Abstract: Abstract
We consider general infinite nanotubes of atoms in
${\mathbb{R}^3}$
where each atom interacts with all the others through a two-body potential. At the equilibrium, the positions of the atoms satisfy a Euler–Lagrange equation. When there are no exterior forces and for a suitable geometry, a particular family of nanotubes is the set of perfect nanotubes at the equilibrium. When exterior forces are applied on the nanotube, we compare the nanotube to nanotubes of the previous family. In part I of the paper, this quantitative comparison is formulated in our first main result as a discrete Saint-Venant principle. As a corollary, we also give a Liouville classification result. Our Saint-Venant principle can be derived for a large class of potentials (including the Lennard-Jones potential), when the perfect nanotubes at the equilibrium are stable. The approach is designed to be applicable to nanotubes that can have general shapes like, for instance, carbon nanotubes or DNA, under the oversimplified assumption that all the atoms are identical. In part II of the paper, we derive from our Saint-Venant principle a macroscopic mechanical model for general nanotubes. We prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting, but no bending. We establish error estimates between the discrete and the continuous solution. More precisely we give two error estimates: one at the microscopic level and one at the macroscopic level. PubDate: 2014-07-01

Abstract: Abstract
In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations
${(-\Delta )^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}}$
with an isolated singularity, where
${\sigma\in (0,1)}$
. We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved by Caffarelli et al. (Comm Pure Appl Math 42:271–297, 1989). PubDate: 2014-07-01

Abstract: Abstract
We prove the global well-posedness of three dimensional compressible Navier–Stokes equations for some classes of large initial data, which may have large oscillation for the density and large energy for the velocity. The proof uses the special structure of the system (especially the effective viscous flux). PubDate: 2014-07-01

Abstract: Abstract
We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian
${-\Delta^D_\Omega}$
in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as
${{\rm e}^{-E_1t} t^{-3/2}}$
, where E
1 is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators
${-\Delta^D_\Omega-V}$
. PubDate: 2014-07-01

Abstract: Abstract
We analyze the Cahn–Hilliard equation with a relaxation boundary condition modeling the evolution of an interface in contact with the solid boundary. An L
∞ estimate is established which enables us to prove the global existence of the solution. We also study the sharp interface limit of the system. The dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations. PubDate: 2014-07-01

Abstract: Abstract
We study, in dimensions N ≥ 3, the family of first integrals of an incompressible flow: these are
${H^{1}_{\rm loc}}$
functions whose level surfaces are tangential to the streamlines of the advective incompressible field. One main motivation for this study comes from earlier results proving that the existence of nontrivial first integrals of an incompressible flow q is the main key that leads to a “linear speed up” by a large advection of pulsating traveling fronts solving a reaction–advection–diffusion equation in a periodic heterogeneous framework. The family of first integrals is not well understood in dimensions N ≥ 3 due to the randomness of the trajectories of q and this is in contrast with the case N = 2. By looking at the domain of propagation as a union of different components produced by the advective field, we provide more information about first integrals and we give a class of incompressible flows which exhibit “ergodic components” of positive Lebesgue measure (and hence are not shear flows) and which, under certain sharp geometric conditions, speed up the KPP fronts linearly with respect to the large amplitude. In the proofs, we establish a link between incompressibility, ergodicity, first integrals and the dimension to give a sharp condition about the asymptotic behavior of the minimal KPP speed in terms of the configuration of ergodic components. PubDate: 2014-07-01

Abstract: Abstract
We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and the existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka–Volterra type, in physical models of Bose–Einstein condensates and in models of chemical reactions. PubDate: 2014-07-01

Abstract: Abstract
We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system:
$$\left\{\begin{array}{ll} -\Delta u = -u \upsilon^2 &\quad {\rm in}\, \mathbb{R}^N\\
-\Delta \upsilon= -u^2 \upsilon &\quad {{\rm in}\, \mathbb{R}^N},\end{array}\right.$$
for every dimension
${N \geqq 2}$
. In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al. PubDate: 2014-07-01

