Local Well-Posedness of Solutions to 2D Magnetic Prandtl Model in the Prandtl-Hartmann Regime

In this chapter, we shall consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by the energy method in a polynomial weighted Sobolev space. On the one hand, we have noted that the x-derivative of the pressure P plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable P ( \(\partial _x P<0\) ) or the case of \(\partial _x P=0\) (led by constant outer flow \(U=\text{constant} \) ) was only considered. While in this chapter, we have no any restriction on the sign of \(\partial _x P\) , which has generalized all previous results and gives definitely rise to a difficulty in mathematical treatments. To overcome this difficulty, we shall use the skill of cancellation mechanism which is valid under the monotonicity assumption. One the other hand, we shall consider the general outer flow \(U\neq \text{constant}\) , leading to the boundary data at \(y=0\) being much more complicated. To deal with these boundary data, some more delicate estimates and mathematical induction method will be used. Moreover, our result has given us a physical understanding that the outer flow has a stabilizing effect on the Prandtl-Hartmann regime boundary layer in mathematics. The content of this chapter is selected from Qin and Wang (Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime, submitted).