Regularity of the Solution of the Prandtl Equation

Solvability and regularity of the solution of the Dirichlet problem for the Prandtl equation
$$ \frac{u(x)}{p(x)}-\frac{1}{2\pi}\int_{-1}^1\frac{u'(t)}{t-x}\,dt=f(x) $$
is studied. Here $p(x)$ is a positive function on $(-1,1)$ such that $\sup(1-x^2)/p(x)<\infty$. We introduce the scale of spaces $\widetilde H^s(-1,1)$ in terms of the special integral transformation on the interval $(-1,1)$. We obtain theorems about the existence and uniqueness of the solution in the classes $\widetilde H^{s}(-1,1)$ with $0\le s\le 1$. In particular, for $s=1$ the result is as follows: if $r^{1/2}f\in L_2$, then $r^{-1/2}u,r^{1/2}u'\in L_2$, where $r(x)=1-x^2$.