### Abstract

Consider the equations of Navier-Stokes on ℝ^{n} with initial data U_{0} of the form U_{0}(x) = u_{0}(x) - Mx, where M is an n x n matrix with constant real entries and u_{0} ∈ L _{σ}
^{p}(ℝ^{n}). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in L_{σ}
^{p}(ℝ^{n}). Moreover, if ∥e ^{tM}∥ ≦ 1 for all t ≧ 0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.

Original language | English |
---|---|

Pages (from-to) | 269-285 |

Number of pages | 17 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 175 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Feb |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mathematics(all)
- Mathematics (miscellaneous)

### Cite this

*Archive for Rational Mechanics and Analysis*,

*175*(2), 269-285. https://doi.org/10.1007/s00205-004-0347-0

**The Navier-Stokes equations in ℝn with linearly growing initial data.** / Hieber, Matthias Georg; Sawada, Okihiro.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*, vol. 175, no. 2, pp. 269-285. https://doi.org/10.1007/s00205-004-0347-0

}

TY - JOUR

T1 - The Navier-Stokes equations in ℝn with linearly growing initial data

AU - Hieber, Matthias Georg

AU - Sawada, Okihiro

PY - 2005/2

Y1 - 2005/2

N2 - Consider the equations of Navier-Stokes on ℝn with initial data U0 of the form U0(x) = u0(x) - Mx, where M is an n x n matrix with constant real entries and u0 ∈ L σ p(ℝn). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in Lσ p(ℝn). Moreover, if ∥e tM∥ ≦ 1 for all t ≧ 0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.

AB - Consider the equations of Navier-Stokes on ℝn with initial data U0 of the form U0(x) = u0(x) - Mx, where M is an n x n matrix with constant real entries and u0 ∈ L σ p(ℝn). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in Lσ p(ℝn). Moreover, if ∥e tM∥ ≦ 1 for all t ≧ 0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.

UR - http://www.scopus.com/inward/record.url?scp=13244298349&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=13244298349&partnerID=8YFLogxK

U2 - 10.1007/s00205-004-0347-0

DO - 10.1007/s00205-004-0347-0

M3 - Article

AN - SCOPUS:13244298349

VL - 175

SP - 269

EP - 285

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -