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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

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We establish local and global well-posedness of the 2D dissipative quasi-geostrophic 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 iteration, which can be adapted 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.

  • AMS Subject Headings

35A01, 35K55, 35K61

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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 quasi-geostrophic 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 iteration, which can be adapted 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 quasi-geostrophic 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 iteration, which can be adapted 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.

Phan , Tuoc and Sire , Yannick. (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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