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1. Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

   https://doi.org/10.1002/cpa.21974  

   Ionescu, Alexandru; Jia, Hao (April 2022, Communications on Pure and Applied Mathematics)

   Shatah, Jalal (Ed.)

   We prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $$t\to\infty$$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping. © 2021 Wiley Periodicals LLC. 

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2. Mach limits in analytic spaces on exterior domains

   https://doi.org/10.3934/dcds.2022027  

   Jang, Juhi; Kukavica, Igor; Li, Linfeng (January 2022, Discrete and Continuous Dynamical Systems)

   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. 

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3. 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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4. Nonlinear inviscid damping near monotonic shear flows

   https://doi.org/10.4310/ACTA.2023.v230.n2.a2  

   Ionescu, Alexandru D.; Jia, Hao (July 2023, Acta Mathematica)

   Tobias Ekholm (Ed.)

   We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $$\mathbb{T}\times[0,1]$$. More precisely, we consider shear flows $(b(y),0)$ given by a function $$b$$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $$u$$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $$u$$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved. 

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5. Hölder regularity of the Boltzmann equation past an obstacle

   https://doi.org/10.1002/cpa.22167  

   Kim, Chanwoo; Lee, Donghyun (April 2024, Communications on Pure and Applied Mathematics)

   Abstract Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)]. 

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