A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

Discrete nonlinear hyperbolic equations. Classification of integrable cases

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ?². The fields are associated with the vertices and an equation of the form Q(x ₁, x ₂, x ₃, x ₄) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ? ^N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.     

Existence and uniqueness for p-Laplace equations involving singular nonlinearities

We consider quasilinear elliptic equations involving the p-Laplacian and singular nonlinearities. We prove comparison principles and we deduce some uniqueness results.     

Exact solutions of a class of second-order nonlocal boundary value problems and applications

In this paper, solutions of a class of second-order differential equations with some multi-point boundary conditions are studied. We give exact expressions of the solutions for the linear m-point boundary problems by the Green’s functions. As applications, we study uniqueness and iteration of the positive solutions for a nonlinear singular second-order m-point boundary value problem.     

Differential-operator representations of Weyl group and singular vectors in Verma modules

Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.     

A partial analog of the integrability theorem for distributions on p-adic spaces and applications

Let X be a smooth real algebraic variety. Let ξ be a distribution on it. One can define the singular support of ξ to be the singular support of the D _X -module generated by ξ (sometimes it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of T*X. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see, e.g., [AG4, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case. We provide two applications of those results:

• a non-Archimedean analog of the results of [Say] concerning Gel’fand property of nice symmetric pairs
• a proof of multiplicity one theorems for GL _n which is uniform for all local fields. This theorem was proven for the non-Archimedean case in [AGRS] and for the Archimedean case in [AG4] and [SZ].

     

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-?verák’s criterion. We also show that if a weak solution u satisfies $\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $ for some 3 < p < ∞, then the number of singular points is finite.     

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.     

Solutions in Hölder spaces of boundary-value problems for parabolic equations with nonconjugate initial and boundary data

The first and second one-dimensional boundary-value problems for parabolic equations are investigated in the case where the conjugation conditions for all required orders are not satisfied. The existence and uniqueness are proved. Estimates of solutions in classical and weighted Hölder spaces are obtained. We prove that the violation of conjugation for the given functions on the boundary of the domain at the initial-time moment causes the appearance of singular solutions. The order of singularity (as a power of t) is found for the singular solutions for t = 0.     

Ray class fields generated by torsion points of certain elliptic curves

We first normalize the derivative Weierstrass ???-function appearing in the Weierstrass equations which give rise to analytic parametrizations of elliptic curves, by the Dedekind ??-function. And, by making use of this normalization of ???, we associate a certain elliptic curve to a given imaginary quadratic field K and then generate an infinite family of ray class fields over K by adjoining to K torsion points of such an elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of ???, as the y-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result towards the Lang?CSchertz conjecture of constructing ray class fields over K by means of the Siegel?CRamachandra invariant.     

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

We consider a possibly degenerate porous media type equation over all of ${\mathbb R^d}$ with d =?1, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution.     

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.     

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.     

On the summability of the formal solutions for some PDEs with irregular singularity

In this Note, we consider some classes of nonlinear partial differential equations with regular singularity with respect to t=0 and irregular one with respect to x=0. Our purpose is to establish a result which is similar to the k-summability property, known in the case of singular ordinary differential equations. We can prove that, except at most a countable set, the formal solution is Borel summable or k-summable with respect to x in all other directions. To cite this article: Z. Luo et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Singular perturbations to semilinear stochastic heat equations

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε?=?0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.     

On the number of solutions of a type of one-dimensional pseudodifferential equations in Sobolev-Slobodetsky space

The paper considers homogeneous, one-dimensional pseudodifferential equations of nonnegative order with symbols of the form Σ _i=1 ^N th(k _i x + ω _i )A _i (ξ). Using a relationship between such equations and the systems of singular equations, some estimates for the number of solutions of pseudodifferential equations in the Sobolev-Slobodetsky space are obtained.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

A stochastic variational approach to viscous Burgers equations

We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are L ^p metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.     

Cyclic compositions and cycles of the hypercube

The partition graph of a positive integer n, \(P_n\), is the graph whose vertices are the cyclic compositions of n and two vertices are adjacent if one composition is obtained from the other one by replacing two cyclically consecutive parts by their sum. In this paper we introduce and investigate the notions of singular cyclic composition and singular edge of \(P_n\). We associate with every singular edge and every cycle of \(P_n\), whose vertices are aperiodic cyclic compositions of n, a cycle or a set of disjoint cycles of equal length of the hypercube \(Q_n\).     

A thermodynamical formalism for Monge–Ampère equations,Moser–Trudinger inequalities and Kähler–Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold X. This functional can be seen as a generalization of Mabuchi?s K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tian?s α-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.