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Well-posedness and analyticity of solutions for the sixth-order Boussinesq equation

Amin Esfahani

https://orcid.org/0000-0001-9901-7934

Department of Mathematics, Nazarbayev University, Astana 010000, Kazakhstan

E-mail Address: saesfahani@gmail.com

E-mail Address: amin.esfahani@nu.edu.kz

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and

Achenef Tesfahun

https://orcid.org/0000-0003-0259-0182

Department of Mathematics, Nazarbayev University, Astana 010000, Kazakhstan

E-mail Address: achenef.tesfahun@nu.edu.kz

Corresponding author.

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https://doi.org/10.1142/S0219199724500056Cited by:1 (Source: Crossref)

Abstract

In this paper, the sixth-order Boussinesq equation is studied. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the “bad” fourth term $Δ u$ in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.

Keywords:

AMSC: 35B30, 35Q53, 76B15, 35L70, 37K10

References

1. H. Bae, A. Biswas and E. Tadmor , Analyticity and decay estimates of the Navier–Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal. 205 (2012) 963–991. Crossref, Web of Science, Google Scholar
2. H. Bae, A. Biswas and E. Tadmor, Analyticity of the subcritical and critical quasi-geostrophic equations in Besov spaces, arXiv:1310.1624. Google Scholar
3. H. Bahouri, J.-Y. Chemin and R. Danchin , Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, Vol. 343 (Springer, Heidelberg, 2011). Crossref, Google Scholar
4. R. F. Barostichi, R. O. Figueira and A. A. Himonas , Well-posedness of the “good” Boussinesq equation in analytic Gevrey spaces and time regularity, J. Differ. Equ. 267 (2019) 3181–3198. Crossref, Web of Science, Google Scholar
5. J. L. Bona, Z. Grujić and H. Kalisch , Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 783–797. Crossref, Web of Science, Google Scholar
6. J. Boussinesq , Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide continu dans 21 ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (1872) 55–108. Google Scholar
7. J.-Y. Chemin , Perfect Incompressible Fluids (Oxford University Press, Oxford, 1998). Crossref, Google Scholar
8. C. I. Christov, G. A. Maugin and M. G. Velarde , Well-posed boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E 54 (1996) 3621–3638. Crossref, Web of Science, Google Scholar
9. Y. Cho and T. Ozawa , On small amplitude solutions to the generalized Boussinesq equations, Discrete Contin. Dyn. Syst. 17 (2007) 691–711. Crossref, Web of Science, Google Scholar
10. Y. Cho and S. Lee , Strichartz estimates in spherical coordinates, Indiana Univ. Math. J. 62 (2013) 991–1020. Crossref, Web of Science, Google Scholar
11. P. Daripa and W. Hu , A numerical method for solving an ill-posed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207. Google Scholar
12. T. Deneke, T. T. Dufera and A. Tesfahun , Dispersive estimates for linearized water wave type equations in $ℝ^{d}$ , to appear in Ann. Henri Poincaré, arXiv:2106.02717. Web of Science, Google Scholar
13. T. Dufera, S. Mebrate and A. Tesfahun , On the persistence of spatial analyticity for the beam equation, J. Math. Anal. Appl. 509 (2022) 126001. Crossref, Web of Science, Google Scholar
14. A. Esfahani and L. G. Farah , Local well-posedness for the sixth-order Boussinesq equation J. Math. Anal. Appl. 385 (2012) 230–242. Crossref, Google Scholar
15. A. Esfahani, L. G. Farah and H. Wang , Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal. 75 (2012) 4325–4338. Crossref, Web of Science, Google Scholar
16. A. Esfahani and S. Levandosky , Stability of solitary waves for the generalized higher-order Boussinesq equation, J. Dyn. Differ. Equ. 24 (2012) 391–425. Crossref, Web of Science, Google Scholar
17. A. Esfahani and H. Wang , A bilinear estimate with application to the sixth-order Boussinesq equation, Differ. Int. Equ. 27 (2014) 401–414. Web of Science, Google Scholar
