修正Kawahara方程的收敛问题与色散爆破

修正Kawahara方程的收敛问题与色散爆破

Convergence Problem and Dispersive Blow-up for the Modified Kawahara Equation

摘要/Abstract

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数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 595-608.

王伟敏(),闫威^*()

河南师范大学数学与信息科学学院河南新乡 430007

收稿日期:2023-08-16 修回日期:2024-01-02 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *闫威, Email: 011133@htu.edu.cn
作者简介:王伟敏, Email: ydn1129@163.com
基金资助:
河南省骨干教师项目(2017GGJS044)

Wang Weimin(),Yan Wei^*()

School of Mathematics and Information Science, Henan Normal University, Henan Xinxiang 453007

Received:2023-08-16 Revised:2024-01-02 Online:2024-06-26 Published:2024-05-17
Supported by:
Young Core Teachers Problem of Henan Province(2017GGJS044)

摘要：

该文主要研究修正 Kawahara 方程的收敛问题与色散爆破. 首先, 利用傅里叶限制范数法, 高低频分解技巧以及 Strichartz 估计, 用三种不同的方法证明在空间 $H^{s}(\mathbb{R})$ $(s\geq\frac{1}{4})$ 中, 对几乎处处的 $x\in\mathbb{R}$ , 当 $t\rightarrow 0$ 时, $u(x,t)\rightarrow u_0(x)$ , 其中 $u(x,t)$ 是修正 Kawahara 方程的解, $u_0(x)$ 是其柯西问题的初值. 其次, 利用三线性估计和傅里叶限制范数法, 证明在空间 $H^{s}(\mathbb{R})$ $(s>0)$ 中, 当 $t\rightarrow 0$ 时, $u(x,t)\rightarrow U(t)u_0(x)$ (与 $x$ 无关). 最后, 给出方程解的色散爆破.

关键词: 修正 Kawahara 方程, 逐点收敛, 一致收敛, 色散爆破

Abstract:

In this paper, we consider the convergence problem and dispersive blow-up for the modified Kawahara equation. Firstly, we prove that $u(x,t)\rightarrow u_0(x),$ a.e. $x\in\mathbb{R}$ as $t\rightarrow 0$ by the Fourier restriction norm method, high-low frequency technique and Strichartz estimate, respectively. Here $u(x,t)$ is the solution of the modified Kawahara equation, and the initial value $u_0(x)\in H^{s}(\mathbb{R})$ $(s\geq\frac{1}{4})$ . Secondly, using the Fourier restriction norm method, we show that $u(x,t)\rightarrow U(t)u_0(x)$ as $t\rightarrow 0$ with $u_0(x)\in H^{s}(\mathbb{R})$ $(s>0)$ . Finally, we establish the dispersive blow-up of the modified Kawahara equation.

Key words: Modified Kawahara equation, Pointwise convergence, Uniform convergence, Dispersive blow-up

中图分类号:

O175.2

王伟敏, 闫威. 修正Kawahara方程的收敛问题与色散爆破[J]. 数学物理学报, 2024, 44(3): 595-608.

Wang Weimin, Yan Wei. Convergence Problem and Dispersive Blow-up for the Modified Kawahara Equation[J]. Acta mathematica scientia,Series A, 2024, 44(3): 595-608.

[1]	Benjamin T B, Bona J L, Mahony J J. Model equations for long waves in nonlinear dispersive systems. Philos Trans Roy Soc London Ser A, 1972, 272: 47-78
[2]	Bona J L, Ponce G, Saut J C, Sparber C. Dispersive blow-up for nonlinear Schr¨odinger equations revisited. J Math Pures Appl, 2014, 102: 782-811
[3]	Bona J L, Saut J C. Dispersive blow-up of solutions of generalized KdV equations. J Differ Equ, 1993, 103: 3-57
[4]	Bona J L, Saut J C. Dispersive blow-up II. Schrödinger equations, optical and oceanic rogue waves. Chin Ann Math Ser B, 2010, 31: 793-818
[5]	Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom Funct Anal, 1993, 3(2): 107-156
[6]	Carleson L. Some analytical problems related to statistical mechanics//Benedetto J J. Euclidean Harmonic Analysis. Heidelberg: Springer, 1980, 779: 5-45
[7]	Chen W G, Li J F, Miao C X, Wu J X. Low regularity solutions of two fifth-order KdV type equations. J Anal Math, 2009, 107: 221-238
[8]	Compaan E. A smoothing estimate for the nonlinear Schrödinger equation. UIUC research experience for graduate students, August 2013
[9]	Compaan E, Lucà R, Staffilani G. Pointwise convergence of the Schrödinger flow. Int Math Res Not 2021, 2021(1): 599-650
[10]	Cui S B, Tao S P. Strichartz estimates for dispersive equations and solvability of the Kawahara equation. J Math Anal Appl, 2005, 304: 683-702
[11]	Du X M, Zhang R X. Sharp $L^2$ estimates of the Schr¨odinger maximal function in higher dimensions. Ann Math, 2019, 189: 837-861
[12]	Gel’fand I M, Shilov G E. Generalized Functions, volume 1, Properties and Operations. New York: academic press, 1964,
[13]	Grafakos L. Modern Fourier Analysis. New York: Springer, 2009
[14]	Jia Y L, Huo Z H. Well-posedness for the fifth-order shallow water equations. J Diff Eqns, 2009, 246: 2448-2467
[15]	Kawahara T. Oscillatory solitary waves in dispersive media. J Phys Soc Japan, 1972, 33: 260-264
[16]	Kenig C E, Ruiz A, Sogge C D. Uniform Sobolev inequalities and unique continuation differential operators. Duke Math, 1987, 55: 329-347
[17]	Linares F, Pastor A, Drumond Silva J. Dispersive blow-up for solutions the Zakharov-Kuznetsov equation. Annales de l’Institut Henri Poincar´e C. Analyse non linèaire, 2021, 38: 281-300
[18]	Linares F, Ramos J P G. Maximal function estimates and local well-posedness for the generalized Zakharov- Kuznetsov equation. SIAM J Math Anal, 2021, 53: 914-936
[19]	Tao S P, Cui S B. Local and global existence of solutions to initial value problems of modified nonlinear Kawahara equation. Acta Math Sin Engl Ser, 2005, 21: 1035-1044
[20]	Yan W, Li Y S. The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity. Math Meth Appl Sci, 2010, 33: 1647-1660
[21]	Yan W, Li Y S. Ill-posedness of modified Kawahara equation and Kaup-Kupershmidt equation. Acta Math Sci, 2012, 32B: 710-716
[22]	Yan W, Li Y S, Yang X Y. The Cauchy problem for the modified Kawahara equation in Sobolev spaces with low regularity. Math Comput Modelling, 2011, 54: 1252-1261
[23]	Yan W, Wang W M, Yan X Q. Convergence problem of the Kawahara equation on the real line. J Math Anal Appl, 2022, 55: 126386
[24]	Yan W, Yan X Q, Duan J Q, Huang J H. The Cauchy problem for the generalized KdV equation with rough data and random data. arXiv: 2011.07128
[25]	Zhang Y T, Yan W, Yan X Q, Zhao Y J. Convergence problem of Schr¨odinger equation and wave equation in low regularity spaces. J Math Anal Appl, 2023, 522: 126921

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