Abstract: The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional. PubDate: 2016-03-01

Abstract: We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile. PubDate: 2016-03-01

Abstract: In this work we study the long time inviscid limit of the two dimensional Navier–Stokes equations near the periodic Couette flow. In particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin’s 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like two dimensional Euler for times
\({{t \lesssim Re^{1/3}}}\)
, and in particular exhibits inviscid damping (for example the vorticity weakly approaches a shear flow). For times
\({{t \gtrsim Re^{1/3}}}\)
, which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales
\({{t \gtrsim Re}}\)
back to the Couette flow. When properly defined, the dissipative length-scale in this setting is
\({{\ell_D \sim Re^{-1/3}}}\)
, larger than the scale
\({{\ell_D \sim Re^{-1/2}}}\)
predicted in classical Batchelor–Kraichnan two dimensional turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re
−1) L
2 function. PubDate: 2016-03-01

Abstract: The asymptotic analysis carried out in this paper for the problem of a multiple scattering in three dimensions of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy’s method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and the taking into account of self-interactions. In this way, it is established that the usual Foldy model may lead to an approximation whose asymptotic behavior is the same than that obtained when the multiple scattering effects are completely neglected. PubDate: 2016-03-01

Abstract: Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex. PubDate: 2016-03-01

Abstract: We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases. PubDate: 2016-03-01

Abstract: We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in
\({\mathbb{R}^3}\)
equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such types of energies arise, for example, in a model for bilayer membranes introduced by Peletier and Röger (Arch Ration Mech Anal 193(3), 475–537, 2009). Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds. PubDate: 2016-03-01

Abstract: We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shrinking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. We also prove results on the concavity properties of solutions. One of the main tools of this paper is the intersection number principle, however in order to deal with solutions with free boundaries, we introduce what we call “the extended intersection number principle”, which turns out to be exceedingly useful in handling curves with moving endpoints. PubDate: 2016-03-01

Abstract: We study the asymptotic behaviour of the resolvents
\({(\mathcal{A}^\varepsilon+I)^{-1}}\)
of elliptic second-order differential operators
\({{\mathcal{A}}^\varepsilon}\)
in
\({\mathbb{R}^d}\)
with periodic rapidly oscillating coefficients, as the period
\({\varepsilon}\)
goes to zero. The class of operators covered by our analysis includes both the “classical” case of uniformly elliptic families (where the ellipticity constant does not depend on
\({\varepsilon}\)
) and the “double-porosity” case of coefficients that take contrasting values of order one and of order
\({\varepsilon^2}\)
in different parts of the period cell. We provide a construction for the leading order term of the “operator asymptotics” of
\({(\mathcal{A}^\varepsilon+I)^{-1}}\)
in the sense of operator-norm convergence and prove order
\({O(\varepsilon)}\)
remainder estimates. PubDate: 2016-03-01

Abstract: In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian n-body problems. In our assumption, the
\({n=2l \geqq 4}\)
particles are invariant under the dihedral rotation group D
l
in
\({\mathbb{R}^3}\)
such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is the variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations. PubDate: 2016-03-01

Abstract: In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities. PubDate: 2016-03-01

Abstract: We derive optimal scaling laws for the macroscopic fracture energy of polymers failing by crazing. We assume that the effective deformation-theoretical free-energy density is additive in the first and fractional deformation-gradients, with zero growth in the former and linear growth in the latter. 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. For this particular geometry, we derive optimal scaling laws for the dependence of the specific fracture energy on cross-sectional area, micromechanical parameters, opening displacement and intrinsic length of the material. In particular, the upper bound is obtained by means of a construction of the crazing type. PubDate: 2016-02-01

Abstract: A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles. PubDate: 2016-02-01

Abstract: We study quantitative compactness estimates in
\({\mathbf{W}^{1,1}_{{\rm loc}}}\)
for the map
\({S_t}\)
,
\({t > 0}\)
that is associated with the given initial data
\({u_0\in {\rm Lip} (\mathbb{R}^N)}\)
for the corresponding solution
\({S_t u_0}\)
of a Hamilton–Jacobi equation
$$u_t+H\big(\nabla_{\!x} u\big)=0, \qquad t\geq 0,\quad x\in\mathbb{R}^N,$$
with a uniformly convex Hamiltonian
\({H=H(p)}\)
. We provide upper and lower estimates of order
\({1/\varepsilon^N}\)
on the Kolmogorov
\({\varepsilon}\)
-entropy in
\({\mathbf{W}^{1,1}}\)
of the image through the map S
t
of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by Lax (Course on Hyperbolic Systems of Conservation Laws. XXVII Scuola Estiva di Fisica Matematica, Ravello, 2002) within the context of conservation laws, and could provide a measure of the order of “resolution” of a numerical method implemented for this equation. PubDate: 2016-02-01

Abstract: Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we discuss two physically meaningful regularization procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows us to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular “effective” velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics. PubDate: 2016-02-01

Abstract: In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well and we also provide a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an L
∞ bound, where we prove that the total variation decreases by projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, we obtain BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also establish some properties of the Wasserstein projection which are interesting in their own right, and allow, for instance, for the proof of the uniqueness of such a projection in a very general framework. PubDate: 2016-02-01

Abstract: We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal one dimensional Euler equations. PubDate: 2016-02-01

Abstract: We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. PubDate: 2016-02-01

Abstract: We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L
2∩ L
∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy. PubDate: 2016-02-01

Abstract: We consider the relationship between three continuum liquid crystal theories: Oseen–Frank, Ericksen and Landau–de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters s, n, and Q, we establish connections between the variational formulations of these theories. We use these results to justify a version of the Oseen–Frank theory using special functions of bounded variation. This proposed model can describe both orientable and non-orientable defects. Finally we study a number of frustrated nematic and cholesteric liquid crystal systems and show that the model predicts the existence of point and surface discontinuities in the director. PubDate: 2016-02-01