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 & Continuous Dynamical Systems - A, 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. 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, (1975), 8.

[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., (). 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. 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. doi: 10.1002/cpa.20365.

[6]

A. J. Chorin and J. E. Marsden, An Introduction to Mathematical Fluid Mechanics,, Springer, (1993).

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., LVII (2004), 481. 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. 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.

[10]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, 2nd. Edition, (1954). doi: 10.1037/e642452011-001.

[11]

E. H. Lieb and M. Loss, Analysis,, Graduate Studies in Mathematics, (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. doi: 10.1093/qjmam/hbr010.

[13]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-viscous Fluids,, Applied Mathematical Sciences 96, (1994). doi: 10.1007/978-1-4612-4284-0.

[14]

W. A. Strauss, Steady water waves,, Bull. Am. Math. Soc., 47 (2010), 671. doi: 10.1090/S0273-0979-2010-01302-1.

[15]

J. F. Toland, Steady periodic hydroelastic waves,, Arch. Rational Mech. Anal., 189 (2008), 325. 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., ().

show all references

References:
[1]

P. Baldi and J. F. Toland, Steady periodic water waves under nonlinear elastic membranes,, J. Reine Angew. Math., 652 (2011), 67. 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, (1975), 8.

[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., (). 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. 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. doi: 10.1002/cpa.20365.

[6]

A. J. Chorin and J. E. Marsden, An Introduction to Mathematical Fluid Mechanics,, Springer, (1993).

[7]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., LVII (2004), 481. 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. 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.

[10]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, 2nd. Edition, (1954). doi: 10.1037/e642452011-001.

[11]

E. H. Lieb and M. Loss, Analysis,, Graduate Studies in Mathematics, (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. doi: 10.1093/qjmam/hbr010.

[13]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-viscous Fluids,, Applied Mathematical Sciences 96, (1994). doi: 10.1007/978-1-4612-4284-0.

[14]

W. A. Strauss, Steady water waves,, Bull. Am. Math. Soc., 47 (2010), 671. doi: 10.1090/S0273-0979-2010-01302-1.

[15]

J. F. Toland, Steady periodic hydroelastic waves,, Arch. Rational Mech. Anal., 189 (2008), 325. 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., ().

[1]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

[2]

Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101

[3]

Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191

[4]

Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483

[5]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

[6]

Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475

[7]

Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114

[8]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045

[9]

Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225

[10]

Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523

[11]

Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016

[12]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[13]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103

[14]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036

[15]

Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure & Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415

[16]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907

[17]

Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial & Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041

[18]

Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465

[19]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

[20]

Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]