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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 $10^{7}$.
In this paper, we present strong numerical evidences that the $3$D incompressible axisymmetric Navier-Stokes equations with degenerate diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-simila
In this paper we consider the inverse electromagnetic scattering for a cavity surrounded by an inhomogeneous medium in three dimensions. The measurements are scattered wave fields measured on some surface inside the cavity, where such scattered wave
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 Eul
We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with
In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in 3D effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coor