Volume 35, Issue 3
Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework

J. Part. Diff. Eq., 35 (2022), pp. 289-306.

Published online: 2022-06

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• Abstract

In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the  global existence and uniqueness of solutions to 2D Prandtl-Hartmann  equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann  equations can be extended up to $T_\varepsilon$ (see Theorem 2.1) when  the strength of the perturbation is of the order of $\varepsilon$. The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of $x$-derivative in the term $v\partial_yu$. To overcome this difficulty, we introduce the Gaussian weighted Poincar$\acute{e}$ inequality (see Lemma 2.3). Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework, either.

• Keywords

Prandtl-Hartmann existence and uniqueness energy method analytic space.

76N20, 35Q35, 35A02

xld0908@163.com (Xiaolei Dong)

yuming_qin@hotmail.com (Yuming Qin)

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@Article{JPDE-35-289, author = {Xiaolei and Dong and xld0908@163.com and 23982 and College of Information Science and Technology, Donghua University, Shanghai 201620, China and Xiaolei Dong and Yuming and Qin and yuming_qin@hotmail.com and 11579 and Department of Applied Mathematics, Donghua University, Shanghai 201620, China and Yuming Qin}, title = {Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {35}, number = {3}, pages = {289--306}, abstract = {

In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the  global existence and uniqueness of solutions to 2D Prandtl-Hartmann  equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann  equations can be extended up to $T_\varepsilon$ (see Theorem 2.1) when  the strength of the perturbation is of the order of $\varepsilon$. The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of $x$-derivative in the term $v\partial_yu$. To overcome this difficulty, we introduce the Gaussian weighted Poincar$\acute{e}$ inequality (see Lemma 2.3). Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework, either.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n3.7}, url = {http://global-sci.org/intro/article_detail/jpde/20777.html} }
TY - JOUR T1 - Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework AU - Dong , Xiaolei AU - Qin , Yuming JO - Journal of Partial Differential Equations VL - 3 SP - 289 EP - 306 PY - 2022 DA - 2022/06 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n3.7 UR - https://global-sci.org/intro/article_detail/jpde/20777.html KW - Prandtl-Hartmann KW - existence and uniqueness KW - energy method KW - analytic space. AB -

In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the  global existence and uniqueness of solutions to 2D Prandtl-Hartmann  equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann  equations can be extended up to $T_\varepsilon$ (see Theorem 2.1) when  the strength of the perturbation is of the order of $\varepsilon$. The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of $x$-derivative in the term $v\partial_yu$. To overcome this difficulty, we introduce the Gaussian weighted Poincar$\acute{e}$ inequality (see Lemma 2.3). Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework, either.

Xiaolei Dong & Yuming Qin. (2022). Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework. Journal of Partial Differential Equations. 35 (3). 289-306. doi:10.4208/jpde.v35.n3.7
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