Gevrey Norm Research Articles

Trend to Equilibrium for Flows With Random Diffusion

Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak–Keller–Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow \infty $.

Mach limits in analytic spaces on exterior domains

<p style='text-indent:20px;'>We address the Mach limit problem for the Euler equations in an exterior domain with an analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extend more generally to Gevrey initial data with convergence in a Gevrey norm.</p>

Mach limits in analytic spaces

We address the Mach limit problem for the Euler equations in the analytic spaces. We prove that, given analytic data, the solutions to the compressible Euler equations are uniformly bounded in a suitable analytic norm and then show that the convergence toward the incompressible Euler solution holds in the analytic norm. We also show that the same results hold more generally for Gevrey data with the convergence in the Gevrey norms.

Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes

In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces Hs,s>12. This complements the seminal work of Foias and Temam (1989) [26] on H1 based Gevrey classes, thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for 12<s<52.

Asymptotic expansions in time for rotating incompressible viscous fluids

We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.

On whether zero is in the global attractor of the 2D Navier–Stokes equations

The set of nonzero external forces for which the zero function is in the global attractor of the two-dimensional Navier–Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of C∞ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.

Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier-Stokes system on an n-torus. This estimate improves or matches previously known estimates provided that certain bounds on the initial data are satisfied. It is argued that for 2D or 3D turbulent flows, the initial data is guaranteed to satisfy these hypothesized bounds on a significant portion of the 2D global attractor or the 3D weak attractor. In these scenarios, the estimate obtained for 3D generalizes and improves upon that of [Doering-Titi], while in 2D, the estimate matches the best known one found in [Kukavica]. A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms.

Sharp analytic-Gevrey regularity estimates down to $t=0$ for solutions to semilinear heat equations

We study the Gevrey regularity down to $t=0$ of solutions to the initial-value problem for the semilinear heat equation $\partial_tu-\Delta u+F[u]=0$ with polynomial non-linearities. The approach is based on suitable iterative fixed point methods in $L^p$-based Banach spaces with anisotropic Gevrey norms with respect to the time and space variables. We also construct explicit solutions uniformly analytic in $t\geq 0$ and $x\in {\mathbb R}^n$ for some conservative non-linear terms with symmetries.

Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in [formula omitted

Motivated by the work of Foias and Temam [C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989) 359–369], we prove the existence and Gevrey regularity of local solutions to the Kuramoto–Sivashinsky equation in R n with initial data in the space of distributions. The control on the Gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data. In the particular case when n = 1 , our analysis allows for initial data that are less smooth than that considered by Grujić and Kukavica [Z. Grujić, I. Kukavica, Space analyticity for the Navier–Stokes and related equations with initial data in L p , J. Funct. Anal. 152 (1998) 447–466].

Local existence and Gevrey regularity of 3-D Navier–Stokes equations with [formula omitted] initial data

We obtain local existence and Gevrey regularity of 3-D periodic Navier–Stokes equations in case the sequence of Fourier coefficients of the initial data is in ℓ p ( p < 3 / 2 ) . The ℓ p norm of the sequence of Fourier coefficients of the solution and its analogous Gevrey norm remains bounded on a time interval whose length depends only on the size of the body force and the ℓ p norm of the Fourier coefficient sequence of the initial data. The control on the Gevrey norm produces explicit estimates on the analyticity radius of the solution as in Foias and Temam (J. Funct. Anal. 87 (1989) 359–369). The results provide an alternate approach in estimating the space-analyticity radius of solutions to Navier–Stokes equations than the one presented by Grujić and Kukavica (J. Funct. Anal. 152 (1998) 447–466).

On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations

The radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity.

Gevrey regularity for a class of dissipative equations with analytic nonlinearity

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global (in time) estimates in Gevrey norms which yields decay of higher order derivatives which are optimal. Applications include (temporal) decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations, nonlinear heat equations with fractional dissipation, a variant of the Burgers' equation with a cubic or higher order nonlinearity, and the generalized Cahn-Hilliard equation. The decay results for the last three cases seem to be new while our approach provides an alternate proof for the recently obtained $L^p\, (1