The role of initial curvature in solutions to the generalized inviscid Proudman-Johnson equation
Authors:
Alejandro Sarria and Ralph Saxton
Journal:
Quart. Appl. Math. 73 (2015), 55-91
MSC (2010):
Primary 35B44, 35B10, 35B65, 35Q35
DOI:
https://doi.org/10.1090/S0033-569X-2015-01378-3
Published electronically:
January 21, 2015
MathSciNet review:
3322726
Full-text PDF Free Access
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Additional Information
Abstract: In Sarria and Saxton (2013), we derived representation formulae for spatially periodic solutions to the generalized, inviscid Proudman-Johnson equation and studied their regularity for several classes of initial data. The purpose of this paper is to extend these results to larger classes of functions including those having arbitrary local curvature near particular points in the domain.
References
- E. W. Barnes, A New Development of the Theory of the Hypergeometric Functions, Proc. London Math. Soc. (2) 6 (1908), 141–177. MR 1575118, DOI https://doi.org/10.1112/plms/s2-6.1.141
- Alberto Bressan and Adrian Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), no. 3, 996–1026. MR 2191785, DOI https://doi.org/10.1137/050623036
- F. Calogero, A solvable nonlinear wave equation, Stud. Appl. Math. 70 (1984), no. 3, 189–199. MR 742587, DOI https://doi.org/10.1002/sapm1984703189
- S. Childress, G. R. Ierley, E. A. Spiegel, and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech. 203 (1989), 1–22. MR 1002875, DOI https://doi.org/10.1017/S0022112089001357
- Chien-Hong Cho and Marcus Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation, J. Differential Equations 249 (2010), no. 2, 392–413. MR 2644121, DOI https://doi.org/10.1016/j.jde.2010.03.013
- Chien-Hong Cho and Marcus Wunsch, Global weak solutions to the generalized Proudman-Johnson equation, Commun. Pure Appl. Anal. 11 (2012), no. 4, 1387–1396. MR 2900790, DOI https://doi.org/10.3934/cpaa.2012.11.1387
- Adrian Constantin and Marcus Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 7, 81–83. MR 2548017, DOI https://doi.org/10.3792/pjaa.85.81
- Constantine M. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Differ. Equ. 8 (2011), no. 1, 159–168. MR 2796054, DOI https://doi.org/10.1142/S0219891611002366
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779
- Joachim Escher and Marcus Wunsch, Restrictions on the geometry of the periodic vorticity equation, Commun. Contemp. Math. 14 (2012), no. 3, 1250016, 13. MR 2943829, DOI https://doi.org/10.1142/S0219199712500162
- Theodore W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. MR 1830078
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719
- John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498–1521. MR 1135995, DOI https://doi.org/10.1137/0151075
- John K. Hunter and Yu Xi Zheng, On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions, Arch. Rational Mech. Anal. 129 (1995), no. 4, 305–353. MR 1361013, DOI https://doi.org/10.1007/BF00379259
- Boris Khesin and Gerard Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003), no. 1, 116–144. MR 1978343, DOI https://doi.org/10.1016/S0001-8708%2802%2900063-4
- Jonatan Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 643–656. MR 2318260, DOI https://doi.org/10.3934/dcds.2007.18.643
- Hisashi Okamoto and Jinghui Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), 2000, pp. 65–103. MR 1757984, DOI https://doi.org/10.11650/twjm/1500407199
- Hisashi Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech. 11 (2009), no. 1, 46–59. MR 2481858, DOI https://doi.org/10.1007/s00021-007-0247-9
- Ian Proudman and Kathleen Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech. 12 (1962), 161–168. MR 136211, DOI https://doi.org/10.1017/S0022112062000130
- Alejandro Sarria and Ralph Saxton, Blow-up of solutions to the generalized inviscid Proudman-Johnson equation, J. Math. Fluid Mech. 15 (2013), no. 3, 493–523. MR 3084321, DOI https://doi.org/10.1007/s00021-012-0126-x
- A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations, Differential and Integral Equations (in press).
