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 $$r(x)=1-x^2$$ .