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Two dimensional invisibility cloaking via transformation optics
Simple waves and pressure delta waves for a Chaplygin gas in twodimensions
1.  Department of Mathematics, Shanghai University, Shanghai 200444, China, China 
2.  Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, United States 
References:
[1] 
S. Bang, Interaction of three and four rarefaction waves of the pressuregradient system,, J. Differential Equations, 246 (2009), 453. Google Scholar 
[2] 
Y. Brenier, Solutions with concentration to the Riemann problem for onedimensional Chaplygin gas equations,, J. Math. Fluid Mech., 7 (2005), 326. doi: 10.1007/s000210050162x. Google Scholar 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream,, J. Differential Equations, 248 (2010), 2931. Google Scholar 
[4] 
G. Q. Chen and H. Liu, Formation of $\delta$shocks and vacuum states in the vanish pressure limit of solutions to the Euler equations for isentropic fliuds,, SIAM J. Math. Anal., 34 (2003), 925. doi: 10.1137/S0036141001399350. Google Scholar 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience, (1948). Google Scholar 
[6] 
Z. Dai ang T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics,, Arch. Rat. Mech. Anal., 155 (2000), 277. doi: 10.1007/s002050000113. Google Scholar 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2003). Google Scholar 
[8] 
L. H. Guo, W. C. Sheng and T. Zhang, The 2D Riemann problem for isentropic Chaplygin gas dynamic system,, Comm. Pure Appl. Anal., 9 (2010), 431. doi: 10.3934/cpaa.2010.9.431. Google Scholar 
[9] 
Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations,, submitted for publication, (2011). Google Scholar 
[10] 
F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117. doi: 10.1007/s002200100506. Google Scholar 
[11] 
F. John, "Partial Differential Equations,", SpringerVerlag, (1982). Google Scholar 
[12] 
B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions,, J. Differential Equations, 118 (1995), 420. Google Scholar 
[13] 
D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions,", thesis, (1977). Google Scholar 
[14] 
N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation,, In, 45 (1986), 81. Google Scholar 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system,, Appl. Math. Mech, 31 (2010), 1. Google Scholar 
[16] 
P. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar 
[17] 
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation,, J. Differential Equations, 236 (2007), 280. Google Scholar 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations,, SIAM J. Appl. Math., 62 (2001), 831. doi: 10.1137/S0036139900361349. Google Scholar 
[19] 
J. Li, Global solution of an initialvalue problem for 2D compressible Euler equations,, J. Differential Equations, 179 (2002), 178. Google Scholar 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics,, Quart. Appl. Math., 59 (2001), 315. Google Scholar 
[21] 
J. Li, Zhicheng Yang and Y. Zheng, Characteristic decompositions and interaction for rarefaction waves of the 2D Euler equations,, to appear in J. Diff. Euqs., (2011). Google Scholar 
[22] 
J. Li and T. Zhang, Generalized RankineHugoniot relations of deltashocks in solutions of transportation equations,, in, (1998), 219. Google Scholar 
[23] 
J. Li, T. Zhang and Y. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,, Commu. Math. Phys, 267 (2006), 1. doi: 10.1007/s0022000600331. Google Scholar 
[24] 
J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D selfsimilar Euler equations,, Arch. Rat. Mech. Anal., 193 (2009), 623. doi: 10.1007/s0020500801406. Google Scholar 
[25] 
M. Li and Y. Zheng, Semihyperbolic patches of solutions of the 2D Euler equations,, to appear in Arch. Rat. Mech. Anal., (2011). Google Scholar 
[26] 
T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", John Wiley and Sons, (1994). Google Scholar 
[27] 
T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems,", Duke University, (1985). Google Scholar 
[28] 
D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rat. Mech. Anal., 191 (2008), 539. doi: 10.1007/s002050080110z. Google Scholar 
[29] 
V. M. Shelkovich, $\delta$ and $\delta'$ wave types of singular solutions of systems of conservation laws and transport and concentration process,, Russian Math. Surveys, 63 (2008), 473. doi: 10.1070/RM2008v063n03ABEH004534. Google Scholar 
[30] 
W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics,, Mem. Amer. Math. Soc. 137, 564 (1999). Google Scholar 
[31] 
K. Song and Y. Zheng, Semihyperbolic patches of solutions of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. doi: 10.3934/dcds.2009.24.1365. Google Scholar 
[32] 
J. H. Spurk and N. Aksel, "Fluid Mechanics,", SpringVerlag Berlin Heidelberg, (2008). Google Scholar 
[33] 
D. Tan and T. Zhang and Y. Zheng, Deltashock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws,, J. Differential Equations, 112 (1994), 1. Google Scholar 
[34] 
T. Zhang and Y. Zheng, Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system,, SIAM J. Math. Anal., 21 (1990), 593. doi: 10.1137/0521032. Google Scholar 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems,", 38 PNLDE, (2001). Google Scholar 
[36] 
Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations,, Disc. Cont. Dyna. Syst., 23 (2009), 605. doi: 10.3934/dcds.2009.23.605. Google Scholar 
