### Abstract

Consider the Cauchy problem for a one-dimensional compressible flow through porous media, v_{t} - u_{x} = 0, x ∈ R, t > 0, u_{t} + p(v)x = -αu, (v, u)|t=0 = (v_{0}, u_{0}) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v_{0}(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥_{L}∞ = O(t_{-1} obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t^{-1/2}). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

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

Pages (from-to) | 177-196 |

Number of pages | 20 |

Journal | Royal Society of Edinburgh - Proceedings A |

Volume | 133 |

Issue number | 1 |

Publication status | Published - 2003 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Royal Society of Edinburgh - Proceedings A*,

*133*(1), 177-196.

**Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media.** / Nishihara, Kenji.

Research output: Contribution to journal › Article

*Royal Society of Edinburgh - Proceedings A*, vol. 133, no. 1, pp. 177-196.

}

TY - JOUR

T1 - Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

AU - Nishihara, Kenji

PY - 2003

Y1 - 2003

N2 - Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

AB - Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

UR - http://www.scopus.com/inward/record.url?scp=0037228601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037228601&partnerID=8YFLogxK

M3 - Article

VL - 133

SP - 177

EP - 196

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1

ER -