Abstract: Abstract
Consider a monokinetic probability measure on the phase space
\({{\bf R}^N_{x} \times {\bf R}^N_{\xi}}\)
, i.e.
\({\mu^{\rm {in}} = \rho^{\rm {in}}(x)\delta(\xi - U^{\rm {in}}(x))}\)
where U
in is a vector field on R
N
and ρ
in a probability density on R
N
. Let Φ
t
be a Hamiltonian flow on R
N
× R
N
. In this paper, we study the structure of the transported measure
\({\mu(t) := \Phi_t\#\mu^{\rm {in}}}\)
and of its integral in the ξ variable denoted ρ(t). In particular, we give estimates on the number of folds in
\({\Phi_t({\rm graph of} U^{\rm {in}})}\)
, on which μ
(
t) is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles U
in with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp. PubDate: 2015-07-01

Abstract: Abstract
We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superposition of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time. More precisely, we consider the evolutionary equation
$$(v_\varepsilon)_t=\frac{1}{\varepsilon}\left( \mathcal{I}v_\varepsilon-\frac{1}{\varepsilon^{2s}}W'(v_\varepsilon)+\sigma(t, x)\right),$$
where
\({v_\varepsilon=v_\varepsilon(t, x)}\)
is the atom dislocation function at time t > 0 at the point
\({x \in \mathbb{R}, {\mathcal{I}_{s}}}\)
is an integro-differential operator of order
\({2s \in (0, 2), W}\)
is a periodic potential,
\({\sigma}\)
is an external stress and
\({\varepsilon > 0}\)
is a small parameter that takes into account the small periodicity scale of the crystal. We suppose that
\({v_\varepsilon(0, x)}\)
is the superposition of N−K transition layers in the positive direction and K in the negative one (with
\({K \in\{0,\dots,N\}}\)
); more precisely, we fix points
\({x_1^0 < \dots < x_N^0}\)
and we take
$$v_\varepsilon(0, x)= \frac{\varepsilon^{2s}}{W''(0)}\sigma(0, x)+\sum_{i=1}^N u\left(\zeta_i\frac{x-x_i^0}{\varepsilon}\right).$$
Here
\({\zeta_i}\)
is either −1 or 1, depending on the orientation of the transition layer u, which in turn solves the stationary equation
\({\mathcal{I}_{s} u=W'(u)}\)
. We show that our problem possesses a unique solution and that, as
\({\varepsilon \to 0^+}\)
, it approaches the sum of Heaviside functions H with different orientations centered at points x
i
(t), namely
$$\sum_{i=1}^N H(\zeta_i(x-x_i(t))).$$
The point x
i
evolves in time from
\({x_i^0}\)
, being subject to the external stress and a singular potential, which may be either attractive or repulsive, according to the different orientation of the transitions; more precisely, the speed
\({\dot x_i}\)
is proportional to
$$\sum_{j\neq i}\zeta_i\zeta_j\frac{x_i-x_j}{2s x_i-x_j ^{1+2s}}-\zeta_i\sigma(t, x_i).$$
The evolution of such a dynamical system may lead to collisions in finite time. We give a detailed description of such collisions when N = 2, 3 and we show that the solution itself keeps track of such collisions; indeed, at the collision time T
c
the two opposite dislocations have the tendency to annihilate each other and make the dislocation vanish, but only outside the coll... PubDate: 2015-07-01

Abstract: Abstract
We consider by a combination of analytical and numerical techniques, some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (1) can inviscid instability occur? (2) can viscous instability not detected by inviscid theory occur? (3) can there occur the—necessarily viscous—effect of Hopf bifurcation, or “galloping instability”? and, perhaps most important from a practical point of view, (4) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (1) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (2) and (3) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (4) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state. PubDate: 2015-07-01

Abstract: Abstract
We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance. PubDate: 2015-07-01

Abstract: Abstract
For a class of linear second order partial differential equations of mixed elliptic-hyperbolic type, which includes a well known model for analyzing possible heating in axisymmetric cold plasmas, we give results on the weak well-posedness of the Dirichlet problem and show that such solutions are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functionals restricted to suitable infinite dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights which is associated to the linear operator and is developed herein. Similar characterizations for the weighted eigenvalue problem and nonlinear variants will also be given. Finally, topological methods are employed to obtain existence results for nonlinear problems including perturbations in the gradient which are then applied to the well-posedness of the linear problem with lower order terms. PubDate: 2015-07-01

Abstract: Abstract
We study the behavior of brittle atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations. PubDate: 2015-07-01

Abstract: Abstract
In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly. PubDate: 2015-07-01

Abstract: Abstract
We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits local minimizers with respect to L
1 perturbations preserving the volume. However, we prove that the ball is stable under small C
1,1 perturbations when the charge is small enough. PubDate: 2015-07-01

Abstract: Abstract
In this paper, we provide a much simplified proof of the main result in
Lin and Zhang (Commun Pure Appl Math 67: 531–580, 2014) concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a three dimensional incompressible complex fluid model under the assumption that the initial data are close to some equilibrium states. Besides the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the L
∞ estimates for time slices to deduce the key L
1 in time estimates. The latter is responsible for the global in time existence. PubDate: 2015-06-01

Abstract: Abstract
Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier–Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L
1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components. PubDate: 2015-06-01

Abstract: Abstract
We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick’s law when connected to particles reservoirs. We consider a box on a d + 1-dimensional lattice and when
\({d\geqq7}\)
, we show that under a diffusive rescaling of space and time, the probability of finding a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in a high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart. PubDate: 2015-06-01

Abstract: Abstract
We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier–Stokes–Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case). PubDate: 2015-06-01

Abstract: Abstract
In this article, we consider parabolic equations on a bounded open connected subset
\({\Omega}\)
of
\({\mathbb{R}^n}\)
. We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of
\({\Omega}\)
of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in Privat et al. (J Eur Math Soc, 2015) for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations. PubDate: 2015-06-01

Abstract: Abstract
Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space. PubDate: 2015-06-01

Abstract: Abstract
This paper is devoted to the proof of uniform Hölder and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscillating coefficients. Our main point is that no structure is assumed on the oscillations of the boundary. In particular, those oscillations are neither periodic, nor quasiperiodic, nor stationary ergodic. We investigate the consequences of our estimates on the large scales of Green and Poisson kernels. Our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries, which is an area of active research. PubDate: 2015-06-01

Abstract: Abstract
Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving. PubDate: 2015-06-01

Abstract: Abstract
We study the d’Alembert equation with a boundary. We introduce the notions of Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations. This allows us to obtain the complete and simple formula of the Green’s functions for the wave equation with the presence of various boundary conditions. We are able to determine whether a Rayleigh surface wave is active or virtual, and study the lacunas of the wave equation in three dimensional with the presence of a boundary in the case of a virtual Rayleigh surface wave. PubDate: 2015-06-01

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: 2015-05-22

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: 2015-05-21

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: 2015-05-21