Existence for the α-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter α with 0<α?1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α=1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.     

A Note on Reflexivity

We prove some new results on Hadwin's general version of reflexivity that reduce the study of E-reflexivity (or E-hyperreflexivity) of a linear subspace to a smaller linear subspace. By applying our abstract results, we present a simple proof of D. Hadwin's theorem, which states that every C^∗-algebra is approximately hyperreflexive. We also prove that the image of any C^∗-algebra under any bounded unital homomorphism into the operators on a Banach space is approximately reflexive. We introduce a new version of reflexivity, called approximate algebraic reflexivity, and study its properties.     

Some Turán type results on the hypercube

For graphs H,G a classical problem in extremal graph theory asks what proportion of the edges of H a subgraph may contain without containing a copy of G. We prove some new results in the case where H is a hypercube. We use a supersaturation technique of Erd?s and Simonivits to give a characterization of a set of graphs such that asymptotically the answer is the same when G is a member of this set and when G is a hypercube of some fixed dimension. We apply these results to a specific set of subgraphs of the hypercube called Fibonacci cubes. Additionally, we use a coloring argument to prove new asymptotic bounds on this problem for a different set of graphs. Finally we prove a new asymptotic bound for the case where G is the cube of dimension 3.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Some fixed point results on L-embedded Banach spaces

In this paper we prove that w-fixed point property and w^★-fixed point property are equivalent concepts for L-embedded Banach spaces which are duals of M-embedded spaces. Similar results will be obtained with respect to the normal structure. These equivalences will be applied to establish new fixed point results for different examples. We will also prove the existence of fixed points for both nonexpansive and asymptotically regular mappings defined on subsets of L-embedded Banach spaces which are sequentially compact for the abstract measure topology. We will check that our results do not hold in the case of the weak topology.     

Gradient estimates via linear and nonlinear potentials

We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by −Δpu=μ. In particular, no matter the nonlinearity of the equations considered, we show that in the case p?2 a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case p?2. In the case p>2 we prove a new gradient estimate employing nonlinear potentials of Wolff type.     

Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

To solve the 1D (linear) convection-diffusion equation, we construct and we analyze two LBM schemes built on the D1Q2 lattice. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L ^∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L ^∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.     

Beale-Kato-Majda type condition for Burgers equation

We consider a multidimensional Burgers equation on the torus Td and the whole space Rd. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of Rd we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.     

On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations

We prove new a priori estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.     

Cluster tilting objects in generalized higher cluster categories

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi-Yau and Hom-finite, arising from an (m+2)-Calabi-Yau dg algebra. This is a generalization of the result for the m=1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension.     

Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity

We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L ² initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139?C166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L ² initial data.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

Regularity and integrability of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x₁,x₂) + Ũ(t,x₁,x₂) + W’(t,x₁,x₂) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f₀. The vector field W’(t,x₁,x₂) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T^*, of regular solutions to classical shallow-water equations with general initial data (T^* → +∞, as 1/f₀ → 0).     

Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences

Standard bases of ideals of the polynomial ring R[X] = R[x ₁, …, x _k ] over a commutative Artinian chain ring R that are concordant with the norm on R have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from R. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family L _R (I), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram $\mathcal{F}$ and a full system of $\mathcal{F}$ -monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Gröbner basis of a monic ideal to a standard basis with the same cardinality.     

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.     

Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.     

Definability in substructure orderings, III: Finite distributive lattices

Let ${{\mathcal D}}$ be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d ^opp} is definable, where d and d ^opp are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identity automorphism of ${{\mathcal D}}$ is the opposite map. Then we apply these results to investigate definability in the closely related lattice of universal classes of distributive lattices. We prove that this lattice has only one non-identity automorphism, the opposite map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each element K of the two subsets, {K, K ^opp} is a definable subset of the lattice.     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.