Journal ArticleDOI
On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations
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In this paper, a criterion of local Holder continuity for suitable weak solutions to Navier-Stokes equations is presented. But the main part of the proof is based on a blow-up procedure and can be applied to other problems in spaces of solenoidal vector fields.Abstract:
We prove a criterion of local Holder continuity for suitable weak solutions to the Navier—Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields.read more
Citations
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Journal ArticleDOI
L3,∞-solutions of the Navier-Stokes equations and backward uniqueness
TL;DR: In this article, it was shown that the L3,∞-solutions of the Cauchy problem for the 3D Navier-Stokes equations are smooth.
Journal ArticleDOI
Nonuniqueness of weak solutions to the Navier-Stokes equation
Tristan Buckmaster,Vlad Vicol +1 more
TL;DR: In this paper, it was shown that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy, and that they can be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions.
Book ChapterDOI
An Introduction to the Navier-Stokes Initial-Boundary Value Problem
TL;DR: The Navier-Stokes equations as discussed by the authors were originally proposed by C.M. Navier and G.H. Stokes in 1822, and they were used by the twenty-six-year old G.L.Stokes 1845 in a quite general way, by means of the theory of continua.
Journal ArticleDOI
Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions
Hao Jia,Vladimír Šverák +1 more
TL;DR: In this paper, it was shown that the classical Cauchy problem for the incompressible 3D Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times.
Journal ArticleDOI
Regularity results for parabolic systems related to a class of non-Newtonian fluids
TL;DR: In this article, a class of parabolic systems of the type: u t − div a(x,t,Du)=0 were considered and various regularity properties: higher integrability, higher differentiability, partial regularity of the spatial gradient, estimates for the (parabolic) Hausdorff dimension of the singular set.
References
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Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Journal ArticleDOI
Partial regularity of suitable weak solutions of the navier‐stokes equations
Journal ArticleDOI
Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet
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Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains
Yoshikazu Giga,Hermann Sohr +1 more
TL;DR: In this article, an abstract perturbation theorem is applied to derive global in time Lq estimates for the Cauchy problem and Lq − Ls estimates for nonstationary Stokes equations in exterior domains.