Gevrey Spaces Research Articles

Complex analytic solutions for the TQG model

Abstract We present a condition under which the thermal quasi-geostrophic (TQG) model possesses a solution that is holomorphic in time with values in the Gevrey space of complex analytic functions. This can be seen as the complex extension of the work by Levermore and Oliver (1997) for the generalised Euler equation but applied to the TQG model.

Propagation of radius of analyticity for solutions to a fourth‐order nonlinear Schrödinger equation

We prove that the uniform radius of spatial analyticity of solution at time to the one‐dimensional fourth‐order nonlinear Schrödinger equation cannot decay faster than for large , given that the initial data are analytic with fixed radius . The main ingredients in the proof are a modified Gevrey space, a method of approximate conservation law, and a Strichartz estimate for free wave associated with the equation.

On the Cauchy problem for p-evolution equations with variable coefficients: A necessary condition for Gevrey well-posedness

In this paper we consider a class of p-evolution equations of arbitrary order with variable coefficients depending on time and space variables (t,x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.

Exact global control of small divisors in rational normal form *

Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

On the radius of analyticity for a Korteweg–de Vries–Kawahara equation with a weak damping term

We consider the Cauchy problem for an equation of Korteweg–de Vries–Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local well-posedness of this problem. By using an approximate conservation law, we extend this to a global result in such a way that the radius of analyticity of solutions is uniformly bounded below by a fixed positive number for all time.

On the persistence of spatial analyticity for generalized KdV equation with higher order dispersion

AbstractPersistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersion where , are integers. For a class of analytic initial data with a fixed radius of analyticity , we show that the uniform radius of spatial analyticity of solutions at time cannot decay faster than as . In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (, ), where they obtained a decay rate of order . Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.

The smoothing effect in sharp Gevrey space for the spatially homogeneous non-cutoff Boltzmann equations with a hard potential

The smoothing effect in sharp Gevrey space for the spatially homogeneous non-cutoff Boltzmann equations with a hard potential

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces.

In this work, we deal with a one-dimensional stress-rate type model for the response of viscoelastic materials, in relation to the strain-limiting theory. The model is based on a constitutive relation of stress-rate type. Unlike classical models in elasticity, the unknown of the model under consideration is uniquely the stress, avoiding the use of the deformation. Here, we treat the case of periodic boundary conditions for a linearized model. We determine an optimal function space that ensures the local existence of solutions to the linearized model around certain steady states. This optimal space is known as the Gevrey-class [Formula: see text], which characterizes the regularity properties of the solutions. The exponent [Formula: see text] in the Gevrey-class reflects the specific dispersion properties of the equation itself. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

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

AbstractIn 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].

Dynamics near Couette flow for the β-plane equation

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T×[−1,1], and asymptotic behavior of Couette flow in Gevrey spaces on T×R for the β-plane equation. Let T>0 be the horizontal period of the channel and α=2πT be the wave number. We obtain a sharp region O in the whole (α,β) half-plane such that non-shear traveling waves do not exist for (α,β)∈O and such traveling waves indeed exist for (α,β) in the remaining regions, near Couette flow for H≥5 velocity perturbation. The borderlines between the region O and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists β⁎>0 such that if |β|≤β⁎, then non-shear traveling waves do not exist for any T>0, while if |β|>β⁎, then there exists a critical period Tβ>0 so that such traveling waves exist for T∈[Tβ,∞) and do not exist for T∈(0,Tβ), near Couette flow for H≥5 velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any β≠0 and T>0, we prove that there exist no non-shear traveling waves with traveling speeds converging in (−1,1) near Couette flow for H≥5 velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for H<52 velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the Coriolis effects. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the β-plane equation on T×R.

On the Cauchy problem of the two-component Novikov-type system with peaked solutions and H1-conservation law

Considered herein is the Cauchy problem for the two-component Novikov-type system with peaked solutions and [Formula: see text]-conservation law. At first, we establish that the solutions maintain corresponding properties at infinity within the lifespan provided that the initial data decay exponentially and algebraically, respectively. Next, the local regularity and analyticity of the solutions to this problem in Sobolev–Gevrey spaces are discussed by a generalized Ovsyannikov theorem in detail.

KdV-type equations in projective Gevrey spaces

We prove well-posedness of the Cauchy problem for a class of third order quasilinear evolution equations with variable coefficients in projective Gevrey spaces. The class considered is connected with several equations in Mathematical Physics as the KdV and KdVB equation and some of their many generalizations.

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $${\mathbb {T}}\times {\mathbb {R}}$$ . The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on $${\mathbb {T}}\times {\mathbb {R}}$$ . The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

Global small solutions of MHD boundary layer equations in Gevrey function space*

In this paper, we obtain global small solutions and decay estimates for the two-dimensional MHD boundary layer in Gevrey space without any structural assumptions, generalizing the results of [20] in analytic space. The proof method is mainly inspired by [14] and [26], using new auxiliary functions and finer structural analysis to overcome the difficulty of the loss of derivatives and then we obtain the global well-posedness of the MHD boundary layer in the Gevrey 32 space.

Long‐Time Instability of the Couette Flow in Low Gevrey Spaces

AbstractWe prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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.

Traveling Waves Near Couette Flow for the 2D Euler Equation

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $$H^s$$ , with $$s<3/2$$ , at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces [see Bedrossian and Masmoudi (Publ Math Inst Hautes Etudes Sci 122:195–300, 2015)], where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $$H^{\frac{3}{2}+}$$ neighborhoods of Couette flow [see Lin and Zeng (Arch Ration Mech Anal 200:1075–1097, 2011)].

Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system

It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.

Normalization, optimal regularity, and solvability in Gevrey classes of vector fields near trapped orbits

We prove a normalization and solvability result in Gevrey spaces for a family of vector fields in a neighborhood of a torus, extending recent work of Meziani [Trans. Amer. Math. Soc. 369 (2017), pp. 3325–3354]. The consideration of Gevrey order allows for a precise characterization of solvability, or lack thereof, for some vector fields of this class, which includes the real-analytic classes considered by Meziani. We also prove semiglobal solvability for a normalized family of real vector fields in the case of classical and non-homogeneous Gevrey spaces.

New blow‐up criteria for local solutions of the 3D generalized MHD equations in Lei–Lin–Gevrey spaces

AbstractThis paper determines the existence of a unique local solution for the 3D generalized magnetohydrodynamics equations. In order to be more precise, our solution is obtained by involving Lei–Lin–Gevrey spaces as well as Lei–Lin spaces. Furthermore, we present five new blow‐up criteria for this same system when the maximal time of existence is finite. It is important to point out that one of these criteria is obtained by assuming fractional Laplacians with equal powers.