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 a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4. PubDate: 2015-11-26

Abstract: Abstract
In this work, we study the Gross–Pitaevskii hierarchy on general—rational and irrational—rectangular tori of dimensions two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in the previous work of Kirkpatrick, Schlein, and Staffilani. PubDate: 2015-11-26

Abstract: Abstract
We prove that solitons (or solitary waves) of the Zakharov–Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg–de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [Proc R Soc Edinburgh 126:89–112, 1996]. Our proofs follow the ideas of Martel [SIAM J Math Anal 157:759–781, 2006] and Martel and Merle [Math Ann 341:391–427, 2008], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV–NLS dynamics. This last Virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition. PubDate: 2015-11-24

Abstract: Abstract
It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem. PubDate: 2015-11-16

Abstract: Abstract
We focus on a special type of domain wall appearing in the Landau–Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free
\({\mathbb{S}^2}\)
-valued transition layers that connect two directions
\({m_\theta^\pm \in \mathbb{S}^2}\)
(differing by an angle
\({2\theta}\)
) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such “asymmetric” domain walls in the limit
\({\theta \downarrow 0}\)
. As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model. PubDate: 2015-11-13

Abstract: Abstract
We consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new W
1,1-trace estimate for the diffuse boundary condition and a delicate construction of an
\({\varepsilon}\)
-tubular neighborhood of the singular set. PubDate: 2015-11-12

Abstract: Abstract
Consider the scattering of electromagnetic waves from a large rectangular cavity embedded in the infinite ground plane. There are two fundamental polarizations for the scattering problem in two dimensions: TM (transverse magnetic) and TE (transverse electric). In this paper, new stability results for the cavity problems are established for large rectangular shape cavities in both polarizations. For the TM cavity problem, an asymptotic property of the solution and a stability estimate with an improved dependence on the high wavenumber are derived. In the TE case, the first stability result is established with an explicit dependence on the wave number. PubDate: 2015-11-09

Abstract: Abstract
Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by
\({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\)
for some small enough constant
\({\eta}\)
and some
\({W^{k+2,p}}\)
domain
\({\Omega_{0}}\)
, with initial vorticity belonging to
\({L^{1} \cap L^{p}}\)
and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain
\({\Omega_0}\)
is preserved by time evolution. PubDate: 2015-11-05