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Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces.
- Source :
-
Journal of Differential Equations . Dec2023, Vol. 376, p71-101. 31p. - Publication Year :
- 2023
-
Abstract
- We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B ˙ p , 1 2 p (R 2) × B ˙ p , q 2 p − 1 (R 2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p < 4 and ill-posed in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p > 4. In this paper, we prove that this system is ill-posed for the critical case p = 4 in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in B ˙ 4 , 1 1 2 (R 2) × B ˙ 4 , q − 1 2 (R 2) for any q ≠ 2. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BESOV spaces
*SHALLOW-water equations
*CAUCHY problem
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 376
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 173097627
- Full Text :
- https://doi.org/10.1016/j.jde.2023.08.012