Local well-posedness of solutions to 2D mixed Prandtl equations in Sobolev space without monotonicity and lower bound (Q6608359)

The paper is devoted to the Prandtl-Shercliff system \N\[\begin{aligned}\N \partial_t u+u\partial_x u+v\partial_y u -\partial_y^2 u & =\partial_x b, \\ \N\partial_x u+\partial_y^2 b& =0, \\ \N\partial_x u+\partial_y v & =0\N\end{aligned}\]\Nin \( R^2_+\times (0,T)\) with the initial datum \( u|_{t=0}=u_0\), the no-slip boundary conditions \( (u,v,b)|_{y=0}=0\) and far field boundary conditions \( (u,b)\to (\tilde u,\tilde b)\) as \( y\to \infty \). The authors prove the unique solvability of the problem under the assumption \( u\in L^\infty ([0,T];H^4(R^2_+))\), \( (v,b)\in L^\infty ([0,T];L^\infty (R_+\times H^3(R)))\), \( u\in L^2([0,T];H^3(R^2_+))\).