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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. Nonlinear inviscid damping and shear‐buoyancy instability in the two‐dimensional Boussinesq equations

   https://doi.org/10.1002/cpa.22123  

   Bedrossian, Jacob; Bianchini, Roberta; Zelati, Michele Coti; Dolce, Michele (July 2023, Communications on Pure and Applied Mathematics)

   Abstract We investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth. 

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3. Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

   https://doi.org/10.1007/s00205-023-01842-3  

   Coti Zelati, Michele; Elgindi, Tarek M.; Widmayer, Klaus (February 2023, Archive for Rational Mechanics and Analysis)

   Abstract We study the behavior of solutions to the incompressible 2dEuler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of new, non-trivial stationary states that are arbitrarily close to the Kolmogorov flow on the square torus$$\mathbb {T}^2$$ $T^2$in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on$$\mathbb {T}^2$$ $T^2$. Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on$$\mathbb {T}^2$$ $T^2$, and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on$${\mathbb T}^2$$ $T^2$. 

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4. Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

   Bedrossian, J; He, S; Iyer, S; Wang, F (January 2025, Communications in Mathematical Physics)

   In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $$\mathbb{T} \times [-1,1]$$, supplemented with Navier boundary conditions $$\omega|_{y = \pm 1} = 0$$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $$\nu$$. On the other hand, the nonzero modes are assumed size $$O(\nu^{\frac12})$$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework. 

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5. 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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