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Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1
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In this article, a class of weak Navier-Stokes equations with bounded kinetic energy, integrable vorticity, and smoothness outside a fractal set of singular times with Hausdorff dimension strictly less than 1 were studied.Citations
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Convex integration and phenomenologies in turbulence
Tristan Buckmaster,Vlad Vicol +1 more
TL;DR: In this article, the authors discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations, and present Isett's recent resolution of the flexible side of the Onsager conjecture.
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Convex integration constructions in hydrodynamics
Tristan Buckmaster,Vlad Vicol +1 more
TL;DR: A review of recent developments in the field of mathematical fluid dynamics which utilize techniques that go under the umbrella name convex integration can be found in this article, where the authors focus their attention on the construction of weak solutions for the incompressible Euler, Navier-Stokes, and magneto-hydrodynamic equations which violate these systems' physical energy laws.
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Convex integration and phenomenologies in turbulence.
Tristan Buckmaster,Vlad Vicol +1 more
TL;DR: In this paper, the authors discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations, and present Isett's recent resolution of the flexible side of the Onsager conjecture.
Journal ArticleDOI
Convex integration solutions to the transport equation with full dimensional concentration
Stefano Modena,Gabriel Sattig +1 more
TL;DR: In this paper, the authors constructed infinitely many incompressible Sobolev vector fields u ∈ C t W x 1, p ˜ on the periodic domain T d for which uniqueness of solutions to the transport equation fails in the class of densities ρ ∈ c t L x p, provided 1 / p + 1/p ˜ > 1 + 1 / d.
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Sharp nonuniqueness for the Navier-Stokes equations
Alexey Cheskidov,Xiaoyutao Luo +1 more
TL;DR: In this paper, the Navier-Stokes equations in the periodic setting were shown to be locally smooth outside a singular set in time of Hausdorff dimension less than Ω(varepsilon).
References
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Partial regularity of suitable weak solutions of the navier‐stokes equations
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On the Navier-Stokes initial value problem. I
TL;DR: In this article, the authors considered the Navier-Stokes equation for 3-dimensional flows and deduced the existence theorems for 3D flows through a Hilbert space approach, making use of the theory of fractional powers of operators.