### Abstract

It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution F(t, x, v) to the Boltzmann equation satisfies locally F/1 + F ∈ W-
_{t,x,v}
^{s,p} for every 1 le; p < D/D - 1 and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

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

Pages (from-to) | 508-548 |

Number of pages | 41 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 65 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Regularity of renormalized solutions in the Boltzmann equation with long-range interactions.** / Arsénio, Diogo; Masmoudi, Nader.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 65, no. 4, pp. 508-548. https://doi.org/10.1002/cpa.21385

}

TY - JOUR

T1 - Regularity of renormalized solutions in the Boltzmann equation with long-range interactions

AU - Arsénio, Diogo

AU - Masmoudi, Nader

PY - 2012/4

Y1 - 2012/4

N2 - It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution F(t, x, v) to the Boltzmann equation satisfies locally F/1 + F ∈ W- t,x,v s,p for every 1 le; p < D/D - 1 and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

AB - It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution F(t, x, v) to the Boltzmann equation satisfies locally F/1 + F ∈ W- t,x,v s,p for every 1 le; p < D/D - 1 and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

UR - http://www.scopus.com/inward/record.url?scp=84856282688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856282688&partnerID=8YFLogxK

U2 - 10.1002/cpa.21385

DO - 10.1002/cpa.21385

M3 - Article

AN - SCOPUS:84856282688

VL - 65

SP - 508

EP - 548

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -