8,175 research outputs found
Nonexistence of self-similar singularities for the 3D incompressible Euler equations
We prove that there exists no self-similar finite time blowing up solution to
the 3D incompressible Euler equations. By similar method we also show
nonexistence of self-similar blowing up solutions to the divergence-free
transport equation in . This result has direct applications to the
density dependent Euler equations, the Boussinesq system, and the
quasi-geostrophic equations, for which we also show nonexistence of
self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of
compact support for the vorticit
Finite time singularities to the 3D incompressible Euler equations for solutions in
We introduce a novel mechanism that reveals finite time singularities within
the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably,
we do not construct our blow up using self-similar coordinates, but build it
from infinitely many regions with vorticity, separated by vortex-free regions
in between. It yields solutions of the 3D incompressible Euler equations in
such that the velocity is in the space
for times
and is not at time 0.Comment: Minor correctio
Remarks on the smoothness of the asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations
We show that the constructions of asymptotically self-similar
singularities for the 3D Euler equations by Elgindi, and for the 3D Euler
equations with large swirl and 2D Boussinesq equations with boundary by
Chen-Hou can be extended to construct singularity with velocity that is not smooth at only one point. The proof is based on a
carefully designed small initial perturbation to the blowup profile, and a
BKM-type continuation criterion for the one-point nonsmoothness. We establish
the criterion using weighted H\"older estimates with weights vanishing near the
singular point. Our results are inspired by the recent work of Cordoba,
Martinez-Zoroa and Zheng that it is possible to construct a
singularity for the 3D axisymmetric Euler equations without swirl and with
velocity .Comment: In the previous version, the initial data is not in the weighted
Holder space. We modify the space and show that the initial data is in the
new weighted Holder space. 20 page
On the locally self-similar singular solutions for the incompressible Euler equations
In this paper we consider the locally backward self-similar solutions for the
Euler system, and focus on the case that the possible nontrivial velocity
profiles have non-decaying asymptotics. We derive the meaningful representation
formula of the pressure profile in terms of velocity profiles in this case, and
by using it and the local energy inequality of profiles, we prove some
nonexistence results and show the energy behavior concerning the possible
velocity profiles.Comment: 18 page
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