Abstract: Abstract
This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. We introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper also allows us to establish new results. PubDate: 2015-12-01

Abstract: Abstract
We study the control and stabilization of the Benjamin-Ono equation in
\({L^2(\mathbb{T})}\)
, the lowest regularity where the initial value problem is well-posed. This problem was already initiated in Linares and Rosier (Trans Am Math Soc 367:4595–4626, 2015) where a stronger stabilization term was used (that makes the equation of parabolic type in the control zone). Here we employ a more natural stabilization term related to the L
2–norm. Moreover, by proving a theorem of controllability in L
2, we manage to prove the global controllability in large time. Our analysis relies strongly on the bilinear estimates proved in Molinet and Pilod (Anal PDE 5:365–395, 2012) and some new extension of these estimates established here. PubDate: 2015-12-01

Abstract: Abstract
This paper is a follow-up of Gérard-Varet and Lacave (Arch Ration Mech Anal 209(1):131–170, 2013), on the existence of global weak solutions to the two dimensional Euler equations in singular domains. In Gérard-Varet and Lacave (Arch Ration Mech Anal 209(1):131–170, 2013), we have established the existence of weak solutions for a large class of bounded domains, with initial vorticity in L
p
(p > 1). For unbounded domains, we have proved a similar result only when the initial vorticity is in
\({L^{p}_{c}}\)
(p > 2) and when the domain is the exterior of a single obstacle. The goal here is to retrieve these two restrictions: we consider general initial vorticity in
\({L^{1} {\cap} L^{p}}\)
(p > 1), outside an arbitrary number of obstacles (not reduced to points). PubDate: 2015-12-01

Abstract: Abstract
We consider the time evolution of a one dimensional n-gradient continuum. Our aim is to construct and analyze discrete approximations in terms of physically realizable mechanical systems, referred to as microscopic because they are living on a smaller space scale. We validate our construction by proving a convergence theorem of the microscopic system to the given continuum, as the scale parameter goes to zero. PubDate: 2015-12-01

Abstract: Abstract
In this paper we study the focusing cubic wave equation in 1 + 5 dimensions with radial initial data as well as the one-equivariant wave maps equation in 1+3 dimensions with the model target manifolds
\({\mathbb{S}^3}\)
and
\({\mathbb{H}^3}\)
. In both cases the scaling for the equation leaves the
\({\dot{H}^{\frac{3}{2}} \times \dot{H}^{\frac{1}{2}}}\)
-norm of the solution invariant, which means that the equation is super-critical with respect to the conserved energy. Here we prove a conditional scattering result: if the critical norm of the solution stays bounded on its maximal time of existence, then the solution is global in time and scatters to free waves as
\({t \to \pm \infty}\)
. The methods in this paper also apply to all supercritical power-type nonlinearities for both the focusing and defocusing radial semi-linear equation in 1+5 dimensions, yielding analogous results. PubDate: 2015-12-01

Abstract: Abstract
We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an L
1 neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiments and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis. PubDate: 2015-12-01

Abstract: Abstract
Given an open bounded subset Ω of
\({\mathbb{R}^n}\)
, which is convex and satisfies an interior sphere condition, we consider the pde
\({-\Delta_{\infty} u = 1}\)
in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C
1(Ω). We then investigate the overdetermined Serrin-type problem, formerly considered in Buttazzo and Kawohl (Int Math Res Not, pp 237–247, 2011), obtained by adding the extra boundary condition
\({ \nabla u = a}\)
on ∂Ω; by using a suitable P-function we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball which touches ∂Ω at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of Ω coincide. In turn, in dimension n = 2, this entails that Ω must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class C
2. PubDate: 2015-12-01

Abstract: Abstract
The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in
\({L^{\infty}}\)
are obtained through the vanishing viscosity method and the compensated compactness framework. The
\({L^{\infty}}\)
uniform estimate and H
−1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in
\({L^{\infty}}\)
to the Gauss-Codazzi equations yield the C
1,1 isometric immersions of surfaces with the given metrics. PubDate: 2015-12-01

Abstract: Abstract
Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in
\({\mathbb{R}^{3}}\)
as the sum of the area integral and an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of
\({\mathbb{S}^{2}}\)
and with fixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For a certain class of anisotropies vanishing at infinity, we prove the existence and nonexistence of volume-constrained,
\({\mathbb{S}^{2}}\)
-type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show the existence of extremals for the full isoperimetric inequality. PubDate: 2015-12-01

Abstract: Abstract
In this paper, we are concerned with the global existence and stability of a smooth supersonic flow with vacuum state at infinity in a three-dimensional infinitely long divergent nozzle. The flow is described by a three-dimensional steady potential equation, which is multi-dimensional quasilinear hyperbolic (but degenerate at infinity) with respect to the supersonic direction, and whose linearized part admits the form
\({{\partial_t^2-\frac{1}{(1+t)^{2(\gamma-1)}}(\partial_1^2+\partial_2^2)+\frac{2(\gamma-1)}{1+t}\partial_t}}\)
for
\({{1 < \gamma < 2}}\)
. From the physical point of view, due to the expansive geometric property of the divergent nozzle and the mass conservation of gases, the moving gases in the nozzle will gradually become rarefactive and tend to vacuum states at infinity, which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no strong resulting compressions in the motion of the flow. We will confirm such a global stability phenomenon by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite parts of the nozzle. PubDate: 2015-12-01

