Gevrey Regularity Research Articles

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications

Band-limited wavelets beyond Gevrey regularity

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.

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.

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Analytic and Gevrey regularity for certain sums of two squares in two variables

Analytic and Gevrey regularity for certain sums of two squares in two variables

Optimal Gevrey Regularity for Supercritical Quasi-Geostrophic Equations

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity endowed with the same decay exponent as the linear part. This settles several open problems in Biswas (J Differ Equ 257(6):1753–1772, 2014), Biswas et al. (J Funct Anal 269(10):3083–3119, 2015).

Gevrey regularity of the semigroup corresponding to an Euler-Bernoulli plate equation with localized structural damping

Gevrey regularity of the semigroup corresponding to an Euler-Bernoulli plate equation with localized structural damping

The Gevrey Regularity for the Vlasov–Poisson–Landau and the Non-cutoff Vlasov–Poisson–Boltzmann Systems

The Gevrey Regularity for the Vlasov–Poisson–Landau and the Non-cutoff Vlasov–Poisson–Boltzmann Systems

Gevrey regularity for the Euler–Bernoulli beam equation with localized structural damping

We study a Euler–Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated C 0 -semigroup ( S ( t ) ) t ≥ 0 is of Gevrey class δ > 24 for t>0, hence immediately differentiable. Moreover, we show that ( S ( t ) ) t ≥ 0 is exponentially stable.

OPTIMAL GEVREY STABILITY OF HYDROSTATIC APPROXIMATION FOR THE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN

Abstract In this paper, we study the hydrostatic approximation for the Navier-Stokes system in a thin domain. When we have convex initial data with Gevrey regularity of optimal index $\frac {3}{2}$ in the x variable and Sobolev regularity in the y variable, we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes/Prandtl system. Due to our method in the paper being independent of $\varepsilon $ , by the same argument, we also obtain the well-posedness of the hydrostatic Navier-Stokes/Prandtl system in the optimal Gevrey space. Our results improve upon the Gevrey index of $\frac {9}{8}$ found in [15, 35].

On the Cauchy problem for a four‐component Novikov system with peaked solutions

Considered herein is the Cauchy problem for a four‐component Novikov system with peaked solutions. We first investigate the local Gevrey regularity and analyticity of the solutions by a generalized Ovsyannikov theorem. Then, based on the local well‐posedness of this problem, the results with respect to the nonuniformly continuous dependence on initial data of the solutions in Besov spaces and are established by constructing new approximate solutions and initial data.

Existence of solutions and their behavior for the anisotropic quasi-geostrophic equation in Sobolev and Sobolev-Gevrey spaces

This paper establishes that the anisotropic quasi-geostrophic equation, with orders of fractional dissipation α and β, admits a unique mild solution θ in homogeneous Sobolev and Sobolev-Gevrey spaces H˙a,σs(R2) (with a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s<1−|α−β|) provided that the initial data θ0 is small enough in these spaces (the critical cases α=12 and β=12 are studied as well). Moreover, this work also studies the behavior of this same solution θ through the following decay rate:limsupt→∞tκ2max⁡{α,β}‖θ(t)‖H˙a,σκ=0, for all κ≥0, where a≥0, σ≥1, α∈(12,34], β∈(12,34] and max⁡{2α−1,2β−1}≤s≤min⁡{2−2α,2−2β}). It is important to emphasize that the limit superior above is a consequence of the Gevrey regularity of θ and of the fact thatlimt→∞⁡‖θ(t)‖L2=0, if it is assumed that θ0∈L2(R2).

Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa–Holm equation

Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa–Holm equation

Regularity of the spatially homogenous fractional Kramers-Fokker-Planck equation

We study the Cauchy problem of the spatially homogenous fractional Kramers-Fokker-Planck equation and show that the solution enjoys Gevrey regularity and decays estimation with an L2 initial datum for positive time.

On echoes in magnetohydrodynamics with magnetic dissipation

We study the long time asymptotic behavior of the inviscid magnetohydrodynamic equations with magnetic dissipation near a combination of Couette flow and a constant magnetic field. Here we show that there exist nearby explicit global in time low frequency solutions, which we call waves. Moreover, the linearized problem around these waves exhibits resonances under high frequency perturbations, called echoes, which result in norm inflation in Gevrey regularity and infinite time blow-up in Sobolev regularity.

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 regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients

Gevrey regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients

Minimal Gevrey Regularity for Hörmander Operators

Abstract We prove a minimal Gevrey regularity theorem for Hörmander’s sum of squares type operators ( 1.1), improving the result of Derridj and Zuily [ 10]. The Gevrey index given here is optimal, in the sense that there are operators of this type that just attain that regularity and not any better.

Regularity of semigroups for exponentially tempered stable processes with drift

In this paper, we study the regularity of semigroups for the exponentially tempered 2β-stable processes. First, we use the Fourier analysis technique to prove that for 1≤p<∞, the semigroup for the tempered stable process without drift is analytic in Lp(Rn). Next, it is shown that the semigroup for the tempered stable process with drift is an analytic semigroup if 1/2≤β<1 and the semigroup is a Gevrey semigroup of order δ with δ>1/(2β) if 0<β<1/2. Finally, we find that the result on Gevrey regularity is optimal in some sense.