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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

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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