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 study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter ɛ and the magnets as classical
\({\pm 1}\)
spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of Γ-convergence, that, up to subsequences, the (continuum) Γ-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the Γ-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic. PubDate: 2015-11-01

Abstract: Abstract
For a class of systems of semi-linear elliptic equations, including
$$-\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\quad i=1,\dots,k,$$
for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform
\({L^\infty}\)
boundedness of the solutions implies uniform boundedness of their Lipschitz norm as
\({\beta \to +\infty}\)
, that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one. PubDate: 2015-11-01

Abstract: Abstract
In this paper, we consider two systems modelling the evolution of a rigid body in an incompressible fluid in a bounded domain of the plane. The first system corresponds to an inviscid fluid driven by the Euler equation whereas the other one corresponds to a viscous fluid driven by the Navier–Stokes system. In both cases we investigate the uniqueness of weak solutions, à la Yudovich for the Euler case, à la Leray for the Navier–Stokes case, as long as no collision occurs. PubDate: 2015-11-01

Abstract: Abstract
We establish well-posedness for the family of thin-film equations
1
$$\left \{\begin{array}{ll}h_t + (h^n h_{xxx})_x \ = \ 0 \quad \ {\rm in } \ \{ h > 0 \},\\ h \ = \ 0, \ h_x \ = \ 1 \quad\quad {\rm on } \ \partial \{ h > 0 \}\end{array}\right. $$
with
\({n \in (0,\frac {14}{5}) \backslash \{ 1, 2 \}}\)
. The model (1) with
\({n \in (0,3]}\)
has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile
\({h \geqq 0}\)
. The family of thin-film equations (1) provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions. The parabolicity of the fourth order parabolic problem degenerates at the free boundary, which leads to a loss of regularity at the moving contact point. Our solutions are regular in terms of the two variables d(x) and d(x)3−n
, where d(x) is the distance to the free boundary. The main technical difficulty in the analysis of (1) is related to the loss of regularity at the contact points. 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

Abstract: Abstract
We consider a variational model related to the formation of islands in heteroepitaxial growth on unbounded domains. We first derive the scaling regimes of the minimal energy in terms of the volume of the film and the amplitude of the crystallographic misfit. For small volumes, non-existence of minimizers is then proven. This corresponds to the experimentally observed wetting effect. On the other hand, we show the existence of minimizers for large volumes. We finally study the asymptotic behavior of the optimal shapes. PubDate: 2015-10-01

Abstract: Abstract
We give a new decay framework for the general dissipative hyperbolic system and the hyperbolic–parabolic composite system, which allows us to pay less attention to the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood–Paley pointwise energy estimates and new time-weighted energy functionals to establish optimal decay estimates on the framework of spatially critical Besov spaces for the degenerately dissipative hyperbolic system of balance laws. Based on the
\({L^{p}(\mathbb{R}^{n})}\)
embedding and the improved Gagliardo–Nirenberg inequality, the optimal
\({L^{p}(\mathbb{R}^{n})-L^{2}(\mathbb{R}^{n})(1\leqq p < 2)}\)
decay rates and
\({L^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})(1\leqq p < 2\leqq q\leqq \infty)}\)
decay rates are further shown. Finally, as a direct application, the optimal decay rates for three dimensional damped compressible Euler equations are also obtained. PubDate: 2015-10-01

Abstract: Abstract
In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space; as consequences we get conservation of Hamiltonian energy and some weak time-regularity of solutions. PubDate: 2015-10-01

Abstract: Abstract
In this paper we construct families of real analytic solutions of the Surface Quasi-Geostrophic equation (SQG) that are locally constant outside a thin neighborhood of a curve of arbitrarily small thickness. Despite the fact that only local existence results are known for SQG, and that our initial conditions have a arbitrarily large gradient we show that solutions exist for a time independent of the thickness of the neighborhood. PubDate: 2015-10-01

Abstract: Abstract
We study the existence, regularity and so-called ‘strict physicality’ of global weak solutions of a Beris–Edwards system which is proposed as a model for the incompressible flow of nematic liquid crystal materials. An important contribution to the dynamics comes from a singular potential introduced by John Ball and Apala Majumdar which replaces the commonly employed Landau-de Gennes bulk potential. This is built into our model to ensure that a natural physical constraint on the eigenvalues of the Q-tensor order parameter is respected by the dynamics of this system. Moreover, by a maximum principle argument, we are able to construct global strong solutions in dimension two. PubDate: 2015-10-01

Abstract: Abstract
We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order. PubDate: 2015-10-01

Abstract: Abstract
We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance as it presents a significant milestone towards understanding, on the atomistic level, phenomena such as melting or plastic behavior. We characterize a set of rotation- and translation-invariant two- and three-body potentials V
2, V
3 such that the energy minimum of
$$\frac{1}{\#Y}E(Y) = \frac{1}{\# Y}
\left(2\sum_{\{y,y'\}
\subset Y}V_2(y, y') + 6\sum_{\{y,y',y''\}
\subset Y} V_3(y,y',y'')\right)$$
over all
\({Y \subset \mathbb{R}^3}\)
, #Y = n, converges to the energy per particle in the face-centered cubic (fcc) lattice as n tends to infinity. The proof involves a careful analysis of the symmetry properties of the fcc lattice. PubDate: 2015-10-01