%0 Journal Article
%T On The Cauchy Problem for A 1D Euler-Alignment System in Besov Spaces
%A Yaojun Yang
%J Journal of Applied Mathematics and Physics
%P 603-631
%@ 2327-4379
%D 2024
%I Scientific Research Publishing
%R 10.4236/jamp.2024.122040
%X 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 <img src=\"Edit_035c7494-63ad-4ad1-9f9b-adfeadf446d0.png\" alt=\"\" /> with 1<span style=\"white-space:nowrap;\">¡Ü<em>p<span style=\"white-space:nowrap;\">&lt;</span><span style=\"white-space:nowrap;\">¡Þ </span></em></span>is established. Next, the ill-posedness of the solutions for this model in Besov spaces <img src=\"Edit_12cc0c90-e28c-48ab-a8bc-f5f02a91e7c2.png\" alt=\"\" /> with <span style=\"white-space:normal;\">1</span>¡Ü<em>p&lt;¡Þ</em> and <img src=\"Edit_7ed88e25-8c1b-4e53-a152-91ba2b03ffaa.png\" alt=\"\" /> is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces <img src=\"Edit_ce5d4a68-2332-4067-990f-0449201e8c46.png\" alt=\"\" /> with <span style=\"white-space:normal;\">1</span>¡Ü<em>p&lt;¡Þ</em> .
%K Euler-Alignment Equations
%K Local Well-Posedness
%K Blow-Up Criteria
%K Ill-Posedness
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=131598