show all references
References:
[1] 
S. Bang, Interaction of three and four rarefaction waves of the pressuregradient system,, J. Differential Equations, 246 (2009), 453. Google Scholar 
[2] 
Y. Brenier, Solutions with concentration to the Riemann problem for onedimensional Chaplygin gas equations,, J. Math. Fluid Mech., 7 (2005), 326. doi: 10.1007/s000210050162x. Google Scholar 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream,, J. Differential Equations, 248 (2010), 2931. Google Scholar 
[4] 
G. Q. Chen and H. Liu, Formation of $\delta$shocks and vacuum states in the vanish pressure limit of solutions to the Euler equations for isentropic fliuds,, SIAM J. Math. Anal., 34 (2003), 925. doi: 10.1137/S0036141001399350. Google Scholar 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves,", Interscience, (1948). Google Scholar 
[6] 
Z. Dai ang T. Zhang, Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics,, Arch. Rat. Mech. Anal., 155 (2000), 277. doi: 10.1007/s002050000113. Google Scholar 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics, (2003). Google Scholar 
[8] 
L. H. Guo, W. C. Sheng and T. Zhang, The 2D Riemann problem for isentropic Chaplygin gas dynamic system,, Comm. Pure Appl. Anal., 9 (2010), 431. doi: 10.3934/cpaa.2010.9.431. Google Scholar 
[9] 
Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations,, submitted for publication, (2011). Google Scholar 
[10] 
F. Huang and Z. Wang, Well posedness for pressureless flow,, Comm. Math. Phys., 222 (2001), 117. doi: 10.1007/s002200100506. Google Scholar 
[11] 
F. John, "Partial Differential Equations,", SpringerVerlag, (1982). Google Scholar 
[12] 
B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions,, J. Differential Equations, 118 (1995), 420. Google Scholar 
[13] 
D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions,", thesis, (1977). Google Scholar 
[14] 
N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation,, In, 45 (1986), 81. Google Scholar 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system,, Appl. Math. Mech, 31 (2010), 1. Google Scholar 
[16] 
P. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. Google Scholar 
[17] 
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation,, J. Differential Equations, 236 (2007), 280. Google Scholar 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations,, SIAM J. Appl. Math., 62 (2001), 831. doi: 10.1137/S0036139900361349. Google Scholar 
[19] 
J. Li, Global solution of an initialvalue problem for 2D compressible Euler equations,, J. Differential Equations, 179 (2002), 178. Google Scholar 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics,, Quart. Appl. Math., 59 (2001), 315. Google Scholar 
[21] 
J. Li, Zhicheng Yang and Y. Zheng, Characteristic decompositions and interaction for rarefaction waves of the 2D Euler equations,, to appear in J. Diff. Euqs., (2011). Google Scholar 
[22] 
J. Li and T. Zhang, Generalized RankineHugoniot relations of deltashocks in solutions of transportation equations,, in, (1998), 219. Google Scholar 
[23] 
J. Li, T. Zhang and Y. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,, Commu. Math. Phys, 267 (2006), 1. doi: 10.1007/s0022000600331. Google Scholar 
[24] 
J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D selfsimilar Euler equations,, Arch. Rat. Mech. Anal., 193 (2009), 623. doi: 10.1007/s0020500801406. Google Scholar 
[25] 
M. Li and Y. Zheng, Semihyperbolic patches of solutions of the 2D Euler equations,, to appear in Arch. Rat. Mech. Anal., (2011). Google Scholar 
[26] 
T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", John Wiley and Sons, (1994). Google Scholar 
[27] 
T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems,", Duke University, (1985). Google Scholar 
[28] 
D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rat. Mech. Anal., 191 (2008), 539. doi: 10.1007/s002050080110z. Google Scholar 
[29] 
V. M. Shelkovich, $\delta$ and $\delta'$ wave types of singular solutions of systems of conservation laws and transport and concentration process,, Russian Math. Surveys, 63 (2008), 473. doi: 10.1070/RM2008v063n03ABEH004534. Google Scholar 
[30] 
W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics,, Mem. Amer. Math. Soc. 137, 564 (1999). Google Scholar 
[31] 
K. Song and Y. Zheng, Semihyperbolic patches of solutions of the pressure gradient system,, Disc. Cont. Dyna. Syst., 24 (2009), 1365. doi: 10.3934/dcds.2009.24.1365. Google Scholar 
[32] 
J. H. Spurk and N. Aksel, "Fluid Mechanics,", SpringVerlag Berlin Heidelberg, (2008). Google Scholar 
[33] 
D. Tan and T. Zhang and Y. Zheng, Deltashock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws,, J. Differential Equations, 112 (1994), 1. Google Scholar 
[34] 
T. Zhang and Y. Zheng, Conjecture on the structure of solution of the Riemann problem for 2D gas dynamics system,, SIAM J. Math. Anal., 21 (1990), 593. doi: 10.1137/0521032. Google Scholar 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems,", 38 PNLDE, (2001). Google Scholar 
[36] 
Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations,, Disc. Cont. Dyna. Syst., 23 (2009), 605. doi: 10.3934/dcds.2009.23.605. Google Scholar 
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