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We investigate the well-posedness of \alpha-surface quasi-geostrophic (\alpha-SQG) equations in the half-plane, where \alpha =0 and \alpha =1 correspond to the 2D Euler and SQG equations respectively. For 0<\alpha \le 1/2, we prove local well-posedness in certain weighted anisotropic Hölder spaces. We also show that such a well-posedness result is sharp: for any 0<\alpha \le 1, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on \alpha) for initial data smooth up to the boundary. [ABSTRACT FROM AUTHOR]

We show that the log-Hölder coefficients of a solution to the transport equation is preserved in time. [ABSTRACT FROM AUTHOR]

For the axi-symmetric incompressible Euler equations, we prove linear in time filamentation near Hill's vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained by the first author with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis in 1986. [ABSTRACT FROM AUTHOR]

The m-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with m-fold rotational symmetry for m ≥ 2 . For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the L 1 norm of the vorticity, for m-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework. [ABSTRACT FROM AUTHOR]

We study two‐dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well‐posedness result that allows for the data and solutions to be scale‐invariant. These scale‐invariant solutions are new and their study seems to have far‐reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two‐dimensional Euler squation. That is, in the well‐known L1∩L∞ theory of Yudovich, the L1‐assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0‐homo‐geneous in space. Such vorticity is shown to satisfy a new one‐dimensional evolution equation on 𝕊1. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi‐periodic solutions to the two‐dimensional Euler equation exhibiting pendulum‐like behavior. Finally, using the analysis of the one‐dimensional equation, we exhibit strong solutions to the two‐dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi‐geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one‐dimensional model that bears great resemblance to the so‐called De Gregorio model. We then show that finite‐time singularity formation for the one‐dimensional model implies finite‐time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite‐energy solutions for long time and have direct bearing on the global regularity problem for finite‐energy solutions. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Abstract We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane. [ABSTRACT FROM AUTHOR]

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in R d with any d ≥ 3 is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li (2021) [5] in the case d = 3. The ill-posedness is shown for certain critical Lorentz spaces as well. [ABSTRACT FROM AUTHOR]

We partially answer a question raised by Kiselev and Zlatos (Int Math Res Not 2005(38):2315-2339, ); in the generalized dyadic model of the Euler equation, a blow-up of $${H^{1/3+\delta}}$$ -norm occurs. We recover a few previous blow-up results for various related dyadic models as corollaries. [ABSTRACT FROM AUTHOR]

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: { (x 1 , x 2 , x 3) : 0 < x 3 < x 2 < x 1 }. In this domain, we prove local well-posedness for C α vorticities not necessarily vanishing on the boundary with any 0 < α < 1 , and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R 3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R 3 with bounded and piecewise smooth vorticities. [ABSTRACT FROM AUTHOR]

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂ t B + V ⋅ ∇ B = B ⋅ ∇ V , V = − ∇ × (− Δ) − a B , ∇ ⋅ B = 0 , where B is a time-dependent vector field in R 3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1 / 2 < a < 1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions. [ABSTRACT FROM AUTHOR]