In this paper, we study the Gevrey and Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff in hard potentials case or Maxwellian molecules case. More explicitly, for the initial data sufficiently small in Hx2Lv2, we prove that the solutions at any positive time are analytic and Gelfand-Shilov regularity G1(Rx3)×S11(Rv3) for strong angular singularity and in Gevrey and Gelfand-Shilov class G1/2s(Rx3)×S1/2s1/2s(Rv3) for mild angular singularity.

In this paper, we investigate an abstract model associated with the Timoshenko system, incorporating fractional dissipative effects. The fractional dissipative effect is characterized by the powers of an arbitrary, strictly positive self-adjoint operator, with domain densely embedded in a Hilbert space. Our first main result demonstrates that the operator, derived by reformulating the abstract system as a first-order system, is the generator of a strongly continuous semigroup. The second main result is that the same semigroup can exhibit properties such as exponential stability, analyticity, or belong to a certain Gevrey class, with its precise behavior dependent on the assigned values of these powers. Furthermore, we establish that the Gevrey class obtained is sharp.

On a Lower Estimate of Alexandrov’s $$ n $$-Width of the Compact Set of Infinitely Smooth Aperiodic Functions in a Gevrey Class

This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier–Stokes equations within the framework of homogeneous Lei–Lin–Gevrey spaces Xa,γρ(R3), where ρ∈[−1,0),a>0, and γ∈(0,1). These function spaces combine the critical scaling structure of the Lei–Lin spaces with the exponential regularity of Gevrey classes, thereby enabling a refined treatment of analytic regularity and frequency localization. The main results are obtained under the assumption of small initial data in the critical Lei–Lin space Xρ(R3), extending previous works and improving regularity thresholds. In particular, we establish that for suitable initial data, the Navier–Stokes system admits unique solutions globally in time. The influence of the Gevrey parameter γ on the high-frequency behavior of solutions is also discussed. This work contributes to a deeper understanding of regularity and decay properties in critical and supercritical regimes.

A new representation of fractional-order Weyl derivatives is given. The Cauchy problem is studied for partial differential equations containing Weyl derivatives. The conditions under which this problem has solutions from the Gevrey classes are found.

On Solvability in Gevrey Classes for the Cauchy Problem for an Equation with Weyl Fractional Derivative

In this article, we investigate the maximal smoothness (infinite differentiable) of the solutions to thermoelastics models, specifically those where the heat equation is of the “phase-lag” of “parabolic” type. We derive optimal regularity results for two distinct models. The first model addresses the transverse oscillations of a fully thermoelastic plate, for which we prove that the semigroup associated with the problem is analytic. The second model considers a partially thermoelastic plate composed of two components: a thermoelastic component with nonzero temperature differences and an elastic component unaffected by temperature variations. For this model, we show that the semigroup S ( t ) belongs to the Gevrey class of order 4, provided the solutions are radial and symmetric. Both analyticity and Gevrey class membership are qualitative properties that intricately link regularity and stablitiy, driven by robust dissipative mechanisms. These properties are significantly stronger than standard regularity conditions, such as belonging to the class C k or a Sobolev space H s .

Abstract We provide an observability inequality in terms of a measurable set for general Gevrey‐regular functions. As an application, we establish an observability estimate from a measurable set for sums of Laplace eigenfunctions in a compact and connected boundaryless Riemannian manifold that belongs to the Gevrey class. The estimate has an explicit dependence on the maximal eigenvalue.

FBI transform in Gevrey classes and Anosov flows

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on \mathbb{T}\times \mathbb{R} . More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as t\to \infty . To our knowledge, this is the first global well-posedness result for the 2-D inhomogeneous incompressible Euler equations.

An alternative definition of fractional-order Weyl derivatives is given and their effect on functions from the Gevrey classes is studied. Conditions for the solvability of the Cauchy problem in Gevrey classes are found for the Weyl partial differential equation.

<p>The paper is concerned with the three-dimensional magnetohydrodynamic equations in the rotational framework concerning with fluid flow of Earth's core and the variation of the Earth's magnetic field. By establishing new balances between the regularizing effects arising from viscosity dissipation and magnetic diffusion with the dispersive effects caused by the rotation of the Earth, we obtain the global existence and uniqueness of solutions of the Cauchy problem of the three-dimensional rotating magnetohydrodynamic equations in Besov spaces. Moreover, the spatial analyticity of solutions is verified by means of the Gevrey class approach.</p>

We discuss nonlinear nonlocal equations with fractional diffusion describing electroconvection phenomena in incompressible viscous fluids. We prove the global well-posedness, global regularity and long time dynamics of the model in bounded smooth domains with Dirichlet boundary conditions. We prove the existence and uniqueness of exponentially decaying in time solutions for H 1 H^1 initial data regardless of the fractional dissipative regularity. In the presence of time independent body forces in the fluid, we prove the existence of a compact finite dimensional global attractor. In the case of periodic boundary conditions, we prove that the unique smooth solution is globally analytic in time, and belongs to a Gevrey class of functions that depends on the dissipative regularity of the model.

On the Asymptotic Behavior of Alexandrov’s $$n $$-Width of the Compact Set of Infinitely Smooth Periodic Functions in a Gevrey Class

Existence and regularity of ultradifferentiable periodic solutions to certain vector fields

Abstract In this paper, we consider a hyperbolic perturbation of the 2D Navier–Stokes equations, which consists in adding the term ε ⁢ u t ⁢ t {\varepsilon u_{tt}} to the Navier–Stokes equations. We prove the global persistence of analyticity and Gevrey-class of solutions. Moreover, we prove that the solution to the perturbed Navier–Stokes equations approximates the solution to the classical Navier–Stokes equations.

We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains D ⊂ ℝd, d = 2, 3. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in D, comprising the countably-normed spaces of I. M. Babuska and B. Q. Guo.As intermediate result, we prove that continuous, piecewise polynomial high order (“p-version”) finite elements with elementwise polynomial degree p ∈ ℕ on arbitrary, regular, simplicial partitions of polyhedral domains D ⊂ ℝd, d ⩾ 2, can be exactly emulated by neural networks combining ReLU and ReLU2 activations.On shape-regular, simplicial partitions of polytopal domains D, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the hp finite element space of I. M. Babuška and B. Q. Guo.

We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyse Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for the Quasi-Monte Carlo integration.

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert [Formula: see text]-function.

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.