No Access

The well-posedness theory for Euler–Poisson fluids with non-zero heat conduction

Jiang Xu

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, P. R. China

https://doi.org/10.1142/S0219891614500210Cited by:0 (Source: Crossref)

Abstract

This paper is devoted to the Euler–Poisson equations for fluids with non-zero heat conduction, arising in semiconductor science. Due to the thermal effect of the temperature equation, the local well-posedness theory by Xu and Kawashima (2014) for generally symmetric hyperbolic systems in spatially critical Besov spaces does not directly apply. To deal with this difficulty, we develop a generalized version of the Moser-type inequality by using Bony's decomposition. With a standard iteration argument, we then establish the local well-posedness of classical solutions to the Cauchy problem for intial data in spatially Besov spaces.

Keywords:

AMSC: 35M10, 35Q35, 76X05

References

H. Abidi , Rev. Mat. Iberoamericana 23 , 537 ( 2007 ) . Crossref, Web of Science, Google Scholar
G. Alì , D. Bini and R. Nionero , SIAM J. Math. Anal. 32 , 572 ( 2000 ) . Crossref, Web of Science, Google Scholar
P. Amster et al. , J. Math. Anal. Appl. 258 , 52 ( 2001 ) . Crossref, Web of Science, Google Scholar
H. Bahouri , J. Y. Chemin and R. Danchin , Fourier Analysis and Nonlinear Partial Differential Equations ( Springer , Berlin , 2011 ) . Crossref, Google Scholar
J.-Y. Chemin , J. Anal. Math. 77 , 25 ( 1999 ) . Crossref, Web of Science, Google Scholar
J.-Y. Chemin and N. Lerner , J. Differential Equations 121 , 314 ( 1995 ) . Crossref, Web of Science, Google Scholar
G. Q. Chen, J. W. Jerome and B. Zhang, Modeling and Computation for Application in Mathematics, Science, and Engineering (Oxford University Press, New York, 1998) pp. 189–215. Google Scholar
R. Danchin , Invent. Math. 141 , 579 ( 2000 ) . Crossref, Web of Science, Google Scholar
R. Danchin , Comm. Partial Differential Equations 26 , 1183 ( 2001 ) . Crossref, Web of Science, Google Scholar
R. Danchin, Fourier analysis methods for PDE's, Lecture note (2005), http://perso-math.univ-mlv.fr/users/danchin.raphael/ . Google Scholar
P. Degond and P. A. Markowich , Ann. Mat. Pura Appl. (4) 165 , 87 ( 1993 ) . Crossref, Web of Science, Google Scholar
D. Donatelli and P. Marcati , Trans. Amer. Math. Soc. 356 , 2093 ( 2004 ) . Crossref, Web of Science, Google Scholar
I. Gamba , Comm. Partial Differential Equations 17 , 553 ( 1992 ) . Web of Science, Google Scholar
I. Gasser and R. Natalini , Quart. Appl. Math. 57 , 269 ( 1999 ) . Crossref, Web of Science, Google Scholar
Y. Guo and W. Strauss , Arch. Ration. Mech. Anal. 179 , 1 ( 2005 ) . Crossref, Web of Science, Google Scholar
L. Hsiao , S. Jiang and P. Zhang , Monatsh. Math. 136 , 269 ( 2002 ) . Crossref, Web of Science, Google Scholar
L. Hsiao and S. Wang , Math. Model Methods Appl. Sci. 12 , 777 ( 2002 ) . Link, Web of Science, Google Scholar
T. Kato , Arch. Ration. Mech. Anal. 58 , 181 ( 1975 ) . Crossref, Web of Science, Google Scholar
Y. P. Li , J. Differential Equations 225 , 134 ( 2006 ) . Crossref, Web of Science, Google Scholar
A. Majda , Compressible Fluid Flow and Conservation laws in Several Space Variables ( Springer , Berlin , 1984 ) . Crossref, Google Scholar
P. Marcati , A. Milani and P. Secchi , Manuscripta Math. 60 , 49 ( 1988 ) . Crossref, Web of Science, Google Scholar
P. Marcati and R. Natalini , Arch. Ration. Mech. Anal. 129 , 129 ( 1995 ) . Crossref, Web of Science, Google Scholar
P. Marcati and B. Rubino , J. Differential Equations 162 , 359 ( 2000 ) . Crossref, Web of Science, Google Scholar
P. A. Markowich , C. Ringhofer and C. Schmeiser , Semiconductor Equations ( Springer , Vienna , 1990 ) . Crossref, Google Scholar
S. Nishibata and M. Suzuki , Arch. Ration. Mech. Anal. 192 , 187 ( 2009 ) . Crossref, Web of Science, Google Scholar
Y. J. Peng , Y. G. Wang and W.-A. Yong , Proc. Roy. Soc. Edinburgh Sect. A 136 , 1013 ( 2006 ) . Crossref, Web of Science, Google Scholar
T. Runst and W. Sickel , Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations ( de Gruyter , Berlin , 1996 ) . Crossref, Google Scholar
D. H. Wang and G. Q. Chen , J. Differential Equations 144 , 44 ( 1998 ) . Crossref, Web of Science, Google Scholar
J. Xu , Comm. Pure Appl. Anal. 8 , 1073 ( 2009 ) . Crossref, Web of Science, Google Scholar
J. Xu , Math. Models Methods Appl. Sci. 20 , 937 ( 2010 ) . Link, Web of Science, Google Scholar
J. Xu , J. Differential Equations 252 , 915 ( 2012 ) . Crossref, Web of Science, Google Scholar
J. Xu and S. Kawashima , Arch. Ration. Mech. Anal. 211 , 513 ( 2014 ) . Crossref, Web of Science, Google Scholar
J. Xu and W.-A. Yong , Discrete Contin. Dyn. Syst. Ser. A 25 , 1319 ( 2009 ) . Crossref, Web of Science, Google Scholar
Y. Zhou , Ann. Inst. Fourier (Grenoble) 54 , 773 ( 2004 ) . Crossref, Web of Science, Google Scholar