Abstract: Abstract
We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function. PubDate: 2015-04-01

Abstract: Abstract
The equatorial shallow water equations at low Froude number form a symmetric hyperbolic system with large terms containing a variable coefficient, the Coriolis parameter f, which depends on the latitude. The limiting behavior of the solutions as the Froude number tends to zero was investigated rigorously a few years ago, using the common approximation that the variations of f with latitude are linear. In that case, the large terms have a peculiar structure, due to special properties of the harmonic oscillator Hamiltonian, which can be exploited to prove strong uniform a priori estimates in adapted functional spaces. It is shown here that these estimates still hold when f deviates from linearity, even though the special properties on which the proofs were based have no obvious generalization. As in the linear case, existence, uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates. PubDate: 2015-04-01

Abstract: Abstract
The non-isentropic Euler system with periodic initial data in
\({{\mathbb{R}}^1}\)
is studied by analyzing wave interactions in a framework of specially chosen Riemann invariants, generalizing Glimm’s functionals and applying the method of approximate conservation laws and approximate characteristics. An
\({{\mathcal O}(\varepsilon^{-2})}\)
lower bound is established for the life span of the entropy solutions with initial data that possess
\({\varepsilon}\)
variation in each period. PubDate: 2015-04-01

Abstract: Abstract
We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, that is, heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of Γ-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts. PubDate: 2015-04-01

Abstract: Abstract
We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward-backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned. PubDate: 2015-04-01

Abstract: Abstract
In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm–Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem, which depends on the mass parameter
\({\beta \in [0,9]}\)
and the eccentricity
\({e \in [0,1)}\)
. Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagrangian solutions. PubDate: 2015-04-01

Abstract: Abstract
We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures. PubDate: 2015-04-01

Abstract: Abstract
We study the optimal sets
\({\Omega^\ast\subseteq\mathbb{R}^d}\)
for spectral functionals of the form
\({F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}\)
, which are bi-Lipschitz with respect to each of the eigenvalues
\({\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}\)
of the Dirichlet Laplacian on
\({\Omega}\)
, a prototype being the problem
$$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ \Omega =1\big\}}.$$
We prove the Lipschitz regularity of the eigenfunctions
\({u_1,\ldots,u_p}\)
of the Dirichlet Laplacian on the optimal set
\({\Omega^\ast}\)
and, as a corollary, we deduce that
\({\Omega^\ast}\)
is open. For functionals depending only on a generic subset of the spectrum, as for example
\({\lambda_k(\Omega)}\)
, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved. PubDate: 2015-04-01

Abstract: Abstract
We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order. PubDate: 2015-03-04

Abstract: Abstract
The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove the existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions. PubDate: 2015-03-04

Abstract: Abstract
This paper considers L
2-asymptotic stability of the spatially inhomogeneous Navier–Stokes–Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes–Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal L
p
-in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal L
2-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal L
p
-in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear. PubDate: 2015-03-01

Abstract: Abstract
A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process. The model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory. An assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional. Triple junction, a phenomenon in which the three components meet at a single point, is a key issue addressed in the construction. Coarsening, an undesirable scenario of excessive growth, is prevented by a lower bound on the long range interaction term in the free energy. The proof involves several ideas: perturbation of double bubbles in a restricted class; use of internal variables to remove nonlinear constraints, local minimization in a restricted class formulated as a nonlinear problem on a Hilbert space; and reduction to finite dimensional minimization. This existence theorem predicts a new morphological phase of a double bubble assembly. PubDate: 2015-03-01

Abstract: Abstract
In this paper, we prove in two dimensions the global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in Imanuvilov and Yamamoto (Global uniqueness in inverse boundary value problems for Navier–Stokes equations and Lamé ststem in two dimensions. arXiv:1309.1694, 2013) to prove the uniqueness of the viscosity for the Stokes equations and for the Navier–Stokes equations. PubDate: 2015-03-01