Abstract: Abstract
In this paper we study the fully nonlinear free boundary problem
$$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ D^{2} u \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$
where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W
2,n
solutions are locally C
1,1 inside B
1. Under the extra condition that
${\Omega \supset \{D{u} \neq 0 \}}$
and a uniform thickness assumption on the coincidence set {D
u = 0}, we also show local regularity for the free boundary
${\partial \Omega \cap B_1}$
. PubDate: 2014-07-01

Abstract: Abstract
We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations. PubDate: 2014-06-01

Abstract: Abstract
We prove the absence of anomalous dissipation of energy for long time averaged solutions of the forced critical surface quasi-geostrophic equation in two spatial dimensions. PubDate: 2014-06-01

Abstract: Abstract
The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case. PubDate: 2014-06-01

Abstract: Abstract
For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in Medvedev (The nonlinear heat equation on dense graphs and graph limits. ArXiv e-prints, 2013) justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs. PubDate: 2014-06-01

Abstract: Abstract
We consider the Navier–Stokes equations for the motion of compressible, viscous flows in a half-space
${\mathbb{R}^n_+,}$
n = 2, 3, with the no-slip boundary conditions. We prove the existence of a global weak solution when the initial data are close to a static equilibrium. The density of the weak solution is uniformly bounded and does not contain a vacuum, the velocity is Hölder continuous in (x, t) and the material acceleration is weakly differentiable. The weak solutions of this type were introduced by D. Hoff in Arch Ration Mech Anal 114(1):15–46, (1991), Commun Pure and Appl Math 55(11):1365–1407, (2002) for the initial-boundary value problem in
${\Omega = \mathbb{R}^n}$
and for the problem in
${\Omega = \mathbb{R}^n_+}$
with the Navier boundary conditions. PubDate: 2014-06-01

Abstract: Abstract
We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data. PubDate: 2014-06-01

Abstract: Abstract
Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution. PubDate: 2014-06-01

Abstract: Abstract
In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids. PubDate: 2014-01-15

Abstract: Abstract
We prove various decay bounds on solutions (f
n
: n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n
ℓ
f
n
in terms of a suitable average of the moments of the initial data for every positive ℓ. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of
${L^p(\mathbb{R}^d \times [0, T])}$
norms of the moments
${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$
, (
${\int_0^{\infty} m^a f_m(x, t)dm}$
in the case of continuous Smoluchowski’s equation) for every
${p \in [1, \infty]}$
. In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function
${\phi(n)}$
that is closely related to the total increase of the diffusion coefficient in the interval (0, n]. PubDate: 2014-01-09

Abstract: Abstract
We study the following nonlinear Stefan problem
$$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu \nabla_{x} u ^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$
where
${\Omega(t) \subset \mathbb{R}^{n}}$
(
${n \geqq 2}$
) is bounded by the free boundary
${\Gamma(t)}$
, with
${\Omega(0) = \Omega_0}$
, μ and d are given positive constants. The initial function u
0 is positive in
${\Omega_0}$
and vanishes on
${\partial \Omega_0}$
. The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary
${\Gamma(t)}$
is smooth outside the closed convex hull of
${\Omega_0}$
, and as
${t \to \infty}$
, either
${\Omega(t)}$
expands to the entire
${\mathbb{R}^n}$
, or it stays bounded. Moreover, in the former case,
${\Gamma(t)}$
converges to the unit sphere when normalized, and in the latter case,
${u \to 0}$
uniformly. When
${g(u) = au - bu^2}$
, we further prove that in the case
${\Omega(t)}$
expands to
${{\mathbb R}^n}$
,
${u \to a/b}$
as
${t \to \infty}$
, and the spreading speed of the free boundary converges to a positive constant; moreover, there exists
${\mu^* \geqq 0}$
such that
${\Omega(t)}$
expands to
${{\mathbb{R}}^n}$
exactly when
${\mu > \mu^*}$
. PubDate: 2014-01-08