Abstract: Abstract
In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation. PubDate: 2016-08-01

Abstract: Abstract
In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the first
non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis of our estimates we conjecture that the eigenvalues of the Neumann–
Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré–Sobolev inequalities, obtained by the authors recently. PubDate: 2016-08-01

Abstract: Abstract
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi–Nash–Moser Hölder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time. PubDate: 2016-08-01

Abstract: The aim of the present paper is twofold:
We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.
We apply this method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W
1. PubDate: 2016-08-01

Abstract: Abstract
In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, that is for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false. PubDate: 2016-08-01

Abstract: Abstract
The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value
\({\rho_{\rm cr}}\)
such that if the incoming density in the upstream is larger than
\({\rho_{\rm cr}}\)
, then there exists a subsonic flow past a wall. Furthermore,
\({\rho_{\rm cr}}\)
is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than
\({\rho_{\rm cr}}\)
. The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states. PubDate: 2016-08-01

Abstract: Abstract
We prove in any dimension
\({d \geqq 1}\)
a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media,
$$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$
when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential
\({W}\)
is a
\({C^2({\mathbb{R}}^d)}\)
radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Modél Math Anal Numér 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential. PubDate: 2016-08-01

Abstract: Abstract
This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in the presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman–Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born–Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions.We also investigate explicit examples for which we can improve
our general theorem: global WKB expansions, quasi-optimal Agmon estimates and
upper bound of the tunelling effect (in symmetric cases).We also apply our strategy
to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide
range of situations analyzed in the last two decades. PubDate: 2016-08-01

Abstract: Abstract
We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by
\({N^{\beta-1}V(N^{\beta}.) where }\)
where
\({\int V \leqslant 0}\)
. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the
\({N \rightarrow \infty}\)
limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system. PubDate: 2016-08-01

Abstract: Abstract
In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding
\({1/3}\)
. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space
\({L_t^\infty B_{3,\infty}^{1/3}}\)
due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than
\({1/5}\)
. The main result of the present paper shows that any given smooth Euler flow can be perturbed in
\({C^{1/5-\epsilon}_{t,x}}\)
on any pre-compact subset of
\({\mathbb{R}\times \mathbb{R}^3}\)
to violate energy conservation. Furthermore, the perturbed solution is no smoother than
\({C^{1/5-\epsilon}_{t,x}}\)
. As a corollary of this theorem, we show the existence of nonzero
\({C^{1/5-\epsilon}_{t,x}}\)
solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting. PubDate: 2016-08-01

Abstract: Abstract
We reformulate the Kohn–Sham density functional theory (KSDFT) as a nested variational problem in the one-particle density operator, the electrostatic potential and a field dual to the electron density. The corresponding functional is linear in the density operator and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, termed spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We prove convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain. PubDate: 2016-08-01

Abstract: Abstract
We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given. PubDate: 2016-08-01

Abstract: Abstract
We construct a singular minimizing map u from
\({\mathbb{R}^{3}}\)
to
\({\mathbb{R}^{2}}\)
of a smooth uniformly convex functional of the form
\({\int_{B_1} F(D{\bf u})\,dx}\)
. PubDate: 2016-07-01

Abstract: Abstract
The strong existence and the pathwise uniqueness of solutions with
\({L^{\infty}}\)
-vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved. PubDate: 2016-07-01

Abstract: Abstract
We study inertial motions of the coupled system,
\({\mathscr{S}}\)
, constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of
\({\mathscr{S}}\)
coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of
\({\mathscr{S}}\)
with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation. PubDate: 2016-07-01

Abstract: Abstract
Consider the flow of a Navier–Stokes liquid past a body rotating with a prescribed constant angular velocity,
\({\omega}\)
, and assume that the motion is steady with respect to a body-fixed frame. In this paper we show that the vorticity field associated to every “weak” solution corresponding to data of arbitrary “size” (Leray Solution) must decay exponentially fast outside the wake region at sufficiently large distances from the body. Our result improves and generalizes in a non-trivial way famous results by Clark (Indiana Univ Math J 20:633–654, 1971) and Babenko and Vasil’ev (J Appl Math Mech 37:651–665, 1973) obtained in the case
\({\omega=0}\)
. PubDate: 2016-07-01

Abstract: Abstract
Consider the three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential. PubDate: 2016-07-01

Abstract: Abstract
This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay. PubDate: 2016-07-01

Abstract: Abstract
We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the 0rate of convergence. PubDate: 2016-07-01

Abstract: Abstract
We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as
\({h \to 0}\)
. We first establish that when the energy per volume scales as the second power of h, the resulting
\({\Gamma}\)
-limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h
2. This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R
1213, R
1223 and R
1212 of G vanish identically. We illustrate our findings with examples; of particular interest is an example where
\({G_{2 \times 2}}\)
, the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for
\({G = Id_{3} + \gamma n \otimes n}\)
given in terms of the inhomogeneous unit director field distribution
\({ n \in \mathbb{R}^3}\)
. PubDate: 2016-07-01