Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

Hyeong Ohk Bae, Hi Jun Choe, Bum Ja Jin

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We first represent the pressure in terms of the velocity in R + 3 . Using this representation we prove that a solution to the Navier-Stokes equations is in L (R + 3 × (0, ∞)) under the critical assumption that u ∈ L loc r, r′ , frac(3, r) + frac(2, r ) ≤ 1 with r ≥ 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L -norm of u.

Original languageEnglish
Pages (from-to)2741-2763
Number of pages23
JournalJournal of Differential Equations
Volume244
Issue number11
DOIs
Publication statusPublished - 2008 Jun 1

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Boundary Regularity
Slip Boundary Condition
Navier Stokes equations
Navier-Stokes Equations
Boundary conditions
Estimate
Reverse Inequality
Weak Solution
Boundedness
Differential equations
Differential equation
Iteration
Norm

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "We first represent the pressure in terms of the velocity in R + 3 . Using this representation we prove that a solution to the Navier-Stokes equations is in L ∞ (R + 3 × (0, ∞)) under the critical assumption that u ∈ L loc r, r′ , frac(3, r) + frac(2, r ′ ) ≤ 1 with r ≥ 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L ∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse H{\"o}lder inequality, we find an integral estimate for L ∞ -norm of u.",
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Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition. / Bae, Hyeong Ohk; Choe, Hi Jun; Jin, Bum Ja.

In: Journal of Differential Equations, Vol. 244, No. 11, 01.06.2008, p. 2741-2763.

Research output: Contribution to journalArticle

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N2 - We first represent the pressure in terms of the velocity in R + 3 . Using this representation we prove that a solution to the Navier-Stokes equations is in L ∞ (R + 3 × (0, ∞)) under the critical assumption that u ∈ L loc r, r′ , frac(3, r) + frac(2, r ′ ) ≤ 1 with r ≥ 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L ∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L ∞ -norm of u.

AB - We first represent the pressure in terms of the velocity in R + 3 . Using this representation we prove that a solution to the Navier-Stokes equations is in L ∞ (R + 3 × (0, ∞)) under the critical assumption that u ∈ L loc r, r′ , frac(3, r) + frac(2, r ′ ) ≤ 1 with r ≥ 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L ∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L ∞ -norm of u.

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