Abstract: Abstract
In this paper we derive a new energy identity for the three-dimensional incompressible Navier–Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier–Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier–Stokes equations whose critical norms can be arbitrarily large. PubDate: 2015-12-01

Abstract: Abstract
We establish the unexpected equality of the optimal volume density of total flux of a linear vector field
\({x \longmapsto Mx}\)
and the least volume fraction that can be swept out by submacroscopic switches, separations, and interpenetrations associated with the purely submacroscopic structured deformation (i, I + M). This equality is established first by identifying a dense set
\({\mathcal{S}}\)
of
\({N{\times}N}\)
matrices M for which the optimal total flux density equals trM , the absolute value of the trace of M. We then use known representation formulae for relaxed energies for structured deformations to show that the desired least volume fraction associated with (i, I + M) also equals trM . We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of
\({x \longmapsto Mx}\)
and the volume fraction swept out by submacroscopic separations alone, with common value (trM)+. Similarly, the optimal volume density of the negative part of the flux of
\({x \longmapsto Mx}\)
and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)−. PubDate: 2015-12-01

Abstract: Abstract
We consider exact nonlinear solitary water waves on a shear flow with an arbitrary distribution of vorticity. Ignoring surface tension, we impose a non-constant pressure on the free surface. Starting from a uniform shear flow with a flat free surface and a supercritical wave speed, we vary the surface pressure and use a continuation argument to construct a global connected set of symmetric solitary waves. This set includes waves of depression whose profiles increase monotonically from a central trough where the surface pressure is at its lowest, as well as waves of elevation whose profiles decrease monotonically from a central crest where the surface pressure is at its highest. There may also be two waves in this connected set with identical surface pressure, only one of which is a wave of depression. PubDate: 2015-11-01

Abstract: Abstract
We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter
\({\varepsilon}\)
. PubDate: 2015-11-01

Abstract: Abstract
In this paper we provide a complete analogy between the Cauchy–Lipschitz and the DiPerna–Lions theories for ODE’s, by developing a local version of the DiPerna–Lions theory. More precisely, we prove the existence and uniqueness of a maximal regular flow for the DiPerna–Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy–Lipschitz theory, this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field. In particular, a global assumption on the divergence is needed to obtain a proper blow-up of the trajectories. PubDate: 2015-11-01

Abstract: Abstract
We investigate the existence of solutions
\({E:\mathbb{R}^3 \to \mathbb{R}^3}\)
of the time-harmonic semilinear Maxwell equation
$$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$
where
\({V:\mathbb{R}^3 \to \mathbb{R}}\)
,
\({V(x) \leqq 0}\)
almost everywhere on
\({\mathbb{R}^3}\)
,
\({\nabla \times}\)
denotes the curl operator in
\({\mathbb{R}^3}\)
and
\({F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}\)
is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the
\({L^{3/2}}\)
-norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and
\({V(x) = -\mu\omega^2 \varepsilon(x)}\)
where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field
\({\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}\)
and
\({\varepsilon}\)
is the linear part of the permittivity in an inhomogeneous medium. PubDate: 2015-11-01

Abstract: Abstract
We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length
\({\psi(b, t)}\)
, which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length
\({\psi_0(b, t)}\)
obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its
\({\mathcal H^1}\)
-elliptic envelope. PubDate: 2015-11-01

Abstract: Abstract
Isoperimetric inequalities for the principal eigenvalues of the Robin-Laplacian are interpreted as free discontinuity problems (of unusual type). We prove a full range of Faber–Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincaré inequalities with trace terms in
\({\mathbb{R}^N}\)
. PubDate: 2015-11-01

Abstract: Abstract
We prove the immediate appearance of a lower bound for mild solutions to the full Boltzmann equation in the torus or a C
2 convex domain with specular boundary conditions, under the sole assumption of continuity away from the grazing set of the solution. These results are entirely constructive if the domain is C
3 and strictly convex. We investigate a wide range of collision kernels, some satisfying Grad’s cutoff assumption and others not. We show that this lower bound is exponential, independent of time and space with explicit constants depending only on the a priori bounds on the solution. In particular, this lower bound is Maxwellian in the case of cutoff collision kernels. A thorough study of characteristic trajectories, as well as a geometric approach of grazing collisions against the boundary are derived. PubDate: 2015-11-01

Abstract: Abstract
We derive the effective energy density of thin membranes of liquid crystal elastomers as the
\({\Gamma}\)
-limit of a widely used bulk model. These membranes can display fine-scale features both due to wrinkling that one expects in thin elastic membranes and due to oscillations in the nematic director that one expects in liquid crystal elastomers. We provide an explicit characterization of the effective energy density of membranes and the effective state of stress as a function of the planar deformation gradient. We also provide a characterization of the fine-scale features. We show the existence of four regimes: one where wrinkling and microstructure reduces the effective membrane energy and stress to zero, a second where wrinkling leads to uniaxial tension, a third where nematic oscillations lead to equi-biaxial tension and a fourth with no fine scale features and biaxial tension. Importantly, we find a region where one has shear strain but no shear stress and all the fine-scale features are in-plane with no wrinkling. PubDate: 2015-11-01