# Boundary Regularity of Weak Solutions of the Navier-Stokes Equations

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

We prove that a solution to Navier-Stokes equations is inL2(0,∞:H2(Ω)) under the critical assumption thatu∈Lr,r′, 3/r+2/r′≤1 withr≥3. A boundaryLestimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate forL-norm ofu. Moreover the solution isC1,αcontinuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure.

Original language English 211-247 37 Journal of Differential Equations 149 2 https://doi.org/10.1006/jdeq.1998.3481 Published - 1998 Nov 1

### Fingerprint

Boundary Regularity
Navier Stokes equations
Weak Solution
Navier-Stokes Equations
Reverse Inequality
Continuous Solution
Estimate
Bootstrap
Derivatives
Regularity
Iteration
Norm
Derivative
Theorem

### All Science Journal Classification (ASJC) codes

• Analysis
• Applied Mathematics

### Cite this

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title = "Boundary Regularity of Weak Solutions of the Navier-Stokes Equations",
abstract = "We prove that a solution to Navier-Stokes equations is inL2(0,∞:H2(Ω)) under the critical assumption thatu∈Lr,r′, 3/r+2/r′≤1 withr≥3. A boundaryL∞estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse H{\"o}lder inequality, we find an integral estimate forL∞-norm ofu. Moreover the solution isC1,αcontinuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure.",
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year = "1998",
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In: Journal of Differential Equations, Vol. 149, No. 2, 01.11.1998, p. 211-247.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Boundary Regularity of Weak Solutions of the Navier-Stokes Equations

AU - Choe, Hi Jun

PY - 1998/11/1

Y1 - 1998/11/1

N2 - We prove that a solution to Navier-Stokes equations is inL2(0,∞:H2(Ω)) under the critical assumption thatu∈Lr,r′, 3/r+2/r′≤1 withr≥3. A boundaryL∞estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate forL∞-norm ofu. Moreover the solution isC1,αcontinuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure.

AB - We prove that a solution to Navier-Stokes equations is inL2(0,∞:H2(Ω)) under the critical assumption thatu∈Lr,r′, 3/r+2/r′≤1 withr≥3. A boundaryL∞estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate forL∞-norm ofu. Moreover the solution isC1,αcontinuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure.

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