# American Institute of Mathematical Sciences

August  2014, 34(8): 3193-3210. doi: 10.3934/dcds.2014.34.3193

## Energy-minimising parallel flows with prescribed vorticity distribution

 1 St John's College, Cambridge, CB2 1TP, United Kingdom

Received  April 2013 Published  January 2014

This note concerns a nonlinear differential equation problem in which both the nonlinearity in the equation and its solution are determined by other constraints. The question under consideration arises from a study of two-dimensional steady parallel-flows of a perfect fluid governed by Euler's equations and a free-boundary condition, when the distribution of vorticity is arbitrary but prescribed.
Citation: J. F. Toland. Energy-minimising parallel flows with prescribed vorticity distribution. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3193-3210. doi: 10.3934/dcds.2014.34.3193
##### References:
 [1] P. Baldi and J. F. Toland, Steady periodic water waves under nonlinear elastic membranes, J. Reine Angew. Math., 652 (2011), 67-112. doi: 10.1515/CRELLE.2011.015. [2] T. B. Benjamin, The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics, in Applications of Methods of Functional Analysis to Problems in Mechanics, (Joint Sympos., IUTAM/IMU, Marseille, 1975), pp. 8-29. Lecture Notes in Math., 503. Springer, Berlin, 1976. [3] B. Buffoni and G. R. Burton, On the stability of travelling waves with vorticity obtained by minimisation, To appear in Nonlinear Differential Equations Appl., http://arxiv.org/abs/1207.7198. doi: 10.1007/s00030-013-0223-4. [4] G. R. Burton, Global nonlinear stability for steady ideal fluid flow in bounded planar domains, Arch. Ration. Mech. Anal., 176 (2005), 149-163. doi: 10.1007/s00205-004-0339-0. [5] G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., LXIV (2011), 975-1007. doi: 10.1002/cpa.20365. [6] A. J. Chorin and J. E. Marsden, An Introduction to Mathematical Fluid Mechanics, Springer, New York, 1993. [7] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., LVII (2004), 481-527. doi: 10.1002/cpa.3046. [8] A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., LX (2007), 911-950. doi: 10.1002/cpa.20165. [9] M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291. [10] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd. Edition, Cambridge University Press, Cambridge, 1954. doi: 10.1037/e642452011-001. [11] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, R. I. 1997. [12] V. Kozlov and N. Kuznetsov, Steady free-surface vortical flows parallel to the horizontal bottom, Quart. J. Mech. Appl. Math., 64 (2011), 371-399. doi: 10.1093/qjmam/hbr010. [13] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-viscous Fluids, Applied Mathematical Sciences 96, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0. [14] W. A. Strauss, Steady water waves, Bull. Am. Math. Soc., New Ser., 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1. [15] J. F. Toland, Steady periodic hydroelastic waves, Arch. Rational Mech. Anal., 189 (2008), 325-362. doi: 10.1007/s00205-007-0104-2. [16] J. F. Toland, Non-existence of global minimisers of energy in Stokes-wave problems, to appear in Discrete Continuous Dynam. Systems - A.

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##### References:
 [1] P. Baldi and J. F. Toland, Steady periodic water waves under nonlinear elastic membranes, J. Reine Angew. Math., 652 (2011), 67-112. doi: 10.1515/CRELLE.2011.015. [2] T. B. Benjamin, The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics, in Applications of Methods of Functional Analysis to Problems in Mechanics, (Joint Sympos., IUTAM/IMU, Marseille, 1975), pp. 8-29. Lecture Notes in Math., 503. Springer, Berlin, 1976. [3] B. Buffoni and G. R. Burton, On the stability of travelling waves with vorticity obtained by minimisation, To appear in Nonlinear Differential Equations Appl., http://arxiv.org/abs/1207.7198. doi: 10.1007/s00030-013-0223-4. [4] G. R. Burton, Global nonlinear stability for steady ideal fluid flow in bounded planar domains, Arch. Ration. Mech. Anal., 176 (2005), 149-163. doi: 10.1007/s00205-004-0339-0. [5] G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., LXIV (2011), 975-1007. doi: 10.1002/cpa.20365. [6] A. J. Chorin and J. E. Marsden, An Introduction to Mathematical Fluid Mechanics, Springer, New York, 1993. [7] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., LVII (2004), 481-527. doi: 10.1002/cpa.3046. [8] A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., LX (2007), 911-950. doi: 10.1002/cpa.20165. [9] M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291. [10] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd. Edition, Cambridge University Press, Cambridge, 1954. doi: 10.1037/e642452011-001. [11] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, R. I. 1997. [12] V. Kozlov and N. Kuznetsov, Steady free-surface vortical flows parallel to the horizontal bottom, Quart. J. Mech. Appl. Math., 64 (2011), 371-399. doi: 10.1093/qjmam/hbr010. [13] C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-viscous Fluids, Applied Mathematical Sciences 96, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0. [14] W. A. Strauss, Steady water waves, Bull. Am. Math. Soc., New Ser., 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1. [15] J. F. Toland, Steady periodic hydroelastic waves, Arch. Rational Mech. Anal., 189 (2008), 325-362. doi: 10.1007/s00205-007-0104-2. [16] J. F. Toland, Non-existence of global minimisers of energy in Stokes-wave problems, to appear in Discrete Continuous Dynam. Systems - A.
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