TITLE:
On The Cauchy Problem for A 1D Euler-Alignment System in Besov Spaces
AUTHORS:
Yaojun Yang
KEYWORDS:
Euler-Alignment Equations, Local Well-Posedness, Blow-Up Criteria, Ill-Posedness
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.2,
February
29,
2024
ABSTRACT: In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagrange coordinate transformation, the local well-posedness of the solutions for the 1D pressureless Euler-alignment in Besov spaces with 1≤p∞ is established. Next, the ill-posedness of the solutions for this model in Besov spaces with 1≤p and is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces with 1≤p .