%0 Journal Article
%T On the Cauchy Problem for Mildly Nonlinear and Non-Boussinesq Case-(ABC) System
%A Luhang Zhou
%J Journal of Applied Mathematics and Physics
%P 1286-1307
%@ 2327-4379
%D 2024
%I Scientific Research Publishing
%R 10.4236/jamp.2024.124079
%X In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution for this system in Besov spaces <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>B</mi>
   <mrow>
    <mi>p</mi><mn>,</mn><mi>r</mi></mrow>
   <mrow>
    <mi>s</mi><mo> </mo><mn>1</mn></mrow>
  </msubsup>
  <mo>×</mo><msubsup>
   <mi>B</mi>
   <mrow>
    <mi>p</mi><mn>,</mn><mi>r</mi></mrow>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>
 with <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mn>1</mn><mo>≤</mo><mi>p</mi><mn>,</mn><mi>r</mi><mo>≤</mo><mi>∞</mi></mrow>
</math>
 and <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>s</mi><mo>></mo><mi>max</mi><mrow><mo>{</mo> <mrow>
   <mn>1</mn><mo> </mo><mfrac>
    <mn>1</mn>
    <mi>p</mi>
   </mfrac>
   <mo>,</mo><mfrac>
    <mn>3</mn>
    <mn>2</mn>
   </mfrac>
   </mrow> <mo>}</mo></mrow></mrow>
</math>
 was firstly established. Next, we consider the continuity of the solution-to-data map, <i>i.e.</i> the ill-posedness of the solution for this system in Besov space <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>B</mi>
   <mrow>
    <mi>p</mi><mn>,</mn><mi>∞</mi></mrow>
   <mrow>
    <mi>s</mi><mo> </mo><mn>1</mn></mrow>
  </msubsup>
  <mo>×</mo><msubsup>
   <mi>B</mi>
   <mrow>
    <mi>p</mi><mn>,</mn><mi>∞</mi></mrow>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>
 was derived. Finally, the Gevrey regularity of the system was presented.
%K Local Well-Posedness
%K Ill-Posedness
%K Gevrey Regularity
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=132835