American Institute of Mathematical Sciences

Advanced
September  2007, 19(3): 515-529. doi: 10.3934/dcds.2007.19.515

On unique continuation for the modified Euler-Poisson equations

A. Alexandrou Himonas 1, , Gerard Misiołek 2, and  Feride Tiǧlay 3,

1. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States
2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, United States
3. 

Department of Mathematics, University of New Orleans, New Orleans, LA 70148, United States

Received  May 2007 Revised  June 2007 Published  July 2007

It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.
Keywords: local well-posedness, exponential decay, unique continuation, Burgers equation, Camassa-Holm equation, Modified Euler-Poisson equations, Sobolev spaces..
Mathematics Subject Classification: Primary: 35Q5.
Citation: A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[1]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[2]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[3]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
[4]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501
[5]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493
[6]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042
[7]

Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007
[8]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[9]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[10]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[11]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[12]

Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090
[13]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[14]

Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011
[15]

Andrea Natale, François-Xavier Vialard. Embedding Camassa-Holm equations in incompressible Euler. Journal of Geometric Mechanics, 2019, 11 (2) : 205-223. doi: 10.3934/jgm.2019011
[16]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[17]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304
[18]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[19]

H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133
[20]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences