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August  2016, 36(8): 4213-4225. doi: 10.3934/dcds.2016.36.4213

The unsteady transonic small disturbance equation: Data on oblique curves

Mary Chern 1, and  Barbara Lee Keyfitz 2,

1. 

Fields Institute, United States
2. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States

Received  May 2015 Revised  November 2015 Published  March 2016

We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.
Keywords: self-similar problems, Degerate elliptic equations, multidimensional hyperbolic systems, Browder-Minty theorem., conservation laws.
Mathematics Subject Classification: Primary: 35J70, 35L65, 35M10; Secondary: 47H05, 47H1.
Citation: Mary Chern, Barbara Lee Keyfitz. The unsteady transonic small disturbance equation: Data on oblique curves. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4213-4225. doi: 10.3934/dcds.2016.36.4213
References:
[1]

M. Brio and J. K. Hunter, Mach reflection for the two dimensional Burgers equation, Physica D, 60 (1992), 194-207. doi: 10.1016/0167-2789(92)90236-G.

[2]

S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, Journal of Differential Equations, 125 (1996), 548-574. 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), 319-340. 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), 71-92.

[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), 1365-1393. doi: 10.1137/S003613999730884X.

[6]

G.-Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Annals of Mathematics, 171 (2010), 1067-1182. doi: 10.4007/annals.2010.171.1067.

[7]

S.-X. Chen, Mach configuration in pseudo-stationary compressible flow, Journal of the American Mathematical Society, 21 (2008), 63-100. doi: 10.1090/S0894-0347-07-00559-0.

[8]

V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge}, Communications on Pure and Applied Mathematics, 61 (2008), 1347-1448. doi: 10.1002/cpa.20231.

[9]

G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti della Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat. Ser. I, 5 (1956), 1-30.

[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 97-120. University of Wisconsin, Madison, 1960.

[11]

J. K. Hunter and A. M. Tesdall, On the self-similar diffraction of a weak shock into an expansion wavefront, SIAM Journal on Applied Mathematics, 72 (2012), 124-143. doi: 10.1137/110834135.

[12]

I. Jegdić and K. Jegdić, Properties of solutions in semi-hyperbolic 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 125-163. CRC Press, Boca Raton, 2007. doi: 10.1201/9781420011159.ch6.

[14]

B. L. Keyfitz, Self-similar solutions of two-dimensional conservation laws, Journal of Hyperbolic Differential Equations, 1 (2004), 445-492. 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), 83-94.

[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), 119-133. doi: 10.1090/S0033-569X-2012-01283-6.

[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, Springer-Verlag, 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), 1874-1892. doi: 10.1063/1.868246.

[20]

A. M. Tesdall and J. K. Hunter, Self-similar solutions for weak shock reflection, SIAM Journal on Applied Mathematics, 63 (2002), 42-61. doi: 10.1137/S0036139901383826.

[21]

A. M. Tesdall and B. L. Keyfitz, A continuous, two-way free boundary in the unsteady transonic small disturbance equations, Journal of Hyperbolic Differential Equations, 7 (2010), 317-338. doi: 10.1142/S0219891610002153.

[22]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM Journal on Applied Mathematics, 68 (2008), 1360-1377. 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), 194-207. doi: 10.1016/0167-2789(92)90236-G.

[2]

S. Čanić and B. L. Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, Journal of Differential Equations, 125 (1996), 548-574. 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), 319-340. 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), 71-92.

[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), 1365-1393. doi: 10.1137/S003613999730884X.

[6]

G.-Q. Chen and M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow, Annals of Mathematics, 171 (2010), 1067-1182. doi: 10.4007/annals.2010.171.1067.

[7]

S.-X. Chen, Mach configuration in pseudo-stationary compressible flow, Journal of the American Mathematical Society, 21 (2008), 63-100. doi: 10.1090/S0894-0347-07-00559-0.

[8]

V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge}, Communications on Pure and Applied Mathematics, 61 (2008), 1347-1448. doi: 10.1002/cpa.20231.

[9]

G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti della Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat. Ser. I, 5 (1956), 1-30.

[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 97-120. University of Wisconsin, Madison, 1960.

[11]

J. K. Hunter and A. M. Tesdall, On the self-similar diffraction of a weak shock into an expansion wavefront, SIAM Journal on Applied Mathematics, 72 (2012), 124-143. doi: 10.1137/110834135.

[12]

I. Jegdić and K. Jegdić, Properties of solutions in semi-hyperbolic 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 125-163. CRC Press, Boca Raton, 2007. doi: 10.1201/9781420011159.ch6.

[14]

B. L. Keyfitz, Self-similar solutions of two-dimensional conservation laws, Journal of Hyperbolic Differential Equations, 1 (2004), 445-492. 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), 83-94.

[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), 119-133. doi: 10.1090/S0033-569X-2012-01283-6.

[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, Springer-Verlag, 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), 1874-1892. doi: 10.1063/1.868246.

[20]

A. M. Tesdall and J. K. Hunter, Self-similar solutions for weak shock reflection, SIAM Journal on Applied Mathematics, 63 (2002), 42-61. doi: 10.1137/S0036139901383826.

[21]

A. M. Tesdall and B. L. Keyfitz, A continuous, two-way free boundary in the unsteady transonic small disturbance equations, Journal of Hyperbolic Differential Equations, 7 (2010), 317-338. doi: 10.1142/S0219891610002153.

[22]

A. M. Tesdall, R. Sanders and B. L. Keyfitz, Self-similar solutions for the triple point paradox in gasdynamics, SIAM Journal on Applied Mathematics, 68 (2008), 1360-1377. doi: 10.1137/070698567.

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