18. L. G. Farah , Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation, Commun. Pure Appl. Anal. 8 (2009) 1521–1539. Crossref, Web of Science, Google Scholar
19. C. Foias and R. Temam , Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989) 359–369. Crossref, Web of Science, Google Scholar
20. D. Geba, A. Himonas and D. Karapetyan , Ill-posedness results for generalized Boussinesq equations, Nonlinear Anal. 95 (2014) 404–413. Crossref, Web of Science, Google Scholar
21. D.-A. Geba and E. Witz , Improved global well-posedness for defocusing sixth-order Boussinesq equations, Nonlinear Anal. 191 (2020) 1–16. Crossref, Web of Science, Google Scholar
22. D.-A. Geba and E. Witz , Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations, Electron. Res. Arch. 28 (2020) 627–649. Crossref, Web of Science, Google Scholar
23. Z. Guo, L. Peng and B. Wang , Decay estimates for a class of waves equations, J. Funct. Anal. 254 (2008) 1642–1660. Crossref, Web of Science, Google Scholar
24. Z. Grujić and H. Kalisch , Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differ. Integral Equ. 15 (2002) 1325–1334. Crossref, Google Scholar
25. N. Hayashi , Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J. 60 (1990) 717–727. Crossref, Web of Science, Google Scholar
26. Y. Katznelson , An Introduction to Harmonic Analysis, Corrected edn. (Dover Publications, Inc., New York, 1976). Google Scholar
27. N. Kishimoto , Sharp local well-posedness for the “good” Boussinesq equation, J. Differ. Equ. 254 (2013) 2393–2433. Crossref, Web of Science, Google Scholar
28. N. Kishimoto and K. Tsugawa , Local well-posedness for quadratic nonlinear Schrödinger equations and the “good” Boussinesq equation, Differ. Int. Equ. 23 (2010) 463–493. Web of Science, Google Scholar
29. F. Linares and G. Ponce , Introduction to Nonlinear Dispersive Equations, 2nd edn. (Springer, New York, 2015). Crossref, Google Scholar
30. G. A. Maugin , Nonlinear Waves in Elastic Crystals (Oxford University Press, Oxford, 1999). Crossref, Google Scholar
31. S. Selberg and A. Tesfahun , On the radius of spatial analyticity for the 1d Dirac-Klein-Gordon equations, J. Differ. Equ. 259 (2015) 4732–4744. Crossref, Web of Science, Google Scholar
32. E. M. Stein , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43 (Princeton University Press, Princeton, NJ, 1993). Google Scholar
33. S. Selberg , Spatial analyticity of solutions to nonlinear dispersive PDE, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, EMS Series of Congress Reports (European Mathematical Society, Zurich, 2018), pp. 437–454. Crossref, Google Scholar
34. S. Selberg , On the radius of spatial analyticity for solutions of the Dirac–Klein–Gordon equations in two space dimensions, Ann. Inst. H. Poincaré Anal. Non Lineairé 36 (2019) 1311–1330. Crossref, Web of Science, Google Scholar
35. T. Tao , Multilinear weighted convolution of $L^{2}$ functions and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001) 839–908. Crossref, Web of Science, Google Scholar
36. T. Tao , Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106 (AMS, Providence, RI, 2006). Crossref, Google Scholar
37. S. K. Turitsyn , Nonstable solitons and sharp criteria for wave collapse, Phys. Rev. E 47 (1993) R13. Crossref, Web of Science, Google Scholar
38. H. Wang and A. Esfahani , Global rough solutions to the sixth-order Boussinesq equation, Nonlinear Anal. 102 (2014) 97–104. Crossref, Web of Science, Google Scholar
39. H. Wang and A. Esfahani , The limit behavior of solutions for the Cauchy problem of the sixth-order Boussinesq equation, Acta Appl. Math. 176 (2021) 1–19. Crossref, Web of Science, Google Scholar
40. Y.-Z. Wang and W. Li , Well-posedness and scattering of solutions to the sixth-order Boussinesq type equation in the framework of modulation spaces, Math Methods Appl. Sci. 43 (2020) 5507–5521. Crossref, Web of Science, Google Scholar
41. V. E. Zakharov , On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP 38 (1974) 108–110. Google Scholar