On the interior regularity of weak solutions of the Navier-Stokes equations

Serrin, James

doi:10.1007/bf00253344

Cited by 913 publications

(582 citation statements)

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“…[1,11]) and for the Navier-Stokes equations(e.g. [2,30,26,29,13,21]), the solutions to the problem still look too far to be seen. On the other hand, in many of the nonlinear partial differential equations where the finite time singularity is searched for, one of the most popular scenario to check is by the self-similar ansatz, consistent with the scaling properties of the equations.…”

Section: Introductionmentioning

confidence: 99%

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Chae

2007

Math. Ann.

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In this paper we rule out the possibility of asymptotically selfsimilar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T . For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm,Ḃ 0 1,∞ (R 3 ). For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z, r)) for some q ∈ [2, ∞), where the ball of radius r is shrinking toward a possible singularity point z at the order of √ T − t as t approaches to T . In the L q (R 3 ) convergence case with q ∈ [3, ∞) we present a simple alternative proof of the similar result in [16]. *

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Section: Introductionmentioning

confidence: 99%

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Chae

2007

Math. Ann.

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“…We are interested in the classical problem of finding sufficient conditions for weak solutions of (1.1) such that they become regular. J. Serrin [10,11] is the pioneer in this direction, and later on, Fabes, Jones and Riviere [4], Giga [5], Sohr [14], Struwe [15] and Takahashi [16] extended and improved Serrin's regularity criterion. They showed that if u is a Leray-Hopf weak solution (see Definition 1.1) of (1.1) belonging to Serrin's class,…”

Section: Introductionmentioning

confidence: 99%

Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations

Chen

Zhang

2006

Proc. Amer. Math. Soc.

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“…On the other hand, it has been long known that various additional assumptions guarantee such smoothness. One important class of such assumptions is the following so-called Prodi-Serrin-Ladyzhenskaya criteria, developed over three decades in [11], [21], [24], [23], [25], [26]. If a Leray-Hopf solution u(x, t) further satisfies…”

Section: Introductionmentioning

confidence: 99%

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

Tran

2017

Journal of Mathematical Physics

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In this article we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical L r (0, T ; L s ) regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u L 3 and u Ḣ1/2 . * cvt1@st-andrews.ac.uk † xinwei2@ualberta.ca 2

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On the interior regularity of weak solutions of the Navier-Stokes equations

Cited by 913 publications

References 3 publications

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

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