JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (6): 56-63.doi: 10.6040/j.issn.1671-9352.0.2019.921

Previous Articles    

Local Gevrey regularity and analyticity of the solutions to the initial value problem associated with the two-component Novikov system

WANG Hai-quan, CHONG Ge-zi   

  1. School of Mathematics, Northwest University, Xian 710127, Shaanxi, China
  • Published:2020-06-01

PDF (PC)

575

Abstract: Considered herein is the initial value problem for the two-component Novikov system. At first, the Gevrey regularity and analyticity of the solutions to this problem in Sobolev-Gevrey spaces G1r,s(R)×G1r,s-1(R)with s>3/2, r≥1 are investigated by making use of the generalized Ovsyannikov theorem. Next, the continuity of the solution map z0→z(t) is discussed. The results can be directly applied to Novikov equation.

Key words: two-component Novikov system, generalized Ovsyannikov theorem, regularity and analyticity, Sobolev-Gevrey spaces

CLC Number: 

  • O175.29
[1] POPOWICZ Z. Doubled extended cubic peakon equation[J]. Physics Letter A, 2015, 379(18/19):1240-1245.
[2] CAMASSA R, HOLM D D. An integrable shallow water equation with peaked solitons[J]. Physical Review Letters, 1993, 71(11/12/13):1661-1664.
[3] LUO Wei, YIN Zhaoyang. Local well-posedness and blow-up criterion for a two-component Novikov system in the critical space[J]. Nonlinear Analysis: Theory, Methods and Applications, 2015, 122(1):1-22.
[4] WANG Haiquan, FU Ying. Non-uniform dependence on initial data for the two-component Novikov system[J]. Journal of Mathematical Physics, 2017, 58(2):021502.
[5] WANG Haiquan, FU Ying. A note on the Cauchy problem for the periodic Novikov system[J]. Applicable Analysis, 2020, 99(6):1042-1065.
[6] NOVIKOV V. Generalizations of the Camassa-Holm equation[J]. Journal of Physics A Mathematical and Theoretical, 2009, 42(34):342002.
[7] HONE A, WANG J. Integrable peakon equations with cubic nonlinearity[J]. Journal of Physics A General Physics, 2008, 41(37):4359-4380.
[8] NI Lidiao, ZHOU Yong. Well-posedness and persistence properties for the Novikov equation[J]. Journal of Differential Equations, 2011, 250(7):3002-3021.
[9] YAN Wei, LI Yongsheng, ZHANG Yimin. The Cauchy problem for the integrable Novikov equation[J]. Journal of Differential Equations, 2012, 253(1):298-318.
[10] OVSYANNIKOV L. Singular operators in Banach spaces scales[J]. Doklady Akademii Nauk SSSR, 1965, 163(4):819-822.
[11] OVSYANNIKOV L. Non-local Cauchy problem in fluid dynamics[J]. Actes du Congrès International des Mathèmaticiens, 1970, 3(5):137-142.
[12] HIMONAS A A, MISIOŁEK G. Analyticity of the Cauchy problem for an integrable evolution equation[J]. Mathematische Annalen, 2003, 327(3):575-584.
[13] YAN Kai, YIN Zhaoyang. Analytic solutions of the Cauchy problem for two-component shallow water systems[J]. Mathematische Zeitschrift, 2011, 269(3):1113-1127.
[14] LUO Wei, YIN Zhaoyang. Gevrey regularity and analyticity for Camassa-Holm type systems[J]. Annali della Scuola normale superiore di Pisa, Classe di scienze, 2018, 18(3):1061-1079.
[15] HE Huijun, YIN Zhaoyang. The global Gevrey regularity and analyticity of a two-component shallow water system with higher-order inertia operators[J]. Journal of Differential Equations, 2019, 267(4):2531-2559.
[16] FOIAS C, TEMAM R. Gevrey class regularity for the solutions of the Navier-Stokes equations[J]. Journal of Functional Analysis, 1989, 87(2):359-369.
[1] DONG Li. Lower bounds for blow up time of two nonlinear wave equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 56-60.
[2] FU Juan, ZHANG Rui, WANG Cai-jun, ZHANG Jing. The stability of a predator-prey diffusion model with Beddington-DeAngelis functional response [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(11): 115-122.
[3] DONG Jian-wei, LOU Guang-pu, WANG Yan-ping. Uniqueness of stationary solutions to a simplified energy-transport model for semiconductors [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 37-41.
[4] LÜ Hong-jie, LIU Jing-jing, QI Jing, LIU Shuo. Blow-up of a weakly dissipative μ-Hunter-Saxton equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(05): 55-59.
[5] DONG Jian-wei, CHENG Shao-hua, WANG Yan-ping. Classical solutions to stationary one-dimensional quantum energy-transport model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 52-56.
[6] CHEN Zhi-hui, WANG Zhen-zhen, CHENG Yong-kuan*. Soliton solutions for quasilinear Schrdinger equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(2): 58-62.
[7] LIU Yan-qin,XU Ming-yu and JIANG Xiao-yun . The fractional nonlinear convection-diffusion equation and its solution [J]. J4, 2007, 42(1): 35-39 .
[8] SHI Chang-guang . Multi-soliton solution of the Faddeev model [J]. J4, 2007, 42(7): 38-40 .
[9] ZHANG Shen-gui. Existence of solutions for fractional Kirchhoff-type equation with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 48-55.
[10] CHEN Peng-yu, MA Wei-feng, Ahmed Abdelmonem. Existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 13-23.
[11] LIN Fu-biao, ZHANG Qian-hong. Solving explicit new travelling wave solutions of KdV-Burgers-Kuramoto equation by Riccati equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 24-31.
[12] LIN Guo-guang, LI Zhuo-xi. Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 1-11.
[13] LI Yuan-fei. Continuous dependence on viscosity coefficient for primitive equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(12): 12-23.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd