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Subsonic irrotational inviscid flow around certain bodies with two protruding corners
On the existence and computation of periodic travelling waves for a 2D water wave model
Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, Colombia |
In this work we establish the existence of at least one weak periodic solution in the spatial directions of a nonlinear system of two coupled differential equations associated with a 2D Boussinesq model which describes the evolution of long water waves with small amplitude under the effect of surface tension. For wave speed $0 < |c| < 1$, the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space $H^{1}_k(\mathbb{R})$ ($k$-periodic functions $f∈ L_k^2(\mathbb{R})$ such that $f' ∈ L_k^2(\mathbb{R})$). For wave speed $|c|>1$, the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a $4× 4$ system with a special Hamiltonian structure. In the case $|c|>1$, we also compute numerical approximations of these travelling waves by using a Fourier spectral discretization of the corresponding 1D travelling wave equations and a Newton-type iteration.
References:
[1] |
U. M. Asher, S. J. Ruuth and B. T. R. Wetton,
Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.
|
[2] |
C. Canuto, M. Y. Hussaini and A. Quarteroni,
Spectral Methods in Fluid Dynamics Series in Computational Physics, 1988, Springer, Berlin. |
[3] |
G. E. Karniadakis, M. Israeli and S. A. Orszag,
High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443.
|
[4] |
T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Proceedings of the symposium at Dundee, Lecture Notes in Mathematics, 448, Springer, (1975), 25-70. |
[5] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Mathematica, 28 (1979), 89-99.
|
[6] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, Advances in Mathematics, Supplementary Studies, 8, Academic Press, (1983), 92-128. |
[7] |
J. Kim and P. Moin,
Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.
|
[8] |
K. R. Meyer, G. R. Hall and D. Offin,
Introduction to Hamiltonian Dynamical Systems and the N-Body problem 2nd ed. Applied Mathematical Sciences, vol. 90,2009, Springer-Verlag. |
[9] |
P. A. Milewski and J. B. Keller,
Three dimensional water waves, Studies Appl. Math., 37 (1996), 149-166.
|
[10] |
L. Paumond,
A rigorous link between KP and a Benney-Luke Equation, Diff. Int. Eq., 16 (2003), 1039-1064.
|
[11] |
J. Quintero,
Solitary water waves for a 2D Boussinesq type system, J. Part. Diff. Eqs., 23 (2010), 251-280.
|
[12] |
J. Quintero,
The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system, Diff. Int. Eqs., 21 (2011), 325-360.
|
[13] |
J. Quintero,
From periodic travelling waves to solitons of a 2D water wave system, Meth. Appl. Anal., 21 (2014), 241-264.
|
[14] |
J. Quintero,
A water wave mixed type problem: existence of periodic travelling waves for a 2D Boussinesq system, Rev. Academia Colombiana de Ciencias Naturales, Físicas y Exactas., 38 (2015), 6-17.
|
[15] |
J. R. Quintero and R. L. Pego,
Two-dimensional solitary waves for a Benney-Luke equation, Physica D., 45 (1999), 476-496.
|
[16] |
J. G. Verwer, J. G. Blom and W. Hundsdorfer,
An implicit-explicit approach for atmospheric transport-chemistry problems, Applied Numerical Mathematics, 20 (1996), 191-209.
|
[17] |
G. B. Whitham,
Linear and Nonlinear Waves Wiley-Interscience, 1974. |
show all references
References:
[1] |
U. M. Asher, S. J. Ruuth and B. T. R. Wetton,
Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.
|
[2] |
C. Canuto, M. Y. Hussaini and A. Quarteroni,
Spectral Methods in Fluid Dynamics Series in Computational Physics, 1988, Springer, Berlin. |
[3] |
G. E. Karniadakis, M. Israeli and S. A. Orszag,
High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414-443.
|
[4] |
T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Proceedings of the symposium at Dundee, Lecture Notes in Mathematics, 448, Springer, (1975), 25-70. |
[5] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Mathematica, 28 (1979), 89-99.
|
[6] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, Advances in Mathematics, Supplementary Studies, 8, Academic Press, (1983), 92-128. |
[7] |
J. Kim and P. Moin,
Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323.
|
[8] |
K. R. Meyer, G. R. Hall and D. Offin,
Introduction to Hamiltonian Dynamical Systems and the N-Body problem 2nd ed. Applied Mathematical Sciences, vol. 90,2009, Springer-Verlag. |
[9] |
P. A. Milewski and J. B. Keller,
Three dimensional water waves, Studies Appl. Math., 37 (1996), 149-166.
|
[10] |
L. Paumond,
A rigorous link between KP and a Benney-Luke Equation, Diff. Int. Eq., 16 (2003), 1039-1064.
|
[11] |
J. Quintero,
Solitary water waves for a 2D Boussinesq type system, J. Part. Diff. Eqs., 23 (2010), 251-280.
|
[12] |
J. Quintero,
The Cauchy problem and stability of solitary waves for a 2D Boussinesq-KdV type system, Diff. Int. Eqs., 21 (2011), 325-360.
|
[13] |
J. Quintero,
From periodic travelling waves to solitons of a 2D water wave system, Meth. Appl. Anal., 21 (2014), 241-264.
|
[14] |
J. Quintero,
A water wave mixed type problem: existence of periodic travelling waves for a 2D Boussinesq system, Rev. Academia Colombiana de Ciencias Naturales, Físicas y Exactas., 38 (2015), 6-17.
|
[15] |
J. R. Quintero and R. L. Pego,
Two-dimensional solitary waves for a Benney-Luke equation, Physica D., 45 (1999), 476-496.
|
[16] |
J. G. Verwer, J. G. Blom and W. Hundsdorfer,
An implicit-explicit approach for atmospheric transport-chemistry problems, Applied Numerical Mathematics, 20 (1996), 191-209.
|
[17] |
G. B. Whitham,
Linear and Nonlinear Waves Wiley-Interscience, 1974. |




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