Volume 57, Issue 8, August 2016

Consider the 3dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for FloquetBloch modes with fixed quasimomentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to appear); [eprint arXiv:1409.6878 (2014)] who show momentum localisation for zero quasimomentum in 2dimensions and is the first result in this direction in 3dimensions.
 ARTICLES

 Partial Differential Equations

A critical nonlinear fractional elliptic equation with saddlelike potential in ℝ^{N}
View Description Hide DescriptionIn this paper, we study the existence of positive solution for the following class of fractional elliptic equation where ϵ, λ > 0 are positive parameters, is the fractional Laplacian, and V is a saddlelike potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.

Nonviscous regularization of the DaveyStewartson equations: Analysis and modulation theory
View Description Hide DescriptionIn the present study, we are interested in the DaveyStewartson equations (DSE) that model packets of surface and capillarygravity waves. We focus on the ellipticelliptic case, for which it is known that DSE may develop a finitetime singularity. We propose three systems of nonviscous regularization to the DSE in a variety of parameter regimes under which the finitetime blowup of solutions to the DSE occurs. We establish the global wellposedness of the regularized systems for all initial data. The regularized systems, which are inspired by the αmodels of turbulence and therefore are called the αregularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the αregularized DSE by using the modulation theory to investigate the mechanism with which the proposed nonviscous regularization prevents the formation of the singularities in the regularized DSE. This is a followup of the work [Cao et al., Nonlinearity 21, 879–898 (2008); Cao et al., Numer. Funct. Anal. Optim. 30, 46–69 (2009)] on the nonviscous αregularization of the nonlinear Schrödinger equation.

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation
View Description Hide DescriptionUsing a sharp GagliardoNirenberg type inequality, wellposedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated.

Existence of solutions for a Schrödinger system with linear and nonlinear couplings
View Description Hide DescriptionWe study an important system of Schrödinger equations with linear and nonlinear couplings arising from BoseEinstein condensates. We use the Nehari manifold to prove the existence of a ground state solution; moreover, we give the sign of the solutions depending on linear coupling; by using index theory and Nehari manifold, we prove that there exist infinitely many positive bound state solutions.

Entropic and gradient flow formulations for nonlinear diffusion
View Description Hide DescriptionNonlinear diffusion ∂t ρ = Δ(Φ(ρ)) is considered for a class of nonlinearities Φ. It is shown that for suitable choices of Φ, an associated Lyapunov functional can be interpreted as thermodynamic entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role.

On the formation of shocks of electromagnetic plane waves in nonlinear crystals
View Description Hide DescriptionAn influential result of F. John states that no genuinely nonlinear strictly hyperbolic quasilinear first order system of partial differential equations in two variables has a global C ^{2}solution for small enough initial data. Inspired by recent work of D. Christodoulou, we revisit John’s original proof and extract a more precise description of the behaviour of solutions at the time of shock. We show that John’s singular first order quantity, when expressed in characteristic coordinates, remains bounded until the final time, which is then characterised by an inverse density of characteristics tending to zero in one point. Moreover, we study the derivatives of second order, showing again their boundedness when expressed in appropriate coordinates. We also recover John’s upper bound for the time of shock formation and complement it with a lower bound. Finally, we apply these results to electromagnetic plane waves in a crystal with no magnetic properties and cubic electric nonlinearity in the energy density, assuming no dispersion.

Schrödinger spectra and the effective Hamiltonian of weak KAM theory on the flat torus
View Description Hide DescriptionIn this paper we investigate the link between the spectrum of some periodic Schrödinger type operators and the effective Hamiltonian of the weak KAM theory. We show that the extension of some local quasimodes is linked to the localization of the Schrödinger spectrum. Such a result provides additional information with respect to the well known BohrSommerfeld quantization rules, here in a more general setting than the integrable or quasiintegrable ones.

On global classical solutions of the three dimensional relativistic VlasovDarwin system
View Description Hide DescriptionWe study the Cauchy problem of the relativistic VlasovDarwin system with generalized variables proposed by SospedraAlfonso et al. [“Global classical solutions of the relativistic VlasovDarwin system with small Cauchy data: the generalized variables approach,” Arch. Ration. Mech. Anal. 205, 827869 (2012)]. We prove global existence of a nonnegative classical solution to the Cauchy problem in three space variables under small perturbation of the initial datum, and as a consequence, we obtain that nearly spherically symmetric solutions with required regularity exist globally in time.

Dynamics of parabolic problems with memory. Subcritical and critical nonlinearities
View Description Hide DescriptionIn this paper, we study the longtime behavior of the solutions of nonautonomous parabolic equations with memory in cases when the nonlinear term satisfies subcritical and critical growth conditions. In order to do this, we show that the family of processes associated to original systems with heat source f(x, t) being translation bounded in is dissipative in higher energy space , 0 < α ≤ 1, and possesses a compact uniform attractor in .

On the emergence of the NavierStokesα model for turbulent channel flows
View Description Hide DescriptionIn a series of papers (see Foias et al. [J. Dyn. Differ. Equations 14(1), 1–35 (2002)] and the pertinent references therein), the 3D NavierStokesα model was shown to be a useful complement to the 3D NavierStokes equations, and in particular, to be a good Reynolds version of the latter equations. In this work, we introduce a simple Reynolds averaging which, due to the wall roughness, transforms the NavierStokes equations into the NavierStokesα model.

