Abstract: Abstract
We prove the existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and the existence of weak solutions in 3D; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in L
1 and a strengthened form of the Ladyzhenskaya inequality
$$\ f \ _{L^{4}} \leqq c \ f \ _{L^{2,\infty}}^{1/2} \ \nabla f\ _{L^{2}}^{1/2},$$
which we derive using the theory of interpolation. The model potentially has applications to the method of magnetic relaxation introduced by Moffatt (J Fluid Mech 159:359–378, 1985) to construct stationary Euler flows with non-trivial topology. PubDate: 2014-11-01

Abstract: Abstract
Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented. PubDate: 2014-11-01

Abstract: Abstract
In this paper, we consider the Hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries. PubDate: 2014-11-01

Abstract: Abstract
We consider weak solutions to a simplified Ericksen–Leslie system of two-dimensional compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity
\({ \nabla\mathbf{d} ^2\mathbf{d}}\)
in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition. PubDate: 2014-11-01

Abstract: Abstract
In the framework of rate-independent systems, a family of elastic-plastic-damage models is proposed through a variational formulation. Since the goal is to account for softening behaviors until the total failure, the dissipated energy contains a gradient damage term in order to limit localization effects. The resulting model owns a great flexibility in the possible coupled responses, depending on the constitutive parameters. Moreover, considering the one-dimensional quasi-static problem of a bar under simple traction and constructing solutions with localization of damage, it turns out that in general a cohesive crack appears at the center of the damage zone before the rupture. The associated cohesive law is obtained in a closed form in terms of the parameters of the model. PubDate: 2014-11-01

Abstract: Abstract
We derive the quantitative modulus of continuity
$$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$
which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980’s state-of-the-art results by Caffarelli– Evans (Arch Rational Mech Anal 81(3):199–220, 1983) and DiBenedetto (Ann Mat Pura Appl 103(4):131–176, 1982), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α(n, p). PubDate: 2014-11-01

Abstract: Abstract
A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger’s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps. PubDate: 2014-11-01

Abstract: Abstract
We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x
0, t
0). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x
0, t
0), intersected with the exterior of a certain space-time paraboloid with vertex at point (x
0, t
0). We make no special assumptions on the solution in the interior of the paraboloid. PubDate: 2014-11-01

Abstract: Abstract
A variational model introduced by Spencer and Tersoff (Appl. Phys. Lett. 96:073114, 2010) to describe optimal faceted shapes of epitaxially deposited films is studied analytically in the case in which there are a non-vanishing crystallographic miscut and a lattice incompatibility between the film and the substrate. The existence of faceted minimizers for every volume of the deposited film is established. In particular, it is shown that there is no wetting effect for small volumes. Geometric properties including a faceted version of the zero contact angle are derived, and the explicit shapes of minimizers for small volumes are identified. PubDate: 2014-11-01

Abstract: Abstract
We consider the Hele-Shaw problem in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domain goes to zero. Specifically, we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic Hele-Shaw problem set in
\({\mathbb{R}^{d}}\)
. The main challenge when deriving the limit lies in controlling the oscillations of the free boundary. This is overcome first by extending De Giorgi–Nash–Moser type estimates to perforated domains and second by proving the almost sure non-degenerate growth of the solution near its free boundary. PubDate: 2014-10-15

Abstract: Abstract
We deal with systems of PDEs, arising in mean field games theory, where viscous Hamilton–Jacobi and Fokker–Planck equations are coupled in a forward-backward structure. We consider the case of local coupling, when the running cost depends on the pointwise value of the distribution density of the agents, in which case the smoothness of solutions is mostly unknown. We develop a complete weak theory, proving that those systems are well-posed in the class of weak solutions for monotone couplings under general growth conditions, and for superlinear convex Hamiltonians. As a key tool, we prove new results for Fokker–Planck equations under minimal assumptions on the drift, through a characterization of weak and renormalized solutions. The results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned. PubDate: 2014-10-15

Abstract: Abstract
This paper considers L
2-asymptotic stability of the spatially inhomogeneous Navier–Stokes–Boussinesq system with general nonlinearity including both power nonlinear terms and convective terms. We construct a local-in-time strong solution of the system by applying semigroup theory on Hilbert spaces and fractional powers of the Stokes–Laplace operator. It is also shown that under some assumptions on an energy inequality the system has a unique global-in-time strong solution when the initial datum is sufficiently small. Furthermore, we investigate the asymptotic stability of the global-in-time strong solution by using an energy inequality, maximal L
p
-in-time regularity for Hilbert space-valued functions, and fractional powers of linear operators in a solenoidal L
2-space. We introduce new methods for showing the asymptotic stability by applying an energy inequality and maximal L
p
-in-time regularity for Hilbert space-valued functions. Our approach in this paper can be applied to show the asymptotic stability of energy solutions for various incompressible viscous fluid systems and the stability of small stationary solutions whose structure is not clear. PubDate: 2014-10-08

Abstract: Abstract
We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function. PubDate: 2014-10-07

Abstract: Abstract
We study the optimal sets
\({\Omega^\ast\subseteq\mathbb{R}^d}\)
for spectral functionals of the form
\({F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}\)
, which are bi-Lipschitz with respect to each of the eigenvalues
\({\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}\)
of the Dirichlet Laplacian on
\({\Omega}\)
, a prototype being the problem
$$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ \Omega =1\big\}}.$$
We prove the Lipschitz regularity of the eigenfunctions
\({u_1,\ldots,u_p}\)
of the Dirichlet Laplacian on the optimal set
\({\Omega^\ast}\)
and, as a corollary, we deduce that
\({\Omega^\ast}\)
is open. For functionals depending only on a generic subset of the spectrum, as for example
\({\lambda_k(\Omega)}\)
, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved. PubDate: 2014-10-03

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