Abstract: Abstract
We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton–Jacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent flame speed s
T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as s
T ≧ O(A/log A). In the second regime where A is slightly above the critical value, homogenization breaks down, and s
T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When A is sufficiently large, the whole flame front ceases to propagate forward due to the flow induced straining. In particular, along directions p = (±1, 0) and (0, ±1), s
T is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player differential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories. PubDate: 2014-10-01

Abstract: Abstract
This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework. PubDate: 2014-10-01

Abstract: Abstract
In this manuscript we are interested in stored energy functionals W defined on the set of d × d matrices, which not only fail to be convex but satisfy
\({{\rm lim}_{\det \xi \rightarrow 0^+}
W(\xi)=\infty.}\)
We initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form
\({E(\mathbf{u})=\int_\Omega (W(\nabla \mathbf{u}) -\mathbf{F}
\cdot \mathbf{u}) {\rm d}x}\)
, as well as their Euler–Lagrange equations. The techniques developed here can be applied to a class of functionals larger than those considered in this manuscript, although we keep our focus on polyconvex stored energy functionals of the form
\({W(\xi)=f(\xi) +h( {\rm det} \xi)}\)
– such that
\({{\rm lim}_{t
\rightarrow 0^+} h(t)=\infty}\)
– which appear in the study of Ogden material. We present a collection of perturbed and relaxed problems for which we prove uniqueness results. Then, we characterize these minimizers by their Euler–Lagrange equations. PubDate: 2014-10-01

Abstract: Abstract
We consider the generalised Burgers equation
$$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0, \,\, t \geqq 0, \,\, x \in S^1,$$
where f is strongly convex and ν is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for the dissipation length scale and the small-scale quantities which characterise the decaying Burgers turbulence, i.e., the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell et al. (J Fluid Mech 238:467–486, 1992). Note that we are dealing with decaying, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval [T
1, T
2], where T
1 and T
2 depend only on f and the initial condition, and do not depend on the viscosity. These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov’s 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form k
−2. These results remain valid in the inviscid limit. PubDate: 2014-10-01

Abstract: Abstract
In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with H
2 initial data. PubDate: 2014-10-01

Abstract: Abstract
We prove existence results concerning equations of the type
\({-\Delta_pu=P(u)+\mu}\)
for p > 1 and F
k
[−u] = P(u) + μ with
\({1 \leqq k < \frac{N}{2}}\)
in a bounded domain Ω or the whole
\({\mathbb{R}^N}\)
, where μ is a positive Radon measure and
\({P(u)\sim e^{au^\beta}}\)
with a > 0 and
\({\beta \geqq 1}\)
. Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form
\({u={\bf W}_{\alpha, p}^R[P(u)]+f}\)
in
\({\mathbb{R}^N}\)
, where
\({0 < R \leqq \infty}\)
and f is a positive integrable function. PubDate: 2014-10-01

Abstract: Abstract
We study the initial-boundary value problem for the Fokker–Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker–Planck equation with absorbing boundary conditions, as well as the Hölder continuity of the solutions up to the singular set. PubDate: 2014-10-01

Abstract: Abstract
Using the variational method, Chenciner and Montgomery (Ann Math 152:881–901, 2000) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver numerically found a similar periodic solution called “super-eight” in the planar four-body problem with equal mass. In this paper we prove the existence of the super-eight orbit by using the variational method. The difficulty of the proof is to eliminate the possibility of collisions. In order to solve it, we apply the scaling technique established by Tanaka (Ann Inst H Poincaré Anal Non Linéaire 10:215–238, 1993), (Proc Am Math Soc 122:275–284, 1994) and investigate the asymptotic behavior of a binary collision. PubDate: 2014-10-01

Abstract: Abstract
The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L
∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C
1. When A is continuous, we also obtain the linear approximation property of the absolute minimizer. PubDate: 2014-10-01

Abstract: Abstract
This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang and Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary. PubDate: 2014-09-01

Abstract: Abstract
We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator,
\({\triangle^{\alpha/2}}\)
for
\({\alpha \in (0,2)}\)
. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on α in the BV-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits
\({\alpha \downarrow 0}\)
and
\({\alpha \uparrow 2}\)
. In the limit
\({\alpha \uparrow 2}\)
,
\({\triangle^{\alpha/2}}\)
converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231–251, 1999) for local degenerate parabolic equations (thus providing an alternative proof). PubDate: 2014-09-01

