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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. Google Scholar 
[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. Google Scholar 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 29312954. 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), 925938. doi: 10.1137/S0036141001399350. Google Scholar 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 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), 277298. doi: 10.1007/s002050000113. Google Scholar 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, SpringerVerlag, 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), 431458. 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), 117146. 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), 420451. 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, Adelphi University, Garden City, NY, 1977. Google Scholar 
[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.. Google Scholar 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system, Appl. Math. Mech, 31 (2010), 112. Google Scholar 
[16] 
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537566. 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), 280292. Google Scholar 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831852. 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), 178194. Google Scholar 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics, Quart. Appl. Math., 59 (2001), 315342. 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 "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219232, World Scientific, Singapore, 1998. 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), 112. 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), 623657. 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), 539577. 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), 473546. 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), 13651380. 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), 132. 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), 593630. doi: 10.1137/0521032. Google Scholar 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. Google Scholar 
[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. 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), 453481. Google Scholar 
[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. Google Scholar 
[3] 
S. X. Chen and A. F. Qu, Interaction of rarefaction waves in jet stream, J. Differential Equations, 248 (2010), 29312954. 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), 925938. doi: 10.1137/S0036141001399350. Google Scholar 
[5] 
R. Courant and K. O. Friedrichs, "Supersonic Flow and Shock Waves," Interscience, New York, 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), 277298. doi: 10.1007/s002050000113. Google Scholar 
[7] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, SpringerVerlag, 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), 431458. 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), 117146. 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), 420451. 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, Adelphi University, Garden City, NY, 1977. Google Scholar 
[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.. Google Scholar 
[15] 
G. Lai and W. C. Sheng, Simple waves for 2D isentropic irrotational selfsimilar Euler system, Appl. Math. Mech, 31 (2010), 112. Google Scholar 
[16] 
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537566. 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), 280292. Google Scholar 
[18] 
J. Li, On the 2D gas expansion for compressible Euler euqations, SIAM J. Appl. Math., 62 (2001), 831852. 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), 178194. Google Scholar 
[20] 
J. Li and H. Yang, Deltashocks as limits of vanishing viscosity fo multidimensional zeropressure gas dynamics, Quart. Appl. Math., 59 (2001), 315342. 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 "Nonlinear PDE and Related Areas" (G. Q. Chen et al. Eds.), pp. 219232, World Scientific, Singapore, 1998. 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), 112. 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), 623657. 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), 539577. 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), 473546. 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), 13651380. 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), 132. 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), 593630. doi: 10.1137/0521032. Google Scholar 
[35] 
Y. Zheng, "Systems of Conservation Laws: 2D Riemann Problems," 38 PNLDE, Birkhäuser, Boston, 2001. Google Scholar 
[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. Google Scholar 
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