Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework

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 Poincaré 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.