Abstract: Abstract
We prove the Eshelby theorem for an ellipsoidal piezoelectric inclusion in an infinite piezoelectric material. Explicit formulas for the link and polarization matrices are derived. Passing to the limits with respect to parameters in the corresponding equations, the result is extended to cases when either the inclusion or the surrounding material is purely elastic. PubDate: 2015-03-01

Abstract: Abstract
We present energetic and strain-threshold models for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. By allowing for anisotropic elastic moduli and multiple damaged states we present the issues for the first time in a truly elastic setting, and show that the threshold methods developed in (Garroni, A., Larsen, C. J., Threshold-based quasi-static brittle damage evolution, Archive for Rational Mechanics and Analysis 194 (2), 585–609, 2009) extend naturally to elastic materials with non-interacting damage. We show the existence of solutions and that energetic evolutions are also threshold evolutions. PubDate: 2015-03-01

Abstract: Abstract
In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W
2, p
estimates and sharp C
1, α
estimates for the potentials, which satisfy a Monge–Ampère type equation. The W
2, p
estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampère equation. PubDate: 2015-03-01

Abstract: Abstract
In this paper, we propose a systematic way of liquid crystal modeling to build connections between microscopic theory and macroscopic theory. In the first part, we propose a new Q-tensor model based on Onsager’s molecular theory for liquid crystals. The Oseen–Frank theory can be recovered from the derived Q-tensor theory by making a uniaxial assumption, and the coefficients in the Oseen–Frank model can be examined. In addition, the smectic-A phase can be characterized by the derived macroscopic model. In the second part, we derive a new dynamic Q-tensor model from Doi’s kinetic theory by the Bingham closure, which obeys the energy dissipation law. Moreover, the Ericksen–Leslie system can also be derived from new Q-tensor system by making an expansion near the local equilibrium. PubDate: 2015-03-01

Abstract: Abstract
In this paper we investigate the limit behavior of the solution to quasi-static Biot’s equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi’s time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the two dimensional Navier’s linear elasticity equations, with elastic moduli depending on Gassmann’s and Biot’s coefficients. The bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. The pressure equation is parabolic only in the vertical direction. As additional terms it contains the time derivative of the in-plane Laplacian of the vertical deflection of the plate and of the elastic in-plane compression term. PubDate: 2015-03-01

Abstract: Abstract
We study Hopf bifurcation from traveling-front solutions in the Cahn–Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of physical phenomena, including pattern formation in chemical deposition and precipitation processes. Technically, we study bifurcation in the presence of an essential spectrum. We contribute a simple and direct functional analytic method and determine bifurcation coefficients explicitly. Our approach uses exponential weights to recover Fredholm properties and spectral flow ideas to compute Fredholm indices. Simple mass conservation helps compensate for negative indices. We also construct an explicit, prototypical example, prove the existence of a bifurcating front, and determine the direction of bifurcation. PubDate: 2015-02-28

Abstract: Abstract
We formulate and study an elliptic transmission-like problem combining local and nonlocal elements. Let
\({\mathbb{R}^n}\)
be separated into two components by a smooth hypersurface Γ. On one side of Γ, a function satisfies a local second-order elliptic equation. On the other, it satisfies a nonlocal one of lower order. In addition, a nonlocal transmission condition is imposed across the interface Γ, which has a natural variational interpretation. We deduce the existence of solutions to the corresponding Dirichlet problem, and show that under mild assumptions they are Hölder continuous, following the method of De Giorgi. The principal difficulty stems from the lack of scale invariance near Γ, which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter. We then turn to the question of optimal regularity and qualitative properties of solutions, and show that (in the case of constant coefficients and flat Γ) they satisfy a kind of transmission condition, where the ratio of “fractional conormal derivatives” on the two sides of the interface is fixed. A perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface. Throughout, we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution, which has an interpretation in the context of quasigeostrophic dynamics. PubDate: 2015-02-25