Abstract: Abstract
This article concerns the time growth of Sobolev norms of classical solutions to the three dimensional incompressible isotropic elastodynamics with small initial displacements. Given initial data in
\({H^k_\Lambda}\)
for a fixed big integer k, the global well-posedness of this Cauchy problem has been established by Sideris and Thomases in (Commun Pure Appl Math 58(6):750–788, 2005) and (J Hyperbolic Differ Equ 3(4):673–690, 2006, Commun Pure Appl Math 60(12):1707–1730, 2007), where the highest-order generalized energy E
k
(t) may have a certain growth in time. Alinhac conjectured that such a growth in time may be a true phenomenon, in (Geometric analysis of hyperbolic differential equations: an introduction, lecture note series: 374. Mathematical Society, London) he proved that E
k
(t) is still uniformly bounded in time only for the three dimensional scalar quasilinear wave equation under a null condition. In this paper, we show that the highest-order generalized energy E
k
(t) is still uniformly bounded for the three dimensional incompressible isotropic elastodynamics. The equations of incompressible elastodynamics can be viewed as nonlocal systems of wave type and are inherently linearly degenerate in the isotropic case. There are three ingredients in our proof: the first is that we still have a decay rate of
\({t^{-\frac{3}{2}}}\)
when we do the highest energy estimate away from the light cone even though in this case the Lorentz invariance is not available. The second one is that the
\({L^\infty}\)
norm of the good unknowns, in particular
\({\nabla(v + G\omega)}\)
, is shown to have a decay rate of
\({t^{-\frac{3}{2}}}\)
near the light cone. The third one is that the pressure is estimated in a novel way as a nonlocal nonlinear term with null structure, as has been recently observed in [16]. The proof employs the generalized energy method of Klainerman, enhanced by weighted L
2 estimates and the ghost weight introduced by Alinhac. PubDate: 2015-05-01

Abstract: Abstract
We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves of finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave is, generically, a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave. PubDate: 2015-05-01

Abstract: Abstract
This paper is concerned with nonlinear elliptic equations in nondivergence form
$$F(D^{2}u, Du, x) = 0 $$
where F has a drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack inequality. This paper presents a new generalization of the Harnack inequality for such equations. As a corollary we obtain the optimal Harnack type of inequality for p(x)-harmonic functions which quantifies the strong minimum principle. PubDate: 2015-05-01

Abstract: Abstract
We study the transition from flat to wrinkled regions in a uniaxially stretched thin elastic film. We set up a model variational problem, and study the energy of its ground state. Using known scaling bounds for the minimal energy, the minimal energy can be written as a minimum of the underlying (convex) relaxed problem plus a term that grows linearly in the thickness of the film.We show that in the limit of vanishing thickness the prefactor in the scaling law for the original problem can be obtained by minimization of simpler scalar constrained variational problems. PubDate: 2015-05-01

Abstract: Abstract
The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling:
$$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g ( w ) w.\end{array}$$
Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains
\({\Lambda_j \subset \mathbb{R}^3}\)
such that
\({-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}\)
,
\({x_j\in\Lambda_j}\)
, then the k-families of solutions
\({w_\hbar^j}\)
concentrate around x
j
as
\({\hbar\to 0}\)
, respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions. PubDate: 2015-05-01

Abstract: We consider a four-parameter family of Boussinesq systems derived by Bona et al. (J Nonlinear Sci 12:283–318, 2002). We establish the existence of the ground states which are solitary waves minimizing the action functional of the systems. We further show that in the presence of large surface tension the ground states are even up to translation. PubDate: 2015-05-01

Abstract: Abstract
Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young’s law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries. PubDate: 2015-05-01

Abstract: Abstract
We introduce a stochastic N-particle system and show that, as N → ∞, an effective description ruled by the homogeneous fermionic Uehling–Uhlenbeck equation is recovered. The particle model we consider is the same as the Kac model for the homogeneous Boltzmann equation with an additional exclusion constraint taking into account the Pauli Exclusion Principle. PubDate: 2015-05-01

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: 2015-04-01

Abstract: Abstract
The equatorial shallow water equations at low Froude number form a symmetric hyperbolic system with large terms containing a variable coefficient, the Coriolis parameter f, which depends on the latitude. The limiting behavior of the solutions as the Froude number tends to zero was investigated rigorously a few years ago, using the common approximation that the variations of f with latitude are linear. In that case, the large terms have a peculiar structure, due to special properties of the harmonic oscillator Hamiltonian, which can be exploited to prove strong uniform a priori estimates in adapted functional spaces. It is shown here that these estimates still hold when f deviates from linearity, even though the special properties on which the proofs were based have no obvious generalization. As in the linear case, existence, uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates. PubDate: 2015-04-01

Abstract: Abstract
The non-isentropic Euler system with periodic initial data in
\({{\mathbb{R}}^1}\)
is studied by analyzing wave interactions in a framework of specially chosen Riemann invariants, generalizing Glimm’s functionals and applying the method of approximate conservation laws and approximate characteristics. An
\({{\mathcal O}(\varepsilon^{-2})}\)
lower bound is established for the life span of the entropy solutions with initial data that possess
\({\varepsilon}\)
variation in each period. PubDate: 2015-04-01

Abstract: Abstract
We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, that is, heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of Γ-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts. PubDate: 2015-04-01

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: 2015-04-01

Abstract: Abstract
In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm–Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem, which depends on the mass parameter
\({\beta \in [0,9]}\)
and the eccentricity
\({e \in [0,1)}\)
. Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagrangian solutions. PubDate: 2015-04-01

Abstract: Abstract
We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures. PubDate: 2015-04-01

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: 2015-04-01

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: 2015-03-13

Abstract: Abstract
We study the infimum of the Ginzburg–Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed to vanish along a smooth curve. We obtain energy asymptotics which are valid when the Ginzburg–Landau parameter is large and the strength of the external field is below the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X.B. Pan–K.H. Kwek. PubDate: 2015-03-11

Abstract: Abstract
We study the structure of shock-free solutions of the compressible Euler equations with large data. We describe conditions under which the rarefactive/ compressive character of solutions changes, and conditions under which the vacuum is formed asymptotically. We present several new examples of shock-free solutions, which demonstrate a large variety of behaviors. PubDate: 2015-03-11