### Abstract

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

Original language | English (US) |
---|---|

Pages (from-to) | 463-491 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 192 |

Issue number | 2 |

DOIs | |

State | Published - 1998 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*192*(2), 463-491. https://doi.org/10.1007/s002200050305

**Zero viscosity limit for analytic solutions of the navier-stokes equation on a half-space. II. Construction of the Navier-stokes solution.** / Sammartino, Marco; Caflisch, Russel.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 192, no. 2, pp. 463-491. https://doi.org/10.1007/s002200050305

}

TY - JOUR

T1 - Zero viscosity limit for analytic solutions of the navier-stokes equation on a half-space. II. Construction of the Navier-stokes solution

AU - Sammartino, Marco

AU - Caflisch, Russel

PY - 1998

Y1 - 1998

N2 - This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

AB - This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

UR - http://www.scopus.com/inward/record.url?scp=84875267000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875267000&partnerID=8YFLogxK

U2 - 10.1007/s002200050305

DO - 10.1007/s002200050305

M3 - Article

AN - SCOPUS:84875267000

VL - 192

SP - 463

EP - 491

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -