Incompressible and ideal 2D flow around a small obstacle with constant velocity at infinity
Authors:
Milton C. Lopes Filho, Huy Hoang Nguyen and Helena J. Nussenzveig Lopes
Journal:
Quart. Appl. Math. 71 (2013), 679-687
MSC (2010):
Primary 35Q31; Secondary 35Q35, 76B03, 76B47
DOI:
https://doi.org/10.1090/S0033-569X-2013-01299-4
Published electronically:
August 28, 2013
MathSciNet review:
3136990
Full-text PDF Free Access
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Additional Information
Abstract: This article is concerned with the limiting behavior of incompressible flow past a small obstacle. Previous work on this problem has dealt with flows with vanishing velocity at infinity. We examine this limit for flows that are constant at infinity in the simplest case, that of two-dimensional, ideal flow past an obstacle. This extends the work in Iftimie, Lopes Filho, and Lopes (2003).
References
- D. J. Acheson, Elementary fluid dynamics, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1990. MR 1069557
- C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40 (1972), 769–790 (French). MR 333488, DOI https://doi.org/10.1016/0022-247X%2872%2990019-4
- Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR 1218879
- Masoumeh Dashti and James C. Robinson, The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 285–312. MR 2781594, DOI https://doi.org/10.1007/s00205-011-0401-7
- Keisuke Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 1, 63–92. MR 700596
- Dragoş Iftimie and James P. Kelliher, Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid, Proc. Amer. Math. Soc. 137 (2009), no. 2, 685–694. MR 2448591, DOI https://doi.org/10.1090/S0002-9939-08-09670-6
- D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Differential Equations 28 (2003), no. 1-2, 349–379. MR 1974460, DOI https://doi.org/10.1081/PDE-120019386
- D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Two-dimensional incompressible viscous flow around a small obstacle, Math. Ann. 336 (2006), no. 2, 449–489. MR 2244381, DOI https://doi.org/10.1007/s00208-006-0012-z
- Dragoş Iftimie, Milton C. Lopes Filho, and Helena J. Nussenzveig Lopes, Incompressible flow around a small obstacle and the vanishing viscosity limit, Comm. Math. Phys. 287 (2009), no. 1, 99–115. MR 2480743, DOI https://doi.org/10.1007/s00220-008-0621-3
- Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200. MR 211057, DOI https://doi.org/10.1007/BF00251588
- Christophe Lacave, Two dimensional incompressible ideal flow around a thin obstacle tending to a curve, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, 1121–1148. MR 2542717, DOI https://doi.org/10.1016/j.anihpc.2008.06.004
- Christophe Lacave, Two-dimensional incompressible viscous flow around a thin obstacle tending to a curve, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 6, 1237–1254. MR 2557320, DOI https://doi.org/10.1017/S0308210508000632
- Christophe Lacave and Evelyne Miot, Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex, SIAM J. Math. Anal. 41 (2009), no. 3, 1138–1163. MR 2529959, DOI https://doi.org/10.1137/080737629
- M. C. Lopes Filho, Vortex dynamics in a two-dimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal. 39 (2007), no. 2, 422–436. MR 2338413, DOI https://doi.org/10.1137/050647967
- Philippe Serfati, Solutions $C^\infty $ en temps, $n$-$\log $ Lipschitz bornées en espace et équation d’Euler, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 5, 555–558 (French, with English and French summaries). MR 1322336
- V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 158189
References
- D. J. Acheson, Elementary fluid dynamics, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press Oxford University Press, New York, 1990. MR 1069557 (93k:76001)
- C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40 (1972), 769–790 (French). MR 0333488 (48 \#11813)
- Alexandre J. Chorin and Jerrold E. Marsden, A mathematical introduction to fluid mechanics, 3rd ed., Texts in Applied Mathematics, vol. 4, Springer-Verlag, New York, 1993. MR 1218879 (94c:76002)
- Masoumeh Dashti and James C. Robinson, The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 285–312. MR 2781594 (2012c:76027), DOI https://doi.org/10.1007/s00205-011-0401-7
- Keisuke Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 1, 63–92. MR 700596 (84g:35151)
- Dragoş Iftimie and James P. Kelliher, Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid, Proc. Amer. Math. Soc. 137 (2009), no. 2, 685–694. MR 2448591 (2009m:35385), DOI https://doi.org/10.1090/S0002-9939-08-09670-6
- D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Differential Equations 28 (2003), no. 1-2, 349–379. MR 1974460 (2004d:76009), DOI https://doi.org/10.1081/PDE-120019386
- D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Two-dimensional incompressible viscous flow around a small obstacle, Math. Ann. 336 (2006), no. 2, 449–489. MR 2244381 (2007d:76050), DOI https://doi.org/10.1007/s00208-006-0012-z
- Dragoş Iftimie, Milton C. Lopes Filho, and Helena J. Nussenzveig Lopes, Incompressible flow around a small obstacle and the vanishing viscosity limit, Comm. Math. Phys. 287 (2009), no. 1, 99–115. MR 2480743 (2009m:35362), DOI https://doi.org/10.1007/s00220-008-0621-3
- Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200. MR 0211057 (35 \#1939)
- Christophe Lacave, Two dimensional incompressible ideal flow around a thin obstacle tending to a curve, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, 1121–1148. MR 2542717 (2010h:76054), DOI https://doi.org/10.1016/j.anihpc.2008.06.004
- Christophe Lacave, Two-dimensional incompressible viscous flow around a thin obstacle tending to a curve, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 6, 1237–1254. MR 2557320 (2010k:76040), DOI https://doi.org/10.1017/S0308210508000632
- Christophe Lacave and Evelyne Miot, Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex, SIAM J. Math. Anal. 41 (2009), no. 3, 1138–1163. MR 2529959 (2010i:76030), DOI https://doi.org/10.1137/080737629
- M. C. Lopes Filho, Vortex dynamics in a two-dimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal. 39 (2007), no. 2, 422–436 (electronic). MR 2338413 (2008i:76013), DOI https://doi.org/10.1137/050647967
- Philippe Serfati, Solutions $C^\infty$ en temps, $n$-$\log$ Lipschitz bornées en espace et équation d’Euler, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 5, 555–558 (French, with English and French summaries). MR 1322336 (96c:35147)
- V. I. Judovič, Non-stationary flows of an ideal incompressible fluid, Z̆. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 1032–1066 (Russian). MR 0158189 (28 \#1415)
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Additional Information
Milton C. Lopes Filho
Affiliation:
Departamento de Matematica, IMECC, Universidade Estadual de Campinas-UNICAMP, Campinas, SP 13083-970, Brazil
Email:
mlopes@ime.unicamp.br
Huy Hoang Nguyen
Affiliation:
Polo de Xerem e Instituto de Matematica, Universidade Federal do Rio de Janeiro, Brazil
Email:
nguyen@im.ufrj.br
Helena J. Nussenzveig Lopes
Affiliation:
Departamento de Matematica, IMECC, Universidade Estadual de Campinas-UNICAMP, Campinas, SP 13083-970, Brazil
Email:
hlopes@ime.unicamp.br
Received by editor(s):
November 14, 2011
Published electronically:
August 28, 2013
Article copyright:
© Copyright 2013
Brown University
The copyright for this article reverts to public domain 28 years after publication.
