Research Article
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Year 2021, Issue: 2, 91 - 100, 30.10.2021
https://doi.org/10.47087/mjm.930045

Abstract

References

  • [1] S. Alinhac and G. Metivier, Propagation de l’analyticite des solutions des systmes hyperboliques non-lin´eaires, Invent. Math., 75, 189-204, 1984.
  • [2] Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differ. Equ.,246, 2448-2467, 2009.
  • [3] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity of the periodic gKdV equation, J.Differ. Equ., 250, 2581-2600, 2011.
  • [4] Z. Huoa and B. Guo, Well-posedness of the Cauchy problem for the Hirota equation in Sobolev spaces Hs, Nonlinear Analysis, 60, 1093-1110, 2005.
  • [5] J. Gorsky, A. Himonas, C. Holliman, G. Petronilho, The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces, J. Math. Anal. Appl., 405, 349-361, 2013.
  • [6] C.E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21, 1993.
  • [7] Barostichi R. F., Figueira R. O., Himonas A. A., Well-posedness of the "good" Boussinesq equation in analytic Gevrey spaces and time regularity, J. Diff. Equ., 267(5), 2019, 3181-3198.
  • [8] C.E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9, 573-603, 1996.
  • [9] X. Yang, Y. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces, J. Diff. Equ., 248, 1458-1472, 2010.
  • [10] Z. Zhang, Z. Liu. M, Sun and S. Li, Low Regularity for the Higher Order Nonlinear Dispersive Equation in Sobolev Spaces of Negative Index, J. Dyn. Diff. Equ., 31, 419-433, 2019.
  • [11] A. Boukarou, Kh. Zennir, K. Guerbati and S. G. Georgiev, Well-posedness of the Cauchy problem of Ostrovsky equation in analytic Gevrey spaces and time regularity, Rend. Circ. Mat. Palermo (2), (2020) https://doi.org/10.1007/s12215-020-00504-7.
  • [12] A. Boukarou, Kh. Zennir, K. Guerbati and S. G. Georgiev, Well-posedness and regularity of the fifth order Kadomtsev-Petviashvili I equation in the analytic Bourgain spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. (2020) https://doi.org/10.1007/s11565-020-00340-8.
  • [13] B.A. Kupershmidt, A super Kortewegde Vries equation: An integrable system, Phys. Lett. A 102 (56) (1984) 213215.
  • [14] E.S. Benilov, R. Grimshaw, E.P. Kuznetsova, The generation of radiating waves in a singularly-perturbed Kortewegde Vries equation, Phys. D 69 (34) (1993) 270278.
  • [15] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D 32 (2) (1988) 253268.
  • [16] X Zhao, BY Zhang,Global controllability and stabilizability of Kawahara equation on a periodic domain, Mathematical Control Related Fields, 2015, pp. 335358

THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES

Year 2021, Issue: 2, 91 - 100, 30.10.2021
https://doi.org/10.47087/mjm.930045

Abstract

The Cauchy problem for the Kawahara equation with data in analytic Gevrey spaces on the circle is considered and its local well-posedness in these spaces is proved. Using Bourgain-Gevrey type analytic spaces and appropriate bilinear estimates, it is shown that local in time wellposedness holds when the initial data belong to an analytic Gevrey spaces of order σ. Moreover, the solution is not necessarily Gσ in time. However, it belongs to G5σ near zero for every x on the circle. We study a Cauchy problem for Kawahara equation . With data in analytic Gevrey spaces on the circal, we prove that the problem is well defined. We also treat the regularity in time which belongs to $G^{5\sigma}$ .

