willmore Energy Under Isoperimetric Constraint
• Abstract: The isoperimetric ratio of an embedded surface in ${\mathbb{R}^3}$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus ${g \geqq 0}$ . The corresponding problem in the case of spherical surfaces, that is g = 0, was recently solved by Schygulla (see Schygulla, Arch Ration Mech Anal 203:901–941, 2012) with different methods.
PubDate: 2013-11-29

• Homogenization of Weakly Coupled Systems of Hamilton–Jacobi Equations with Fast Switching Rates
• Abstract: Abstract We consider homogenization for weakly coupled systems of Hamilton–Jacobi equations with fast switching rates. The fast switching rate terms force the solutions to converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and the rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.
PubDate: 2013-11-20

• Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow
• Abstract: Abstract We investigate the (slightly) super-critical two-dimensional Euler equations. The paper consists of two parts. In the first part we prove well-posedness in C s spaces for all s > 0. We also give growth estimates for the C s norms of the vorticity for ${0 < s \leqq 1}$ . In the second part we prove global regularity for the vortex patch problem in the super-critical regime. This paper extends the results of Chae et al. where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi–Constantin on the vortex patch problem to the slightly supercritical case. Both problems we study in the setting of the whole space.
PubDate: 2013-11-14

• Upper Bounds on Waiting Times for the Thin-Film Equation: The Case of Weak Slippage
• Abstract: Abstract We derive upper bounds on the waiting time of solutions to the thin-film equation in the regime of weak slippage ${n\in [2,\frac{32}{11})}$ . In particular, we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary. For ${n\in (2,\frac{32}{11})}$ , our estimates are sharp, for n = 2, they are sharp up to a logarithmic correction term. Note that the case n = 2 corresponds—with a grain of salt—to the assumption of the Navier slip condition at the fluid-solid interface. We also obtain results in the regime of strong slippage ${n \in (1,2)}$ ; however, in this regime we expect them not to be optimal. Our method is based on weighted backward entropy estimates, Hardy’s inequality and singular weight functions; we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long.
PubDate: 2013-11-09

• Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two
• Abstract: Abstract In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C 1-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .
PubDate: 2013-11-09

• Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations
• Abstract: Abstract This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D 2 u) =  f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L n norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C 1,α regularity estimates are also delivered when ${f\in L^{n+\varepsilon}}$ . The limiting upper borderline case, ${f \in L^\infty}$ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that ${u \in C^{1,{\rm Log-Lip}}}$ , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the ${C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}$ norm of u based on the L n-ε norm of f, where ɛ is the Escauriaza universal constant. The exponent ${\frac{n-2\varepsilon}{n-\varepsilon}}$ is optimal. When the source function f lies in L q , n > q > n−ε, we also obtain the exact, improved sharp Hölder exponent of continuity.
PubDate: 2013-11-09

• Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell
• Abstract: Abstract In this paper we analyse the interaction of an incompressible Newtonian fluid with a linearly elastic Koiter shell whosemotion is restricted to transverse displacements. The middle surface of the shell constitutes the mathematical boundary of the three-dimensional fluid domain. We show that weak solutions exist as long as the magnitude of the displacement stays below some (possibly large) bound that rules out selfintersections of the shell.
PubDate: 2013-11-07

• Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations
• Abstract: Abstract We study the energy cascade problematic for some nonlinear Schrödinger equations on ${\mathbb{T}^2}$ in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one ${ u ^2u}$ . Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).
PubDate: 2013-11-05

• Optimal Scaling in Solids Undergoing Ductile Fracture by Void Sheet Formation
• Abstract: Abstract This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.
PubDate: 2013-11-05

• Desingularization of Vortex Rings and Shallow Water Vortices by a Semilinear Elliptic Problem
• Abstract: Abstract Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problem \left\{ \begin{aligned} -{\rm div} \left(\frac{\nabla u_{\varepsilon}}{b}\right) & = \frac{1}{\varepsilon^2} b f \left(u_{\varepsilon} - \log \tfrac{1}{\varepsilon} q \right) & & \text{ in } \; \Omega, \\u_\varepsilon & = 0 & & \text{ on } \; \partial \Omega, \end{aligned}\right. for small values of ${\varepsilon > 0}$ .
PubDate: 2013-11-01

• On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap
• Abstract: Abstract We consider the dynamics of the three-dimensional N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N 3β-1 V(N β x). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for ${\beta \in (0, 2/7]}$ by adapting and simplifying an argument in Chen and Pavlović (Annales Henri Poincaré, 2013) which solves the problem for ${\beta \in (0, 1/4)}$ in the absence of a trap.
PubDate: 2013-11-01

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