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LOCAL SMOOTH SOLUTIONS OF A THIN SPRAY MODEL WITH COLLISIONS

JULIEN MATHIAUD

CEA, DAM, DIF, F-91297 Arpajon, France

CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue President Wilson, F-94230 Cachan, France

https://doi.org/10.1142/S0218202510004192Cited by:19 (Source: Crossref)

Abstract

Sprays are complex flows made of liquid droplets surrounded by a gas. The aim of this paper is to study the local in time well-posedness of a collisional thin spray model, that is a coupling between Euler equations for a perfect gas and a Vlasov–Boltzmann equation for the droplets. We prove the existence and uniqueness of (local in time) solutions for this problem as soon as initial data are smooth enough.

Keywords:

References

S. Alinhac and P. Gérard , Oprateurs Pseudo-Diffrentiels et Thorme de Nash–Moser ( Interditions/Editions du CNRS , 1991 ) . Crossref, Google Scholar
C. Baranger and L. Desvillettes, J. Hyper. Diff. Eqns. 3, 1 (2006), DOI: 10.1142/S0219891606000707. Link, Web of Science, Google Scholar
N. Bellomo and G. Toscani, J. Math. Phys. 26, 334 (1985), DOI: 10.1063/1.526664. Crossref, Web of Science, Google Scholar
L. Boudin, L. Desvillettes and R. Motte, Commun. Math. Sci. 1, 657 (2003). Crossref, Google Scholar
H. Brezis , Analyse Fonctionnelle ( Masson , 1983 ) . Google Scholar
R. Caflisch and G. C. Papanicolaou, SIAM J. Appl. Math. 43, 885 (1983), DOI: 10.1137/0143057. Crossref, Web of Science, Google Scholar
C. Cercignani , The Boltzmann Equation and Its Applications ( Springer-Verlag , 1988 ) . Crossref, Google Scholar
R. J. DiPerna and P.-L. Lions, Ann. Math. 130, 321 (1989), DOI: 10.2307/1971423. Crossref, Web of Science, Google Scholar
K. Domelevo and J.-M. Roquejoffre, Comm. Partial Diff. Eqns. 24, 61 (1999), DOI: 10.1080/03605309908821418. Crossref, Web of Science, Google Scholar
Y. Guo, Comm. Math. Phys. 231, 391 (2002), DOI: 10.1007/S00220-002-0729-9. Crossref, Web of Science, Google Scholar
Y. Guo, Arch. Rational Mech. Anal. 169, 305 (2003), DOI: 10.1007/S00205-003-0262-9. Crossref, Web of Science, Google Scholar
K. Hamdache, Jpn. J. Indust. Appl. Math. 15, 1 (1998), DOI: 10.1007/BF03167396. Google Scholar
R. Illner and M. Shinbrot, Comm. Math. Phys. 95, 217 (1984), DOI: 10.1007/BF01468142. Crossref, Web of Science, Google Scholar
P.-L. Lions, JMKU 48, 539 (1994). Web of Science, Google Scholar
A. Majda , Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables ( Springer-Verlag , 1984 ) . Crossref, Google Scholar
J. Mathiaud, Etude de systme de type gaz-particules, Ph.D. thesis, CMLA, Ecole normale suprieure de Cachan, 2006 . Google Scholar
S. Mischler and B. Perthame, SIAM J. Math. Anal. 28, 1015 (1997), DOI: 10.1137/S0036141096298102. Crossref, Web of Science, Google Scholar
D. Serre , Systmes de Lois de Conservation I ( Diderot , 1996 ) . Google Scholar
D. Serre , Systmes de Lois de Conservation II ( Diderot , 1996 ) . Google Scholar
M. Taylor , Partial Differential Equations, Basic Theory ( Springer , 1996 ) . Crossref, Google Scholar
S. Ukai and K. Asano, Publ. Res. Inst. Math. Sci. 18, 477 (1982), DOI: 10.2977/prims/1195183569. Crossref, Google Scholar
F. A. Williams , Combustion Theory ( Benjamin Cummings , 1985 ) . Google Scholar