Abstract: Abstract
In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations
$$- \varepsilon^2 M \left(\varepsilon^{2-N}\!\!\int_{\mathbf{R}^N} \nabla u ^2\,\mathrm{d} x \right) \Delta u \!+\! V(x) u \!=\! f(u) \quad{\rm in}\ \mathbf{R}^N, \quad u \!\in\! H^1(\mathbf{R}^N), \ N \!\geqq\!1,$$
where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions
\({(u_\varepsilon)_{\varepsilon \in (0,\tilde{\varepsilon}]}}\)
which concentrates at a local minimum of V after extracting a subsequence (ε
k
). PubDate: 2014-09-01

Abstract: Abstract
In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain
\({\mathcal{D}}\)
in the ambient space such that the weak solution is smooth if and only if its graph lies in
\({\mathcal{D}}\)
. PubDate: 2014-09-01

Abstract: Abstract
It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter
\({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\)
and the eccentricity
\({e \in [0, 1)}\)
. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed
\({e\in [0, 1)}\)
, we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are −1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail. PubDate: 2014-09-01

Abstract: Abstract
Following ideas in Maderna and Venturelli (Arch Ration Mech Anal 194:283–313, 2009), we prove that the Busemann function of the parabolic homotetic motion for a minimal central coniguration of the N-body problem is a viscosity solution of the Hamilton–Jacobi equation and that its calibrating curves are asymptotic to the homotetic motion. PubDate: 2014-09-01

Abstract: Abstract
We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model. PubDate: 2014-09-01

Abstract: Abstract
We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient
\({a(\cdot)}\)
. The model case is given by the functional
$$ w \mapsto \int ( Dw ^p + a(x) Dw ^q) \, {\rm d}x,$$
where q > p and
\({a(\cdot) \geqq 0}\)
. PubDate: 2014-08-15

Abstract: This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in L
p
-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In L
p
-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow one to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby differential constraints on the functions in the sequence can also be investigated easily—a distinct advantage over Young measure theory. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar’s framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a “geometric” way. After developing several aspects of the abstract theory, we consider three applications; for lamination microstructures, the hierarchy of oscillations is reflected in the MCF. The directional information retained in an MCF is harnessed in the relaxation theory for anisotropic integral functionals. Finally, we indicate how the theory pertains to the study of propagation of singularities in certain systems of PDEs. The proofs combine measure theory, Young measures, and harmonic analysis. PubDate: 2014-08-15

Abstract: Abstract
We study the spectrum of a large system of N identical bosons interacting via a two-body potential with strength 1/N. In this mean-field regime, Bogoliubov’s theory predicts that the spectrum of the N-particle Hamiltonian can be approximated by that of an effective quadratic Hamiltonian acting on Fock space, which describes the fluctuations around a condensed state. Recently, Bogoliubov’s theory has been justified rigorously in the case that the low-energy eigenvectors of the N-particle Hamiltonian display complete condensation in the unique minimizer of the corresponding Hartree functional. In this paper, we shall justify Bogoliubov’s theory for the high-energy part of the spectrum of the N-particle Hamiltonian corresponding to (non-linear) excited states of the Hartree functional. Moreover, we shall extend the existing results on the excitation spectrum to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. In particular, the latter covers the case of rotating Bose gases, when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices in the Hartree minimizer. PubDate: 2014-08-12

Abstract: Abstract
We find solutions
\({E : \Omega \to \mathbb{R}^3}\)
of the problem
$$\left\{\begin{aligned}&\nabla \times(\nabla \times E) + \lambda E = \partial_E F(x, E) &&\quad \text{in }\quad \Omega \\& \nu \times E = 0 &&\quad \text{on }\quad \partial \Omega \end{aligned} \right.$$
on a simply connected, smooth, bounded domain
\({\Omega \subset \mathbb{R}^3}\)
with connected boundary and exterior normal
\({\nu : \partial \Omega \to \mathbb{R}^3}\)
. Here
\({\nabla \times}\)
denotes the curl operator in
\({\mathbb{R}^3}\)
, the nonlinearity
\({F : \Omega \times \mathbb{R}^3 \to \mathbb{R}}\)
is superquadratic and subcritical in E. The model nonlinearity is of the form
\({F(x, E)=\Gamma(x) E ^p}\)
for
\({\Gamma \in L^\infty(\Omega)}\)
positive, some 2 < p < 6. It need not be radial nor even in the E-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for Ω surrounded by a perfect conductor. PubDate: 2014-08-12