Details
| Original language | English |
|---|---|
| Pages (from-to) | 377-395 |
| Number of pages | 19 |
| Journal | Communications in Partial Differential Equations |
| Volume | 33 |
| Issue number | 3 |
| Publication status | Published - 3 Mar 2008 |
Abstract
In this paper we study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa-Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa-Holm equation on a compact interval possesses no nontrivial global classical solutions.
Keywords
- Blow-up and global existence, Global weak solutions, Initial boundary value problems, Local well-posedness, The Camassa-Holm equation
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Communications in Partial Differential Equations, Vol. 33, No. 3, 03.03.2008, p. 377-395.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Initial boundary value problems of the Camassa-Holm equation
AU - Escher, Joachim
AU - Yin, Zhaoyang
N1 - Funding information: The second author is now visiting the institute of applied mathematics of Hanover University in Germany. The financial support of the Alexander von Humboldt Foundation is gratefully acknowledged. This work was also partially supported by the NNSF of China (No. 10531040), the SRF for ROCS, SEM and the NSF of Guangdong Province. The authors thank the referee for valuable comments and suggestions.
PY - 2008/3/3
Y1 - 2008/3/3
N2 - In this paper we study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa-Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa-Holm equation on a compact interval possesses no nontrivial global classical solutions.
AB - In this paper we study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa-Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa-Holm equation on a compact interval possesses no nontrivial global classical solutions.
KW - Blow-up and global existence
KW - Global weak solutions
KW - Initial boundary value problems
KW - Local well-posedness
KW - The Camassa-Holm equation
UR - http://www.scopus.com/inward/record.url?scp=40249094441&partnerID=8YFLogxK
U2 - 10.1080/03605300701318872
DO - 10.1080/03605300701318872
M3 - Article
AN - SCOPUS:40249094441
VL - 33
SP - 377
EP - 395
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 3
ER -