- Ralph Saxton and Feride Tiğlay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal. 40 (2008), no. 4, 1499–1515. MR 2466165, DOI https://doi.org/10.1137/080713768
- Feride Tiğlay, The periodic Cauchy problem of the modified Hunter-Saxton equation, J. Evol. Equ. 5 (2005), no. 4, 509–527. MR 2201524, DOI https://doi.org/10.1007/s00028-005-0215-x
- Hermann Weyl, On the differential equations of the simplest boundary-layer problems, Ann. of Math. (2) 43 (1942), 381–407. MR 6294, DOI https://doi.org/10.2307/1968875
- Marcus Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech. 13 (2011), no. 1, 147–154. MR 2784901, DOI https://doi.org/10.1007/s00021-009-0004-3
- Zhaoyang Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal. 36 (2004), no. 1, 272–283. MR 2083862, DOI https://doi.org/10.1137/S0036141003425672
References
- E. W. Barnes, A New Development of the Theory of the Hypergeometric Functions, Proc. London Math. Soc. S2-6, no. 1, 141. MR 1575118, DOI https://doi.org/10.1112/plms/s2-6.1.141
- Alberto Bressan and Adrian Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), no. 3, 996–1026. MR 2191785 (2006j:35203), DOI https://doi.org/10.1137/050623036
- F. Calogero, A solvable nonlinear wave equation, Stud. Appl. Math. 70 (1984), no. 3, 189–199. MR 742587 (85f:35170)
- S. Childress, G. R. Ierley, E. A. Spiegel, and W. R. Young, Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form, J. Fluid Mech. 203 (1989), 1–22. MR 1002875 (90e:76054), DOI https://doi.org/10.1017/S0022112089001357
- Chien-Hong Cho and Marcus Wunsch, Global and singular solutions to the generalized Proudman-Johnson equation, J. Differential Equations 249 (2010), no. 2, 392–413. MR 2644121 (2011e:35392), DOI https://doi.org/10.1016/j.jde.2010.03.013
- Chien-Hong Cho and Marcus Wunsch, Global weak solutions to the generalized Proudman-Johnson equation, Commun. Pure Appl. Anal. 11 (2012), no. 4, 1387–1396. MR 2900790, DOI https://doi.org/10.3934/cpaa.2012.11.1387
- Adrian Constantin and Marcus Wunsch, On the inviscid Proudman-Johnson equation, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 7, 81–83. MR 2548017 (2010i:35318), DOI https://doi.org/10.3792/pjaa.85.81
- Constantine M. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Differ. Equ. 8 (2011), no. 1, 159–168. MR 2796054 (2012f:35460), DOI https://doi.org/10.1142/S0219891611002366
- A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vol. I, McGraw-Hill, 1981, 56-119. MR 0698779 (84h:33001a)
- Joachim Escher and Marcus Wunsch, Restrictions on the geometry of the periodic vorticity equation, Commun. Contemp. Math. 14 (2012), no. 3, 1250016, 13. MR 2943829, DOI https://doi.org/10.1142/S0219199712500162
- Theodore W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. MR 1830078 (2002h:30001)
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., with a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. MR 2128719 (2006d:33028)
- John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498–1521. MR 1135995 (93a:76005), DOI https://doi.org/10.1137/0151075
- John K. Hunter and Yu Xi Zheng, On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions, Arch. Rational Mech. Anal. 129 (1995), no. 4, 305–353. MR 1361013 (96m:35215), DOI https://doi.org/10.1007/BF00379259
- Boris Khesin and Gerard Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003), no. 1, 116–144. MR 1978343 (2004h:37125), DOI https://doi.org/10.1016/S0001-8708%2802%2900063-4
- Jonatan Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 643–656. MR 2318260 (2008c:37120), DOI https://doi.org/10.3934/dcds.2007.18.643
- Hisashi Okamoto and Jinghui Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), Taiwanese J. Math. 4 (2000), no. 1, 65–103. MR 1757984 (2001i:76033)
- Hisashi Okamoto, Well-posedness of the generalized Proudman-Johnson equation without viscosity, J. Math. Fluid Mech. 11 (2009), no. 1, 46–59. MR 2481858 (2010b:35409), DOI https://doi.org/10.1007/s00021-007-0247-9
- Ian Proudman and Kathleen Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech. 12 (1962), 161–168. MR 0136211 (24 \#B2249)
- Alejandro Sarria and Ralph Saxton, Blow-up of solutions to the generalized inviscid Proudman–Johnson equation, J. Math. Fluid Mech. 15 (2013), no. 3, 493–523. MR 3084321, DOI https://doi.org/10.1007/s00021-012-0126-x
- A. Sarria, Regularity of stagnation point-form solutions to the two-dimensional Euler equations, Differential and Integral Equations (in press).
- Ralph Saxton and Feride Tiğlay, Global existence of some infinite energy solutions for a perfect incompressible fluid, SIAM J. Math. Anal. 40 (2008), no. 4, 1499–1515. MR 2466165 (2010a:35202), DOI https://doi.org/10.1137/080713768
- Feride Tiğlay, The periodic Cauchy problem of the modified Hunter-Saxton equation, J. Evol. Equ. 5 (2005), no. 4, 509–527. MR 2201524 (2006i:35325), DOI https://doi.org/10.1007/s00028-005-0215-x
- Hermann Weyl, On the differential equations of the simplest boundary-layer problems, Ann. of Math. (2) 43 (1942), 381–407. MR 0006294 (3,284d)
- Marcus Wunsch, The generalized Proudman-Johnson equation revisited, J. Math. Fluid Mech. 13 (2011), no. 1, 147–154. MR 2784901 (2012d:35299), DOI https://doi.org/10.1007/s00021-009-0004-3
- Zhaoyang Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal. 36 (2004), no. 1, 272–283 (electronic). MR 2083862 (2005e:35034), DOI https://doi.org/10.1137/S0036141003425672
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Additional Information
Alejandro Sarria
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309
Email:
alejandro.sarria@colorado.edu
Ralph Saxton
Affiliation:
Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148
Email:
rsaxton@uno.edu
Received by editor(s):
December 21, 2012
Received by editor(s) in revised form:
August 9, 2013
Published electronically:
January 21, 2015
Article copyright:
© Copyright 2015
Brown University