August  2019, 39(8): 4455-4469. doi: 10.3934/dcds.2019182

Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

1. 

Fakultät für Mathematik, Universität Wien, Austria

2. 

Department of Mathematics, Gakushuin University, Tokyo, Japan

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  July 2018 Revised  October 2018 Published  May 2019

Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution.

Citation: Günther Hörmann, Hisashi Okamoto. Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4455-4469. doi: 10.3934/dcds.2019182
References:
[1]

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2011.

[2]

X. Chen and H. Okamoto, Global existence of solutions to the generalized Proudman–Johnson equation, Proc. Japan Acad. Ser. A, 78 (2002), 136-139. doi: 10.3792/pjaa.78.136.

[3]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[4]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, 2016. doi: 10.1007/978-3-662-49451-6.

[6]

K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Springer, 2006.

[7]

K. Fellner and C. Schmeiser, Burgers-Poisson: A nonlinear dispersive model equation, SIAM J. Appl. Math., 64 (2004), 1509-1525. doi: 10.1137/S0036139902410345.

[8]

B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. R. Soc. Lond., 289 (1978), 373-404. doi: 10.1098/rsta.1978.0064.

[9]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.

[10]

S. Haziot, Wave breaking for the Fornberg-Whitham equation, J. Diff. Equ., 263 (2017), 8178-8185. doi: 10.1016/j.jde.2017.08.037.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Math., 1981.

[12]

J. Holmes, Well-posedness of the Fornberg-Whitham equation on the circle, J. Diff. Equ., 260 (2016), 8530-8549. doi: 10.1016/j.jde.2016.02.030.

[13]

J. Holmes and R. C. Thompson, Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces, J. Diff. Equ., 263 (2017), 4355-4381. doi: 10.1016/j.jde.2017.05.019.

[14]

G. Hörmann, Wave breaking of periodic solutions to the Fornberg-Whitham equation, Disc. Cont. Dynam. Sys., 38 (2018), 1605-1613. doi: 10.3934/dcds.2018066.

[15]

G. Hörmann, Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation, Jour. Diff. Equs., 265 (2018), 2825-2841. doi: 10.1016/j.jde.2018.04.056.

[16]

K. Itasaka, Wave-breaking phenomena and global existence for the generalzed Fornberg-Whitham equation, preprint at arXiv: 1802.00641v1 (2018).

[17]

D. Jackson, Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, Jour. Math. Anal. Appl., 172 (1993), 256-265. doi: 10.1006/jmaa.1993.1022.

[18]

F. John, Partial differential equations, Mathematics Applied to Physics, 229–315, Springer, New York, 1970.

[19]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.

[20]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243.

[21]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. L. Seliger, A note on the breaking of waves, Proc. R. Soc. Lond. A, 303 (1968), 493-496.

[24]

M. E. Taylor, Partial Differential Equations III, Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[25]

G. B. Whitham, Variational methods and applications to water waves, Proc. Royal Soc. London A, 299 (1967), 6-25. doi: 10.1007/978-3-642-87025-5_16.

[26]

J. Zhou and L. Tian, Solitons, peakons and periodic cusp wave solutions for the Fornberg–Whitham equation, Nonlinear Analysis: Real World Appl., 11 (2010), 356-363. doi: 10.1016/j.nonrwa.2008.11.014.

show all references

References:
[1]

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2011.

[2]

X. Chen and H. Okamoto, Global existence of solutions to the generalized Proudman–Johnson equation, Proc. Japan Acad. Ser. A, 78 (2002), 136-139. doi: 10.3792/pjaa.78.136.

[3]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[4]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, 2016. doi: 10.1007/978-3-662-49451-6.

[6]

K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Springer, 2006.

[7]

K. Fellner and C. Schmeiser, Burgers-Poisson: A nonlinear dispersive model equation, SIAM J. Appl. Math., 64 (2004), 1509-1525. doi: 10.1137/S0036139902410345.

[8]

B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. R. Soc. Lond., 289 (1978), 373-404. doi: 10.1098/rsta.1978.0064.

[9]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.

[10]

S. Haziot, Wave breaking for the Fornberg-Whitham equation, J. Diff. Equ., 263 (2017), 8178-8185. doi: 10.1016/j.jde.2017.08.037.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Math., 1981.

[12]

J. Holmes, Well-posedness of the Fornberg-Whitham equation on the circle, J. Diff. Equ., 260 (2016), 8530-8549. doi: 10.1016/j.jde.2016.02.030.

[13]

J. Holmes and R. C. Thompson, Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces, J. Diff. Equ., 263 (2017), 4355-4381. doi: 10.1016/j.jde.2017.05.019.

[14]

G. Hörmann, Wave breaking of periodic solutions to the Fornberg-Whitham equation, Disc. Cont. Dynam. Sys., 38 (2018), 1605-1613. doi: 10.3934/dcds.2018066.

[15]

G. Hörmann, Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation, Jour. Diff. Equs., 265 (2018), 2825-2841. doi: 10.1016/j.jde.2018.04.056.

[16]

K. Itasaka, Wave-breaking phenomena and global existence for the generalzed Fornberg-Whitham equation, preprint at arXiv: 1802.00641v1 (2018).

[17]

D. Jackson, Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, Jour. Math. Anal. Appl., 172 (1993), 256-265. doi: 10.1006/jmaa.1993.1022.

[18]

F. John, Partial differential equations, Mathematics Applied to Physics, 229–315, Springer, New York, 1970.

[19]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.

[20]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10 (1970), 217-243.

[21]

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

R. L. Seliger, A note on the breaking of waves, Proc. R. Soc. Lond. A, 303 (1968), 493-496.

[24]

M. E. Taylor, Partial Differential Equations III, Applied Mathematical Sciences, 117. Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[25]

G. B. Whitham, Variational methods and applications to water waves, Proc. Royal Soc. London A, 299 (1967), 6-25. doi: 10.1007/978-3-642-87025-5_16.

[26]

J. Zhou and L. Tian, Solitons, peakons and periodic cusp wave solutions for the Fornberg–Whitham equation, Nonlinear Analysis: Real World Appl., 11 (2010), 356-363. doi: 10.1016/j.nonrwa.2008.11.014.

Figure 1.  The solution from data1. $ 0 \le t \le 0.65 $
Figure 2.  data2
Figure 3.  traveling wave $ U $; $ c = 0.025, 0.0255, 0.026, 0.0269 $
Figure 4.  The time dependent solution with the traveling wave as the initial data
Figure 5.  The time dependent solution with $ d $ as in the second case of (12). The points $ ( u_{300}^n,u_{600}^n) $ with $ n $ corresponding to $ 0 \le t \le 300 $ are plotted
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