Gevrey Functions Research Articles

An Introduction to Extended Gevrey Regularity

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.

Ridgelet transform on Gelfand–Shilov‐type spaces

We construct Gelfand–Shilov spaces of type and discuss the continuity and boundedness of the ridgelet transform. We then extend the analysis to generalized functions and explore its application to the space of tempered ultradistributions. Finally, we extend our findings to Gevrey functions of compact support. We also discuss an application of the ridgelet transform in image processing.

Global Gevrey-2 solutions of the 3D axially symmetric Prandtl equations

In this paper, we prove the global existence of small Gevrey-2 solutions to the 3D axially symmetric Prandtl equations. The index 2 is the optimal index for well-posedness result in smooth Gevrey function spaces for data without monotonic assumptions. The novelty of our paper lies in two aspects: one is the tangentially weighted energy construction to match the [Formula: see text] weight in the incompressibility and the other is introducing of the new linearly good unknowns to obtain the fast decay of the lower order Gevrey-2 norms of the solutions and auxiliary functions.

Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption

Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption

Semiclassical Gevrey operators in the complex domain

We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is $\leq 2$.

Gevrey Well-posedness of the MHD boundary layer system with temperature

We study the MHD boundary layer system with temperature and establish the existence and uniqueness of the solutions in Gevrey function space with Gevrey index σ∈(1,32] without any structural assumption. Our proof is based on a new type of cancellation mechanism in this system to overcome the difficulty caused by the loss of derivatives.

Global well-posedness of a Prandtl model from MHD in Gevrey function spaces

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index 2. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator.

The Baouendi-Treves approximation theorem for Gevrey classes and applications

In this work we show how to extend the seminal Baouendi-Treves approximation theorem for Gevrey functions and ultradistributions. As applications we present a Gevrey version of the approximate Poincar\'e Lemma and study ultradistributions vanishing on maximally real submanifolds.

Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

AbstractWe show that if $X$ is an $m$-dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$, the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$-parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$. Moreover, these maps are real analytic and this bound is uniform for a definable family.

Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption

AbstractWe establish the well‐posedness in Gevrey function space with optimal class of regularity 2 for the three‐dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system. This shows that the three‐dimensional instabilities in the system leading to ill‐posedness are not worse than the two‐dimensional ones. © 2021 Wiley Periodicals LLC.

Isentropic Approximation and Gevrey Regularity for the Full Compressible Euler Equations in $${\mathbb {R}}^{N}$$

The article is devoted to the study of isentropic approximation and Gevrey regularity for the full compressible Euler system in $${\mathbb {R}}^{N}$$ (or $${\mathbb {T}}^{N}$$ ) with any dimension $$N\ge 1$$ . We first establish the existence and uniqueness of solution in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ , then with the definition modulus of continuity, we show that the solution of Euler system is continuously dependent of the initial data $$v_{0}$$ in $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ . Finally, the isentropic approximation is investigated in Banach spaces $${\mathcal {B}}_{T}^\nu ({\mathbb {R}}^{N})$$ , provided the initial entropy $$S_{0}(x)$$ changes closing a constant $${\bar{S}}$$ in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ .

Global Trace Formula for Ultra-Differentiable Anosov Flows

Adapting tools that we introduced in Jézéquel (J Spectr Theory 10(1):185–249, 2020) to study Anosov flows, we prove that the trace formula conjectured by Dyatlov and Zworski in (Ann. Sci. Éc. Norm. Supér. (4) 49(3):543–577, 2016) holds for Anosov flows in a certain class of regularity (smaller than $${\mathcal {C}}^\infty $$ but larger than the class of Gevrey functions). The main ingredient of the proof is the construction of a family of anisotropic Hilbert spaces of generalized distributions on which the generator of the flow has discrete spectrum.

Well-posedness of the MHD Boundary Layer System in Gevrey Function Space without Structural Assumption

We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption. Compared to the classical Prandtl equation, the loss of tangential derivative comes from both the velocity and magnetic fields that are coupled with each other. By observing a new type of cancellation mechanism in the system for overcoming the loss derivative degeneracy, we show that the MHD boundary layer system is well-posed with Gevrey index up to 3/2 in both two- and three-dimensional spaces.

Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces

In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field of shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Gevrey function space provided that the initial velocity shear flow is non-degenerately critical at the same point.

On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

Decay of singular values for infinite-dimensional systems with Gevrey regularity

We consider the decay rate of the singular values of the input map, the output map and the Hankel operator for a class of infinite-dimensional systems. This class is characterized by the control operator (or the observation operator) having a smoothing effect. We capture this in the definition of a Gevrey operator (which generalizes the known concept of a Gevrey vector). In applications to PDEs, this abstract assumption on the control operator is typically satisfied when the input is multiplied by a function which is a compactly supported Gevrey function in the spatial variable. Using the theory of polynomial approximation (in particular: truncated Chebyshev expansions), we obtain that the singular values decay exponentially in a root of the approximation dimension. The power of the root depends on the order of the Gevrey operator and on whether the underlying semigroup is nilpotent, exponentially stable or polynomially stable.

Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points

We study the well-posedness of the Prandtl system without monotonicity and analyticity assumption. Precisely, for any index \sigma\in[3/2, 2], we obtain the local in time well-posedness in the space of Gevrey class G^\sigma in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised by D. Gérard-Varet and N. Masmoudi [Ann. Sci. École Norm. Sup. (4) 48 (2015), no. 6, 1273–1325], who solved the case \sigma=7/4 .

Global L^q Gevrey functions, Paley-Weiner theorems, and the FBI transform

Global L^q Gevrey functions, Paley-Weiner theorems, and the FBI transform

Microglobal regularity and the global wavefront set

In this paper, we begin the study of regularity of partial differential equations in the space of global $$L^q$$ Gevrey functions, recently introduced in Adwan et al. (J Geom Anal 27(3):1874–1913, 2017) and Hoepfner and Raich (Indiana Univ Math J, forthcoming) and in a generalized and new function space called the space of global $$L^q$$ Denjoy–Carleman functions. We develop a wedge approach similar to Bony’s theorem (Bony in Seminaire Goulaouic–Schwartz (1976/1977), Equations aux derivees partielles et analyse fonctionnelle, Exp No 3. Centre Math, Ecole Polytech, Palaiseau, 1977) and prove three main theorems. The first establishes the existence of boundary values of continuous functions on a wedge. Next, we borrow the FBI transform approach from Hoepfner and Raich (forthcoming) to define global wavefront sets and prove a relationship between the inclusion of a direction in the global wavefront set and the existence of boundary values of sums of weighted $$L^p$$ functions defined in wedges. The final result is an application in which we prove a global version of a classical result: namely, the relationship between the global characteristic set of a partial differential operator P and the microglobal wavefront sets of u and Pu.

Ultradifferential operators in the study of Gevrey solvability and regularity

AbstractIn this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.