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Transonic flows with shocks past curved wedges for the full Euler equations
The unsteady transonic small disturbance equation: Data on oblique curves
1.  Fields Institute, United States 
2.  Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States 
References:
[1] 
M. Brio and J. K. Hunter, Mach reflection for the two dimensional Burgers equation, Physica D, 60 (1992), 194207. doi: 10.1016/01672789(92)90236G. 
[2] 
S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, Journal of Differential Equations, 125 (1996), 548574. doi: 10.1006/jdeq.1996.0040. 
[3] 
S. Čanić and B. L. Keyfitz, A smooth solution for a Keldysh type equation, Communications in Partial Differential Equations, 21 (1996), 319340. doi: 10.1080/03605309608821186. 
[4] 
S. Čanić, B. L. Keyfitz and E. H. Kim, A free boundary problem for a quasilinear degenerate elliptic equation: Regular reflection of weak shocks, Communications on Pure and Applied Mathematics, LV (2002), 7192. 
[5] 
S. Čanić and D. Mirković, A numerical study of shock reflection modeled by the unsteady transonic small disturbance equation, SIAM Journal on Applied Mathematics, 58 (1998), 13651393. doi: 10.1137/S003613999730884X. 
[6] 
G.Q. Chen and M. Feldman, Global solutions of shock reflection by largeangle wedges for potential flow, Annals of Mathematics, 171 (2010), 10671182. doi: 10.4007/annals.2010.171.1067. 
[7] 
S.X. Chen, Mach configuration in pseudostationary compressible flow, Journal of the American Mathematical Society, 21 (2008), 63100. doi: 10.1090/S0894034707005590. 
[8] 
V. Elling and T.P. Liu, Supersonic flow onto a solid wedge}, Communications on Pure and Applied Mathematics, 61 (2008), 13471448. doi: 10.1002/cpa.20231. 
[9] 
G. Fichera, Sulle equazioni differenziali lineari ellitticoparaboliche del secondo ordine, Atti della Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat. Ser. I, 5 (1956), 130. 
[10] 
G. Fichera, On a unified theory of boundary value problems for elliptic parabolic equations of second order, In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 97120. University of Wisconsin, Madison, 1960. 
[11] 
J. K. Hunter and A. M. Tesdall, On the selfsimilar diffraction of a weak shock into an expansion wavefront, SIAM Journal on Applied Mathematics, 72 (2012), 124143. doi: 10.1137/110834135. 
[12] 
I. Jegdić and K. Jegdić, Properties of solutions in semihyperbolic patches for the unsteady transonic small disturbance equation,, Electronic Journal of Differential Equations, (). 
[13] 
K. Jegdić, B. L. Keyfitz and S. Čanić, Transonic regular reflection for the unsteady transonic small disturbance equation  details of the subsonic solution, In R. Glowinski and J. P. Zolesio, editors, Free and Moving Boundaries: Analysis, Simulation and Control, pages 125163. CRC Press, Boca Raton, 2007. doi: 10.1201/9781420011159.ch6. 
[14] 
B. L. Keyfitz, Selfsimilar solutions of twodimensional conservation laws, Journal of Hyperbolic Differential Equations, 1 (2004), 445492. doi: 10.1142/S0219891604000160. 
[15] 
B. L. Keyfitz, The Fichera function and nonlinear equations, Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I, 30 (2006), 8394. 
[16] 
B. L. Keyfitz, A. Tesdall, K. R. Payne and N. I. Popivanov, The sonic line as a free boundary, Quarterly of Applied Mathematics, 71 (2013), 119133. doi: 10.1090/S0033569X2012012836. 
[17] 
O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, American Mathematical Society, Providence, 1973. 
[18] 
M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, SpringerVerlag, New York, 1993. 
[19] 
E. G. Tabak and R. R. Rosales, Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection, Physics of Fluids A, 6 (1994), 18741892. doi: 10.1063/1.868246. 
[20] 
A. M. Tesdall and J. K. Hunter, Selfsimilar solutions for weak shock reflection, SIAM Journal on Applied Mathematics, 63 (2002), 4261. doi: 10.1137/S0036139901383826. 
[21] 
A. M. Tesdall and B. L. Keyfitz, A continuous, twoway free boundary in the unsteady transonic small disturbance equations, Journal of Hyperbolic Differential Equations, 7 (2010), 317338. doi: 10.1142/S0219891610002153. 
