On smoothness of L3,?-solutions to the Navier?Stokes equations up to boundary

Seregin, Gregory

doi:10.1007/s00208-004-0625-z

Cited by 52 publications

(47 citation statements)

References 12 publications

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“…The endpoint case (p, q) = (3, ∞) was recently resolved [7] (also see the references in [34,7] for earlier results in subclasses). Similar regularity criteria have been established near the boundary [36,12,28].…”

Section: Introductionsupporting

confidence: 56%

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Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Gustafson

Kang

Tsai

2007

Commun. Math. Phys.

106

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We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled L p,q x,t -norm of the velocity with 3/p+2/q ≤ 2, 1 ≤ q ≤ ∞, or the L p,q x,t -norm of the vorticity with 3/p+2/q ≤ 3, 1 ≤ q < ∞, or the L p,q x,t -norm of the gradient of the vorticity with 3/p+2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z.

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Section: Introductionsupporting

confidence: 56%

“…The second term in (28) is just N r|(∇h) r |. For the first term in (28), use the Poincaré-Sobolev inequality, the mean-value property, and an interior estimate:…”

Section: Proof Of Theorem 11 (Ii)mentioning

confidence: 99%

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Gustafson

Kang

Tsai

2007

Commun. Math. Phys.

106

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“…However, there is an interesting marginal case s = 3, in which no estimate of type (1.5) is known. In papers [3,14,17], it has been shown that lim sup t↑T v(·, t) L 3 ( ) = ∞ (1.6) for = R 3 , = R 3 + := {x = (x i ) ∈ R 3 : x 3 > 0} and for being a bounded domain with sufficiently smooth boundary.…”

Section: 4)mentioning

confidence: 99%

A necessary condition of potential blowup for the Navier–Stokes system in half-space

Barker

Seregin

2016

Math. Ann.

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“…In particular, this required the unique solvability theorem for the inverse heat conduction equation with lower terms in the half-space (see [18]- [22]). …”

Section: §1 Introductionmentioning

confidence: 99%

New version of the Ladyzhenskaya–Prodi–Serrin condition

Seregin

2006

St. Petersburg Math. J.

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Abstract.A new local version of the Ladyzhenskaya-Prodi-Serrin regularity condition for weak solutions of the nonstationary 3-dimensional Navier-Stokes system is proved. The novelty is in that the energy of the solution is not assumed to be finite. §1. IntroductionIn this paper we present a new local version of the Ladyzhenskaya-Prodi-Serrin condition (LPS condition), which ensures the regularity of weak solutions of the nonstationary 3-dimensional Navier-Stokes system. It is known that, in the regularity theory for this system, there is a considerable distinction between the local and the global version. What is usually meant by global regularity is the differentiability properties of solutions of initial-boundary value problems for the Navier-Stokes equations, treated in dependence on the initial and boundary data and the right-hand sides of the equations. In contrast, the local theory describes the property of the Navier-Stokes equations to smooth out their solutions under the assumption that these solutions possess a certain regularity from the outset.Usually, the distinction mentioned above is illustrated by Serrin's example in which the velocity field v(x, t) = c(t)∇h(x) and the pressure field p(x, t) = −c (t)h(x) − and T is a positive parameter, provided h is a function harmonic in Ω. It is seen that this solution is not smooth in t if the function c is not. Formally, this is explained by the fact that, locally, ∂ t v and ∇p may compensate each other and cannot be estimated separately. However, in the case of initial-boundary value problems, one usually starts with attempts to estimate ∂ t v (globally), after which the properties of ∇p can easily be obtained from the equations.In the present paper we shall not touch on the global regularity, referring the readerIn its turn, in the local theory, two cases are distinguished: interior and boundary. First, we discuss the simpler case of interior regularity. The first result in this direction was established by Serrin in [5]; by the way, this was the starting point of the local theory of nonstationary Navier-Stokes equations. Later, it was generalized by Struwe in [6]. The next theorem is stated in Struwe's wording.2000 Mathematics Subject Classification. Primary 35Q30.

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On smoothness of L3,?-solutions to the Navier?Stokes equations up to boundary

Abstract: We show that L 3,∞ -solutions to the three-dimensional NavierStokes equations near a flat part of the boundary are smooth.1991 Mathematical subject classification (Amer. Math. Soc.): 35K, 76D.

Cited by 52 publications

References 12 publications

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

A necessary condition of potential blowup for the Navier–Stokes system in half-space

New version of the Ladyzhenskaya–Prodi–Serrin condition

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