Gevrey Type Research Articles

Double‐Scale Expansions for a Logarithmic Type Solution to a q‐Analog of a Singular Initial Value Problem

We examine a linear q−difference differential equation, which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q−analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm logt and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q−difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey‐type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q−Gevrey type growth.

The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations

In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey type asymptotic expansion with respect to the parameter.

Gevrey Type Regularity of the Riesz–Feller Operator Perturbed by Gradient in $$L^p(\mathbb {R})$$

Gevrey Type Regularity of the Riesz–Feller Operator Perturbed by Gradient in $$L^p(\mathbb {R})$$

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.

Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

In this paper we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping, and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of a point spectrum for the 1D exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has positive Lebesgue measure.

On the Representation and the Uniform Polynomial Approximation of Polyanalytic Functions of Gevrey Type on the Unit Disk

On the Representation and the Uniform Polynomial Approximation of Polyanalytic Functions of Gevrey Type on the Unit Disk

On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities

A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

The Surface Quasi-geostrophic Equation With Random Diffusion

Abstract Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by a recent work of Glatt-Holtz and Vicol [16].

Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball—Preasymptotics, asymptotics, and tractability

In this paper, we investigate optimal linear approximations (n-approximation numbers) of the embeddings from the Sobolev spaces Hr(r>0) for various equivalent norms and the Gevrey type spaces Gα,β(α,β>0) on the sphere Sd and on the ball Bd, where the approximation error is measured in the L2-norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasize that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems Id:Hr→L2 are weakly tractable if and only if r>1, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α,β>0, the approximation problems Id:Gα,β→L2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α≥1.

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding ${id}:\ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identically to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrodinger equation, where mixed regularity is present.

Exponentially sparse representations of Fourier integral operators

We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order s>1 or analytic ( s=1 ), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ( s<1 ) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

We analyze the theoretical properties of an adaptive Legendre–Galerkin method in the multidimensional case. After the recent investigations for Fourier–Galerkin methods in a periodic box and for Legendre–Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/ $$hp$$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $$H^1$$ , based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type.

On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

Contraction and optimality properties of adaptive Legendre–Galerkin methods: The one-dimensional case

As a first step towards a mathematically rigorous understanding of adaptive spectral/hp discretizations of elliptic boundary-value problems, we study the performance of adaptive Legendre–Galerkin methods in one space dimension. These methods offer unlimited approximation power only restricted by solution and data regularity. Our investigation is inspired by a similar study that we recently carried out for Fourier–Galerkin methods in a periodic box. We first consider an “ideal” algorithm, which we prove to be convergent at a fixed rate. Next we enhance its performance, consistently with the expected fast error decay of high-order methods, by activating a larger set of degrees of freedom at each iteration. We guarantee optimality (in the non-linear approximation sense) by incorporating a coarsening step. Optimality is measured in terms of certain sparsity classes of the Gevrey type, which describe a (sub-)exponential decay of the best approximation error.

The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces

This paper studies the periodic Cauchy problem for a KdV equation whose dispersion is of order m=2j+1, where j is a positive integer, (KdVm). Using Bourgain–Gevrey type analytic spaces and appropriate bilinear estimates, it is shown that local in time well-posedness holds when the initial data belong to an analytic Gevrey spaces of order σ. This implies that in the space variable the regularity of the solution remains the same with that of the initial data. It also implies that the size of the uniform radius of analyticity is preserved. Moreover, the solution is not necessarily Gσ in time. However, it belongs to Gmσ(R) near zero for every x on the circle.

Structure and Gevrey asymptotic of solutions representing topological defects to some partial differential equations

We consider the positive real-valued solutions of a particular type of ordinary differential equations that arise when considering defect solutions to semilinear partial differential equations. We provide sufficient conditions on the nonlinear term to ensure the existence, uniqueness and monotonicity of solutions to the ordinary differential equation with the prescribed boundary conditions. We then focus on the behaviour of such solutions at infinity and we prove that there is a unique formal expansion at infinity of the Gevrey type, i.e. the coefficients of the expansion grow as a power of a factorial. Moreover, we show that the actual solution is indeed asymptotic 1-Gevrey to this formal expansion. We also present a numerical algorithm to compute the solution for arbitrary values of the degree n in the particular case of the Ginzburg–Landau equation. In particular, we address the difficulty in the numerical computations when n is relatively large due to the fact that the shooting parameter becomes exponentially small for the whole class of nonlinearities considered in this work.

Formal Solutions of Second Order Evolution Equations

We study the initial value problem for a second order evolution equation ∂_tu = F(x,u,∇_xu,∇^2_xu),\ u|_{t=0} = u_0 , where F(x, u, p, q) is a polynomial function in variables u \in \mathbb R , p\in \mathbb R^d , q\in \mathbb R^{d^2} with coefficients analytic on a domain Ω \subset \mathbb R^d , d ≥ 1 and u_0 is analytic on Ω . We construct a formal power series solution \hat u(t, x) = \sum^∞_{n=0} φ_n(x)t^n of the equation and prove that it satisfies Gevrey type estimates |φ_n(x)| ≤ C^{n+1}n! for x\in K ⋐ Ω and n \in \mathbb N_0 , where C does not depend on n . The proof is based on some combinatorial identities and estimates which may be of independent interest.

Regularization of some classes of ultradistribution semigroups and sines

We systematically analyze regularization of different kinds of ultradistribution semigroups and sines, in general, with nondensely defined generators and contemplate several known results concerning the regularization of Gevrey type ultradistribution semigroups. We prove that, for every closed linear operator A which generates an ultradistribution semigroup (sine), there exists a bounded injective operator C such that A generates a global differentiable C-semigroup (C-cosine function) whose derivatives possess some expected properties of operator valued ultradifferentiable functions. With the help of regularized semigroups, we establish the new important characterizations of abstract Beurling spaces associated to nondensely defined generators of ultradistribution semigroups (sines). The study of regularization of ultradistribution sines also enables us to perceive significant ultradifferentiable properties of higher-order abstract Cauchy problems.

The Asymptotic Behavior of Singular Solutions of Some Nonlinear Partial Differential Equations in the Complex Domain

Let u(t, x) ( (t, x) \in ℂ × ℂ^d ) be a solution of a nonlinear partial differential equation in a neighborhood of the origin, which is not necessarily holomorphic on \{t = 0\} . We study the asymptotic behavior of u(t,x) as t → 0 and give its asymptotic terms with remainder estimate of Gevrey type.

One dimensional invariant manifolds of Gevrey type in real-analytic maps

In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of one dimensional parabolic manifolds of degenerate fixed points, i.e. invariant manifolds tangent to the eigenspace of 1, which is assumed to be a simple eigenvalue. We use the parameterization method, reducing the dynamics on the parabolic manifold to a polynomial. We prove that the asymptotic expansions of the parabolic manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence $C^\infty$.