[22] 
A. M. Tesdall, R. Sanders and B. L. Keyfitz, Selfsimilar solutions for the triple point paradox in gasdynamics, SIAM Journal on Applied Mathematics, 68 (2008), 13601377. doi: 10.1137/070698567. 
show all references
References:
[1] 
M. Brio and J. K. Hunter, Mach reflection for the two dimensional Burgers equation, Physica D, 60 (1992), 194207. doi: 10.1016/01672789(92)90236G. 
[2] 
S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, Journal of Differential Equations, 125 (1996), 548574. doi: 10.1006/jdeq.1996.0040. 
[3] 
S. Čanić and B. L. Keyfitz, A smooth solution for a Keldysh type equation, Communications in Partial Differential Equations, 21 (1996), 319340. doi: 10.1080/03605309608821186. 
[4] 
S. Čanić, B. L. Keyfitz and E. H. Kim, A free boundary problem for a quasilinear degenerate elliptic equation: Regular reflection of weak shocks, Communications on Pure and Applied Mathematics, LV (2002), 7192. 
[5] 
S. Čanić and D. Mirković, A numerical study of shock reflection modeled by the unsteady transonic small disturbance equation, SIAM Journal on Applied Mathematics, 58 (1998), 13651393. doi: 10.1137/S003613999730884X. 
[6] 
G.Q. Chen and M. Feldman, Global solutions of shock reflection by largeangle wedges for potential flow, Annals of Mathematics, 171 (2010), 10671182. doi: 10.4007/annals.2010.171.1067. 
[7] 
S.X. Chen, Mach configuration in pseudostationary compressible flow, Journal of the American Mathematical Society, 21 (2008), 63100. doi: 10.1090/S0894034707005590. 
[8] 
V. Elling and T.P. Liu, Supersonic flow onto a solid wedge}, Communications on Pure and Applied Mathematics, 61 (2008), 13471448. doi: 10.1002/cpa.20231. 
[9] 
G. Fichera, Sulle equazioni differenziali lineari ellitticoparaboliche del secondo ordine, Atti della Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat. Ser. I, 5 (1956), 130. 
[10] 
G. Fichera, On a unified theory of boundary value problems for elliptic parabolic equations of second order, In R. E. Langer, editor, Boundary Problems in Differential Equations, pages 97120. University of Wisconsin, Madison, 1960. 
[11] 
J. K. Hunter and A. M. Tesdall, On the selfsimilar diffraction of a weak shock into an expansion wavefront, SIAM Journal on Applied Mathematics, 72 (2012), 124143. doi: 10.1137/110834135. 
[12] 
I. Jegdić and K. Jegdić, Properties of solutions in semihyperbolic patches for the unsteady transonic small disturbance equation,, Electronic Journal of Differential Equations, (). 
[13] 
K. Jegdić, B. L. Keyfitz and S. Čanić, Transonic regular reflection for the unsteady transonic small disturbance equation  details of the subsonic solution, In R. Glowinski and J. P. Zolesio, editors, Free and Moving Boundaries: Analysis, Simulation and Control, pages 125163. CRC Press, Boca Raton, 2007. doi: 10.1201/9781420011159.ch6. 
[14] 
B. L. Keyfitz, Selfsimilar solutions of twodimensional conservation laws, Journal of Hyperbolic Differential Equations, 1 (2004), 445492. doi: 10.1142/S0219891604000160. 
[15] 
B. L. Keyfitz, The Fichera function and nonlinear equations, Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I, 30 (2006), 8394. 
[16] 
B. L. Keyfitz, A. Tesdall, K. R. Payne and N. I. Popivanov, The sonic line as a free boundary, Quarterly of Applied Mathematics, 71 (2013), 119133. doi: 10.1090/S0033569X2012012836. 
[17] 
O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, American Mathematical Society, Providence, 1973. 
[18] 
M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, SpringerVerlag, New York, 1993. 
[19] 
E. G. Tabak and R. R. Rosales, Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection, Physics of Fluids A, 6 (1994), 18741892. doi: 10.1063/1.868246. 
[20] 
A. M. Tesdall and J. K. Hunter, Selfsimilar solutions for weak shock reflection, SIAM Journal on Applied Mathematics, 63 (2002), 4261. doi: 10.1137/S0036139901383826. 
[21] 
A. M. Tesdall and B. L. Keyfitz, A continuous, twoway free boundary in the unsteady transonic small disturbance equations, Journal of Hyperbolic Differential Equations, 7 (2010), 317338. doi: 10.1142/S0219891610002153. 
[22] 
A. M. Tesdall, R. Sanders and B. L. Keyfitz, Selfsimilar solutions for the triple point paradox in gasdynamics, SIAM Journal on Applied Mathematics, 68 (2008), 13601377. doi: 10.1137/070698567. 
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