Local and global solutions of ChernSimons gauged O(3) sigma equations in one space dimension
View Description Hide DescriptionWe study an initial value problem of the ChernSimons gauged O(3) sigma model in one space dimension. The global existence of solutions to the model is proved for high regularity initial data. Moreover we study low regularity local wellposedness, observing null forms of the system and applying bilinear estimates for waveSobolev space . As a byproduct, a finite energy global solution is constructed.

Cauchy problem for the ellipsoidalBGK model of the Boltzmann equation
View Description Hide DescriptionEllipsoidal BGK model (ESBGK) is a generalized version of the original BGK model designed to reproduce the physically correct Prandtl number in the NavierStokes limit. In this paper, we study the Cauchy problem for the ESBGK model under the condition of finite initial mass, energy, and entropy. Equivalence type estimates for the temperature tensor are crucially used.

Existence and uniqueness of small energy weak solution to multidimensional compressible NavierStokes equations with large external potential force
View Description Hide DescriptionWe study the 3D compressible NavierStokes equations with an external potential force and a general nondecreasing pressure. We prove the globalintime existence of weak solutions with smallenergy initial data and with densities being nonnegative and essentially bounded. A solution may have large oscillations and contain vacuum states. No smallness assumption is made on the external force nor the initial perturbation in L ^{∞} for density. Initial velocity u 0 is taken to be bounded in L^{q} for some q > 6 and no further regularity assumption is imposed on u 0. Finally, we discuss the uniqueness of weak solutions.

Wellposedness for compressible MHD systems with highly oscillating initial data
View Description Hide DescriptionIn this paper, a unique local solution for compressible magnetohydrodynamics systems has been constructed in the critical Besov space framework by converting the system in Euler coordinates to a system in Lagrange coordinates. Our results improve the range of the Lebesgue exponent in the Besov space from [2, N) to [2, 2N), where N denotes the space dimension. Then, we give a lower bound for the maximal existence time, which is important for our construction of global solutions. Based on the lower bound, we use the effective viscous flux and Hoff’s energy method to obtain the unique global solution, which allows the initial velocity field and the magnetic field to have large energies and allows the initial density to exhibit large oscillations on a set of small measure.

Ground states of nonlinear Choquard equations with multiwell potentials
View Description Hide DescriptionIn this paper, we study minimizers of the Hartreetype energy functional under the mass constraint , where with α ∈ (0, N) for N ≥ 2 is the mass critical exponent. Here I α denotes the Riesz potential and the trapping potential satisfies . We prove that minimizers exist if and only if a satisfies , where Q is a positive radially symmetric ground state of in ℝ^{N}. The uniqueness of positive minimizers holds if a > 0 is small enough. The blowup behavior of positive minimizers as a↗a ^{∗} is also derived under some general potentials. Especially, we prove that minimizers must blow up at the central point of the biggest inscribed sphere of the set Ω ≔ {x ∈ ℝ^{N}, V(x) = 0} if .

Global existence and explosion of the stochastic viscoelastic wave equation driven by multiplicative noises
View Description Hide DescriptionIn this paper, we discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term and a source term of the type . We first establish the local existence and uniqueness of solution by the iterative technique truncation function method. Moreover, we also show that the solution is global for q ≥ p. Lastly, by modifying the energy functional, we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for p > q.

Decay of the 3D viscous liquidgas twophase flow model with damping
View Description Hide DescriptionWe establish the optimal L^{p} − L ^{2}(1 ≤ p < 6/5) time decay rates of the solution to the Cauchy problem for the 3D viscous liquidgas twophase flow model with damping and analyse the influences of the damping on the qualitative behaviors of solution. It is observed that the fraction effect of the damping affects the dispersion of fluids and enhances the time decay rate of solution. Our method of proof consists of Hodge decomposition technique, L^{p} − L ^{2} estimates for the linearized equations, and delicate energy estimates.

Asymptotic and spectral analysis of the gyrokineticwaterbag integrodifferential operator in toroidal geometry
View Description Hide DescriptionAchieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phasespace. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magneticfieldinduced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested onedimensional problems: an integral equation in the poloidal variable, followed by a onedimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finitedimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokineticwaterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original gyrokineticVlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.
 Representation Theory and Algebraic Methods

Infinite rank SchrödingerVirasoro type Lie conformal algebras
View Description Hide DescriptionMotivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of SchrödingerVirasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV(a, b), where a, b are complex numbers. The conformal derivations of CSV(a, b) are uniformly determined. The rank one conformal modules and ℤgraded free intermediate series modules over CSV(a, b) are classified. Corresponding results of the conformal subalgebra CHV(a, b) of CSV(a, b) are also presented.
 Quantum Mechanics

On the phasespace distribution of Bloch eigenmodes for periodic point scatterers
View Description Hide DescriptionConsider the 3dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for FloquetBloch modes with fixed quasimomentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to appear); [eprint arXiv:1409.6878 (2014)] who show momentum localisation for zero quasimomentum in 2dimensions and is the first result in this direction in 3dimensions.