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Register a new userOn the stabilizing effect of rotation in the 3d Euler equations
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While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in $mathbb{R}^3$ with a fixed speed of rotation. We show that for any $M>0$, axisymmetric initial data of sufficiently small size $varepsilon$ lead to solutions that exist for a long time at least $varepsilon^{-M}$ and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.
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Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blowup scenario revealed by Luo-Hou in cite{luo2014potentially,luo2014toward}, which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in cite{Hou-Huang-2021}. One important difference between these two blowup scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in cite{Hou-Huang-2021}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier--Stokes equations. We will present strong numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data. Finally, we present some preliminary results to demonstrate that the 3D Navier--Stokes equations using the same initial condition develop nearly singular behavior with maximum vorticity increased by a factor of $10^{7}$.
The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result for weak solutions to the Navier- Stokes equations.
Inspired by the numerical evidence of a potential 3D Euler singularity cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in cite{luo2014potentially,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in cite{luo2014potentially,luo2013potentially-2} share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in cite{chen2019finite} to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $C^gamma$ norm of the density $theta$ with $gammaapprox 1/3$ is uniformly bounded up to the singularity time.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works(cite{Ch02, Ch03, ChMiZh10}). The proof relies on the classical Bona-Smith method as cite{GuLiYi18}, where similar result was obtained in critical Besov spaces $B^1_{infty,1}$.
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