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, existence and uniqueness, energy method, 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 PoincarĂ© 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.