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On wellposedness of the DegasperisProcesi equation
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), 453481. 
[2] 
Y. Brenier, Solutions with concentration to the Riemann problem for onedimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), 326331. doi: 10.1007/s000210050162x. 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 29312954. 
[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), 925938. doi: 10.1137/S0036141001399350. 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 1948. 
[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), 277298. doi: 10.1007/s002050000113. 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, SpringerVerlag, 2003. 
[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), 431458. doi: 10.3934/cpaa.2010.9.431. 
[9] 
Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations, submitted for publication, 2011. 
[10] 
F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117146. doi: 10.1007/s002200100506. 
[11] 
F. John, "Partial Differential Equations," SpringerVerlag, 1982. 
[12] 
B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), 420451. 
[13] 
D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions," thesis, Adelphi University, Garden City, NY, 1977. 
[14] 
N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, In "Nonlinear Functional Analysis and its Applications," Proc. Symp. Pure Math., 45 (1986), 8190, Providence, Amer. Math. Soc.. 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system, Appl. Math. Mech, 31 (2010), 112. 
[16] 
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537566. doi: 10.1002/cpa.3160100406. 
[17] 
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280292. 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831852. doi: 10.1137/S0036139900361349. 
[19] 
J. Li, Global solution of an initialvalue problem for 2D compressible Euler equations, J. Differential Equations, 179 (2002), 178194. 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics, Quart. Appl. Math., 59 (2001), 315342. 
[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. 
[22] 
J. Li and T. Zhang, Generalized RankineHugoniot relations of deltashocks in solutions of transportation equations, in "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219232, World Scientific, Singapore, 1998. 
[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), 112. doi: 10.1007/s0022000600331. 
[24] 
J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D selfsimilar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623657. doi: 10.1007/s0020500801406. 
[25] 
M. Li and Y. Zheng, Semihyperbolic patches of solutions of the 2D Euler equations, to appear in Arch. Rat. Mech. Anal., 2011. 
[26] 
T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley and Sons, 1994. 
[27] 
T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems," Duke University, 1985. 
[28] 
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rat. Mech. Anal., 191 (2008), 539577. doi: 10.1007/s002050080110z. 
[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), 473546. doi: 10.1070/RM2008v063n03ABEH004534. 
[30] 
W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137, 564 (1999). 
[31] 
K. Song and Y. Zheng, Semihyperbolic patches of solutions of the pressure gradient system, Disc. Cont. Dyna. Syst., 24 (2009), 13651380. doi: 10.3934/dcds.2009.24.1365. 
[32] 
J. H. Spurk and N. Aksel, "Fluid Mechanics," SpringVerlag Berlin Heidelberg, 2008. 
[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), 132. 
[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), 593630. doi: 10.1137/0521032. 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. 
[36] 
Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations, Disc. Cont. Dyna. Syst., 23 (2009), 605616. doi: 10.3934/dcds.2009.23.605. 
show all references
References:
[1] 
S. Bang, Interaction of three and four rarefaction waves of the pressuregradient system, J. Differential Equations, 246 (2009), 453481. 
[2] 
Y. Brenier, Solutions with concentration to the Riemann problem for onedimensional Chaplygin gas equations, J. Math. Fluid Mech., 7 (2005), 326331. doi: 10.1007/s000210050162x. 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 29312954. 
[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), 925938. doi: 10.1137/S0036141001399350. 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 1948. 
[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), 277298. doi: 10.1007/s002050000113. 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, SpringerVerlag, 2003. 
[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), 431458. doi: 10.3934/cpaa.2010.9.431. 
[9] 
Y. B. Hu, J. Q. Li and W. C. Sheng, Interaction of rarefaction waves for 2D isothermal Euler equations, submitted for publication, 2011. 
[10] 
F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117146. doi: 10.1007/s002200100506. 
[11] 
F. John, "Partial Differential Equations," SpringerVerlag, 1982. 
[12] 
B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118 (1995), 420451. 
[13] 
D. J. Korchinski, "Solutions of a Riemann Problem for A 2 $\times$ 2 System of Conservation Laws Prosssing No Classical Solutions," thesis, Adelphi University, Garden City, NY, 1977. 
[14] 
N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation, In "Nonlinear Functional Analysis and its Applications," Proc. Symp. Pure Math., 45 (1986), 8190, Providence, Amer. Math. Soc.. 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system, Appl. Math. Mech, 31 (2010), 112. 
[16] 
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537566. doi: 10.1002/cpa.3160100406. 
[17] 
Z. Lei and Y. Zheng, A complete global solution to the pressure gradient equation, J. Differential Equations, 236 (2007), 280292. 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831852. doi: 10.1137/S0036139900361349. 
[19] 
J. Li, Global solution of an initialvalue problem for 2D compressible Euler equations, J. Differential Equations, 179 (2002), 178194. 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics, Quart. Appl. Math., 59 (2001), 315342. 
[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. 
[22] 
J. Li and T. Zhang, Generalized RankineHugoniot relations of deltashocks in solutions of transportation equations, in "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219232, World Scientific, Singapore, 1998. 
[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), 112. doi: 10.1007/s0022000600331. 
[24] 
J. Li and Y. Zheng, Interaction of rarefaction waves of the 2D selfsimilar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623657. doi: 10.1007/s0020500801406. 
[25] 
M. Li and Y. Zheng, Semihyperbolic patches of solutions of the 2D Euler equations, to appear in Arch. Rat. Mech. Anal., 2011. 
[26] 
T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," John Wiley and Sons, 1994. 
[27] 
T. Li and W. Yu, "Boundary Value Problem for Quasilinear Hyperbolic Systems," Duke University, 1985. 
[28] 
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rat. Mech. Anal., 191 (2008), 539577. doi: 10.1007/s002050080110z. 
[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), 473546. doi: 10.1070/RM2008v063n03ABEH004534. 
[30] 
W. Sheng and T. Zhang, The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137, 564 (1999). 
[31] 
K. Song and Y. Zheng, Semihyperbolic patches of solutions of the pressure gradient system, Disc. Cont. Dyna. Syst., 24 (2009), 13651380. doi: 10.3934/dcds.2009.24.1365. 
[32] 
J. H. Spurk and N. Aksel, "Fluid Mechanics," SpringVerlag Berlin Heidelberg, 2008. 
[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), 132. 
[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), 593630. doi: 10.1137/0521032. 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. 
[36] 
Y. Zheng, Absorption of characteristics by sonic curves of the 2D Euler equations, Disc. Cont. Dyna. Syst., 23 (2009), 605616. doi: 10.3934/dcds.2009.23.605. 
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