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 ε (see Theorem 2.1) when the strength of the perturbation is of the order of ε . The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of x -derivative in the term v ∂ y u . To overcome this difficulty, we introduce the Gaussian weighted Poincar´ e inequality (see Lemma 2.3). Com-pared 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.