您的位置:山东大学 -> 科技期刊社 -> 《山东大学学报(理学版)》

《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (6): 56-63.doi: 10.6040/j.issn.1671-9352.0.2019.921

• • 上一篇    

两分量Novikov系统初值问题解的局部Gevrey正则性与解析性

王海权,种鸽子   

  1. 西北大学数学学院, 陕西 西安 710127
  • 发布日期:2020-06-01
  • 作者简介:王海权(1991— ),男,博士研究生,研究方向为偏微分方程. E-mail:hqwangmath@163.com
  • 基金资助:
    陕西省自然科学基金资助项目(2019JM007)

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

摘要: 利用一个推广的Ovsyannikov定理,讨论了两分量Novikov系统Cauchy问题解在 Sobolev-Gevrey空间G1r,s(R)×G1r,s-1(R)中的正则性与解析性,其中s>3/2, r≥1,并研究了该问题解映射z0→z(t)的连续性。此结论可以直接应用到Novikov方程。

关键词: 两分量Novikov系统, 推广的Ovsyannikov定理, 正则性与解析性, Sobolev-Gevrey空间

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

中图分类号: 

  • 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] 董莉. 两类非线性波动方程解的爆破时间的下确界[J]. 山东大学学报(理学版), 2017, 52(4): 56-60.
[2] 付娟,张睿,王彩军,张婧. 具有Beddington-DeAngelis功能反应项的捕食-食饵扩散模型的稳定性[J]. 山东大学学报(理学版), 2016, 51(11): 115-122.
[3] 董建伟,娄光谱,王艳萍. 一个半导体简化能量输运模型稳态解的唯一性[J]. 山东大学学报(理学版), 2016, 51(2): 37-41.
[4] 吕红杰, 刘静静, 齐静, 刘硕. 弱耗散μ-Hunter-Saxton方程的爆破[J]. 山东大学学报(理学版), 2015, 50(05): 55-59.
[5] 董建伟, 程少华, 王艳萍. 一维稳态量子能量输运模型的古典解[J]. 山东大学学报(理学版), 2015, 50(03): 52-56.
[6] 陈志辉,王真真,程永宽*. 拟线性薛定谔方程的孤立解[J]. 山东大学学报(理学版), 2014, 49(2): 58-62.
[7] 刘艳芹,徐明瑜,蒋晓芸 . 分数阶非线性对流-扩散方程及其解[J]. J4, 2007, 42(1): 35-39 .
[8] 石长光 . Faddeev模型中的多孤立子解[J]. J4, 2007, 42(7): 38-40 .
[9] 张申贵. 变指数基尔霍夫型分数阶方程解的存在性[J]. 《山东大学学报(理学版)》, 2020, 55(6): 48-55.
[10] 陈鹏玉,马维凤,Ahmed Abdelmonem. 一类分数阶随机发展方程非局部问题mild解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(10): 13-23.
[11] 林府标,张千宏. 用Riccati方程求KdV-Burgers-Kuramoto方程的显式新行波解[J]. 《山东大学学报(理学版)》, 2019, 54(12): 24-31.
[12] 林国广,李卓茜. 一类高阶非线性Kirchhoff方程吸引子族及其维数[J]. 《山东大学学报(理学版)》, 2019, 54(12): 1-11.
[13] 李远飞. 原始方程组对黏性系数的连续依赖性[J]. 《山东大学学报(理学版)》, 2019, 54(12): 12-23.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!