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Blow-up for the incompressible 3D-Euler equations with uniform $C^{1,\frac{1}{2}-\epsilon}\cap L^2$ force
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abstract
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space $C^{3,\frac12}\cap L^2$ for the time interval $[0, T)$, where $T > 0$ is finite, subject to a uniform force in $C^{1,\frac12 -\epsilon}\cap L^2$. Remarkably, our methodology results in a blow-up: as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity, all while preserving the solution's smoothness throughout, except at the origin. In the process of our blow-up construction, self-similar coordinates are not utilized and we are able to treat solutions beyond the $C^{1,\frac13+}$ threshold regularity of axy-symmetric solutions without swirl.
Forward citations
Cited by 4 Pith papers
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Blow-up and uniqueness of Leray-Hopf solutions to forced Navier-Stokes equations
Explicit forces in L^2_loc L^{6/5} and L^{5/4}_loc L^2 produce unique, energy-conserving global Leray-Hopf solutions of forced 3D Navier-Stokes with finite-time norm blow-up at one or countably many points.
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Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity
For every d≥3 and every α<1−2/d, axisymmetric swirl-free incompressible Euler admits self-similar blow-up solutions with C^{1,α} initial velocity that is smooth away from the origin.
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Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon}\cap L^2$ force
There exist compactly supported, smooth-before-blow-up solutions of the forced 2D Boussinesq equation that blow up in finite time with a uniformly C^{1,alpha} cap L^2 force for every alpha < sqrt(4/3)-1.
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Analytic finite-rank corrections for singularly weighted estimates in a computer-assisted proof of 3D Euler singularity
Analytic low-rank corrections convert numerically determined global basis functions into exactly vanishing local modes, enforcing |x|^3 vanishing conditions needed for singular weighted stability estimates in computer...
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