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Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity
1D VlasovPoisson equations with electron sheet initial data
1.  Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States 
References:
[1] 
F. Bouchut, Global weak solution of the VlasovPoisson system for small electrons mass, Comm PDEs, 16 (1991), 13371365. Google Scholar 
[2] 
D. Chae and E. Tadmor, On the finite time blowup of the EulerPoisson equations in R2, Commun. Math. Sci., 6 (2008), 785789. Google Scholar 
[3] 
B. Cheng and E. Tadmor, Long time existence of smooth solutions for the rapidly rotating shallowwater and Euler equations, SIAM J. Math. Anal., 39 (2008), 16681685. Google Scholar 
[4] 
R. S. Dziurzynski, "Patches of Electrons and Electron Sheets for the 1D VlasovPoisson Equation," Ph.D. dissertation (supervised by A. Majda), Dept. of Math., University of California, Berkeley, CA, 1987. Google Scholar 
[5] 
S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in EulerPoisson equations, Indiana Univ. Math. J., 50 (2001), 109157. Google Scholar 
[6] 
S. Jin and X. T. Li, Multiphase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 182 (2003), 4685. Google Scholar 
[7] 
S. Jin, X. M. Liao and X. Yang, The VlasovPoisson equations as the semiclassical Limit of the SchrödingerPoisson equations: A numerical study, Journal of Hyperbolic Differential Equations, 5 (2008), 569587. Google Scholar 
[8] 
X. T. Li, J. G. Wohlbier, S. Jin and J. H. Booske, An Eulerian method for computing multivalued solutions of the EulerPoisson equaqtions and applications to wave breaking in klystrons, Phys. Rev. E., 70 (2004), 016502. Google Scholar 
[9] 
P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3dimensional VlasovPoisson system, Inventiones Mathematicae, 105 (1991), 415430. Google Scholar 
[10] 
H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435466. Google Scholar 
[11] 
H. Liu and E. Tadmor, Critical thresholds in 2D restricted EulerPoisson equations, SIAM J. Appl. Math., 63 (2003), 18891910. Google Scholar 
[12] 
H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 12251231. Google Scholar 
[13] 
A. Majda, G. Majda and Y. X. Zheng, Concentrations in the onedimensional VlasovPoisson equation I: Temporal development and nonunique weak solutions in the single component case, Physica D, 74 (1994), 268300. Google Scholar 
[14] 
A. Majda, G. Majda and Y. X. Zheng, Concentrations in the onedimensiolal VlasovPoisson equations. II: Screening and the necessity for measurevalued solutions in the two component case, Physica D, 79 (1994), 4176. Google Scholar 
[15] 
E. Tadmor and D. Wei, On the global regularity of subcritical EulerPoisson equations with pressure, J. European Math. Society, 10 (2008), 757769. Google Scholar 
[16] 
Y. X. Zheng and A. Majda, Existence of global weak solutions to onecomponent VlasovPoisson and FokkerPlanckPoisson systems in one space dimension with messures as initial data, Comm. on Pure and Applied Math., XLVII (1994), 13651401. Google Scholar 
show all references
References:
[1] 
F. Bouchut, Global weak solution of the VlasovPoisson system for small electrons mass, Comm PDEs, 16 (1991), 13371365. Google Scholar 
[2] 
D. Chae and E. Tadmor, On the finite time blowup of the EulerPoisson equations in R2, Commun. Math. Sci., 6 (2008), 785789. Google Scholar 
[3] 
B. Cheng and E. Tadmor, Long time existence of smooth solutions for the rapidly rotating shallowwater and Euler equations, SIAM J. Math. Anal., 39 (2008), 16681685. Google Scholar 
[4] 
R. S. Dziurzynski, "Patches of Electrons and Electron Sheets for the 1D VlasovPoisson Equation," Ph.D. dissertation (supervised by A. Majda), Dept. of Math., University of California, Berkeley, CA, 1987. Google Scholar 
[5] 
S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in EulerPoisson equations, Indiana Univ. Math. J., 50 (2001), 109157. Google Scholar 
[6] 
S. Jin and X. T. Li, Multiphase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 182 (2003), 4685. Google Scholar 
[7] 
S. Jin, X. M. Liao and X. Yang, The VlasovPoisson equations as the semiclassical Limit of the SchrödingerPoisson equations: A numerical study, Journal of Hyperbolic Differential Equations, 5 (2008), 569587. Google Scholar 
[8] 
X. T. Li, J. G. Wohlbier, S. Jin and J. H. Booske, An Eulerian method for computing multivalued solutions of the EulerPoisson equaqtions and applications to wave breaking in klystrons, Phys. Rev. E., 70 (2004), 016502. Google Scholar 
[9] 
P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3dimensional VlasovPoisson system, Inventiones Mathematicae, 105 (1991), 415430. Google Scholar 
[10] 
H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435466. Google Scholar 
[11] 
H. Liu and E. Tadmor, Critical thresholds in 2D restricted EulerPoisson equations, SIAM J. Appl. Math., 63 (2003), 18891910. Google Scholar 
[12] 
H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 12251231. Google Scholar 
[13] 
A. Majda, G. Majda and Y. X. Zheng, Concentrations in the onedimensional VlasovPoisson equation I: Temporal development and nonunique weak solutions in the single component case, Physica D, 74 (1994), 268300. Google Scholar 
[14] 
A. Majda, G. Majda and Y. X. Zheng, Concentrations in the onedimensiolal VlasovPoisson equations. II: Screening and the necessity for measurevalued solutions in the two component case, Physica D, 79 (1994), 4176. Google Scholar 
[15] 
E. Tadmor and D. Wei, On the global regularity of subcritical EulerPoisson equations with pressure, J. European Math. Society, 10 (2008), 757769. Google Scholar 
[16] 
Y. X. Zheng and A. Majda, Existence of global weak solutions to onecomponent VlasovPoisson and FokkerPlanckPoisson systems in one space dimension with messures as initial data, Comm. on Pure and Applied Math., XLVII (1994), 13651401. Google Scholar 
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