Regularity of the solution of the Prandtl equation

Abstract

Solvability and regularity of the solution of the Dirichlet problem for the Prandtl equation $$ {u(x)\over p(x)}- {1\over 2\pi}\int_{-1}^1 {u'(t) \over t-x} \,dt = f(x) $$ is studied. It is assumed that $p(x)$ is a positive function on $(-1,1)$ such that $\sup \frac{(1-x^2)}{ p(x)} < \infty$. 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 theorem about 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 $r(x)=1-x^2$.