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Volume 36, Issue 1
Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces

Achraf Azanzal, Chakir Allalou, Said Melliani & Adil Abbassi

DOI: 10.4208/jpde.v36.n1.1

J. Part. Diff. Eq., 36 (2023), pp. 1-21.

Published online: 2022-12

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  • Abstract

In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists $v$ a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces  $ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}$. Moreover, we show the asymptotic behavior of the global solution $v$. i.e., $\|v(t)\|_{ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}}$ decays to zero as time goes to infinity.

  • Keywords

2D quasi-geostrophic equation subcritical dissipation Littlewood-Paley theory global well-posedness long time behavior of the solution Fourier-Besov-Morrey spaces.

  • AMS Subject Headings

35A01, 35A02, 35k30, 35K08, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

achraf0665@gmail.com (Achraf Azanzal)

chakir.allalou@yahoo.fr (Chakir Allalou)

saidmelliani@gmail.com (Adil Abbassi)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-36-1, author = {Azanzal , AchrafAllalou , ChakirMelliani , Said and Abbassi , Adil}, title = {Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {36}, number = {1}, pages = {1--21}, abstract = {

In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists $v$ a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces  $ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}$. Moreover, we show the asymptotic behavior of the global solution $v$. i.e., $\|v(t)\|_{ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}}$ decays to zero as time goes to infinity.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n1.1}, url = {http://global-sci.org/intro/article_detail/jpde/21290.html} }
TY - JOUR T1 - Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces AU - Azanzal , Achraf AU - Allalou , Chakir AU - Melliani , Said AU - Abbassi , Adil JO - Journal of Partial Differential Equations VL - 1 SP - 1 EP - 21 PY - 2022 DA - 2022/12 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n1.1 UR - https://global-sci.org/intro/article_detail/jpde/21290.html KW - 2D quasi-geostrophic equation KW - subcritical dissipation KW - Littlewood-Paley theory KW - global well-posedness KW - long time behavior of the solution KW - Fourier-Besov-Morrey spaces. AB -

In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists $v$ a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces  $ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}$. Moreover, we show the asymptotic behavior of the global solution $v$. i.e., $\|v(t)\|_{ \mathcal{F} {\mathcal{N}}_{p, \lambda, q}^{3-2 \alpha+\frac{\lambda-2}{p}}}$ decays to zero as time goes to infinity.

Azanzal , AchrafAllalou , ChakirMelliani , Said and Abbassi , Adil. (2022). Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces. Journal of Partial Differential Equations. 36 (1). 1-21. doi:10.4208/jpde.v36.n1.1
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