References

  • [1] S. Alinhac and G. Metivier, Propagation de l’analyticite des solutions des systmes hyperboliques non-lin´eaires, Invent. Math., 75, 189-204, 1984.
  • [2] Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differ. Equ.,246, 2448-2467, 2009.
  • [3] H. Hannah, A. Himonas, G. Petronilho, Gevrey regularity of the periodic gKdV equation, J.Differ. Equ., 250, 2581-2600, 2011.
  • [4] Z. Huoa and B. Guo, Well-posedness of the Cauchy problem for the Hirota equation in Sobolev spaces Hs, Nonlinear Analysis, 60, 1093-1110, 2005.
  • [5] J. Gorsky, A. Himonas, C. Holliman, G. Petronilho, The Cauchy problem of a periodic higher order KdV equation in analytic Gevrey spaces, J. Math. Anal. Appl., 405, 349-361, 2013.
  • [6] C.E. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1-21, 1993.
  • [7] Barostichi R. F., Figueira R. O., Himonas A. A., Well-posedness of the "good" Boussinesq equation in analytic Gevrey spaces and time regularity, J. Diff. Equ., 267(5), 2019, 3181-3198.
  • [8] C.E. Kenig, G. Ponce, L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9, 573-603, 1996.
  • [9] X. Yang, Y. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces, J. Diff. Equ., 248, 1458-1472, 2010.
  • [10] Z. Zhang, Z. Liu. M, Sun and S. Li, Low Regularity for the Higher Order Nonlinear Dispersive Equation in Sobolev Spaces of Negative Index, J. Dyn. Diff. Equ., 31, 419-433, 2019.
  • [11] A. Boukarou, Kh. Zennir, K. Guerbati and S. G. Georgiev, Well-posedness of the Cauchy problem of Ostrovsky equation in analytic Gevrey spaces and time regularity, Rend. Circ. Mat. Palermo (2), (2020) https://doi.org/10.1007/s12215-020-00504-7.
  • [12] A. Boukarou, Kh. Zennir, K. Guerbati and S. G. Georgiev, Well-posedness and regularity of the fifth order Kadomtsev-Petviashvili I equation in the analytic Bourgain spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. (2020) https://doi.org/10.1007/s11565-020-00340-8.
  • [13] B.A. Kupershmidt, A super Kortewegde Vries equation: An integrable system, Phys. Lett. A 102 (56) (1984) 213215.
  • [14] E.S. Benilov, R. Grimshaw, E.P. Kuznetsova, The generation of radiating waves in a singularly-perturbed Kortewegde Vries equation, Phys. D 69 (34) (1993) 270278.
  • [15] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D 32 (2) (1988) 253268.
  • [16] X Zhao, BY Zhang,Global controllability and stabilizability of Kawahara equation on a periodic domain, Mathematical Control Related Fields, 2015, pp. 335358
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Aissa Boukarou

Kaddour Guerbati 0000-0003-4256-3760

Publication Date October 30, 2021
Acceptance Date October 21, 2021
Published in Issue Year 2021 Issue: 2

Cite

APA Boukarou, A., & Guerbati, K. (2021). THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES. Maltepe Journal of Mathematics, 3(2), 91-100. https://doi.org/10.47087/mjm.930045
AMA Boukarou A, Guerbati K. THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES. Maltepe Journal of Mathematics. October 2021;3(2):91-100. doi:10.47087/mjm.930045
Chicago Boukarou, Aissa, and Kaddour Guerbati. “THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES”. Maltepe Journal of Mathematics 3, no. 2 (October 2021): 91-100. https://doi.org/10.47087/mjm.930045.
EndNote Boukarou A, Guerbati K (October 1, 2021) THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES. Maltepe Journal of Mathematics 3 2 91–100.
IEEE A. Boukarou and K. Guerbati, “THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES”, Maltepe Journal of Mathematics, vol. 3, no. 2, pp. 91–100, 2021, doi: 10.47087/mjm.930045.
ISNAD Boukarou, Aissa - Guerbati, Kaddour. “THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES”. Maltepe Journal of Mathematics 3/2 (October 2021), 91-100. https://doi.org/10.47087/mjm.930045.
JAMA Boukarou A, Guerbati K. THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES. Maltepe Journal of Mathematics. 2021;3:91–100.
MLA Boukarou, Aissa and Kaddour Guerbati. “THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES”. Maltepe Journal of Mathematics, vol. 3, no. 2, 2021, pp. 91-100, doi:10.47087/mjm.930045.
Vancouver Boukarou A, Guerbati K. THE CAUCHY PROBLEM OF A PERIODIC KAWAHARA EQUATION IN ANALYTIC GEVREY SPACES. Maltepe Journal of Mathematics. 2021;3(2):91-100.

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