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1. Gevrey estimates for asymptotic expansions of Tori in weakly dissipative systems*

   https://doi.org/10.1088/1361-6544/ac5cd5  

   Bustamante, Adrián P; De la Llave, Rafael (May 2022, Nonlinearity)

   Abstract We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift . It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asymptotic expansions for the quasiperiodic solutions and the counterterm satisfy Gevrey estimates. That is, the n th term of the expansion is bounded by a power of n !. The Gevrey class (the power of n !) depends only on the Diophantine condition of the frequency and the order of the friction coefficient in powers of the perturbative parameter. The method of proof we introduce may be of interest beyond the problem considered here. We consider a modified Newton method in a space of power expansions. As is custumary in KAM theory, each step of the method is estimated in a smaller domain. In contrast with the KAM results, the domains where we control the Newton method shrink very fast and the Newton method does not prove that the solutions are analytic. On the other hand, by examining carefully the process, we can obtain estimates on the coefficients of the expansions and conclude the series are Gevrey. 

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2. Remarks on the inviscid limit problem for the Navier-Stokes equations

   Igor Kukavica, Vlad Vicol (June 2022, Pure and applied functional analysis)

   For data which are analytic only close to the boundary of the domain, we prove that in the inviscid limit the Navier-Stokes solution converges to the corresponding Euler solution. Compared to earlier results, in this paper we only require boundedness of an integrable analytic norm of the initial data, with respect to the normal variable, thus removing the uniform in viscosity boundedness assumption on the vorticity. As a consequence, we may allow the initial vorticity to be unbounded close to the set $y=0$, which we take as the boundary of the domain; in particular the vorticity can grow with the rate $$1/y^{1-\delta}$$ for $$y$$ close to $$0$$, for any $$\delta>0$$. 

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3. Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

   https://doi.org/10.1007/s00205-021-01697-6  

   Deng, Yu; Zillinger, Christian (July 2021, Archive for Rational Mechanics and Analysis)

   Abstract In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity. 

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4. Counting eigenvalues of Schrödinger operators with fast decaying complex potentials

   Borichev, Alexander; Frank, Rupert; Volberg, Alexander (February 2022, Advances in mathematics)

   We give a sharp estimate of the number of zeros of analytic functions in the unit disc belonging to analytic quasianalytic Carleman–Gevrey classes. As an application, we estimate the number of the eigenvalues for discrete Schrödinger operators with rapidly decreasing complex-valued potentials, and, more generally, for non-symmetric Jacobi matrices. 

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5. Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

   https://doi.org/10.1002/mana.202300273  

   Latushkin, Yuri; Pogan, Alin (July 2024, Mathematische Nachrichten)

   Abstract We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology. 

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