Volume 36, Issue 2
On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces

Tuoc Phan & Yannick Sire

Anal. Theory Appl., 36 (2020), pp. 111-127.

Published online: 2020-06

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  • Abstract

We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's interation, which can be apdated to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.

  • Keywords

Local well-posedness, global well-posedness, dissipative quasi-geostrophic equation, fractional heat equation, mixed-norm Lebesgue spaces.

  • AMS Subject Headings

35A01, 35K55, 35K61

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-36-111, author = {Phan , Tuoc and Sire , Yannick}, title = {On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {2}, pages = {111--127}, abstract = {

We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's interation, which can be apdated to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0018}, url = {http://global-sci.org/intro/article_detail/ata/17111.html} }
TY - JOUR T1 - On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces AU - Phan , Tuoc AU - Sire , Yannick JO - Analysis in Theory and Applications VL - 2 SP - 111 EP - 127 PY - 2020 DA - 2020/06 SN - 36 DO - http://doi.org/10.4208/ata.OA-0018 UR - https://global-sci.org/intro/article_detail/ata/17111.html KW - Local well-posedness, global well-posedness, dissipative quasi-geostrophic equation, fractional heat equation, mixed-norm Lebesgue spaces. AB -

We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's interation, which can be apdated to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.

Tuoc Phan & Yannick Sire. (2020). On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces. Analysis in Theory and Applications. 36 (2). 111-127. doi:10.4208/ata.OA-0018
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