Regularity of the solution of the Prandtl equation

Abstract

Solvability and regularity of the solution of the Dirichlet problem for the Prandtl equation $$\frac{u(x)}{p(x)}-\frac{1}{2\pi }{\int }_{-1}^1\frac{u^{\prime }(t)}{t-x}dt=f(x)$$ 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 ${\tilde{H}}^s(-1,1)$ 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 ${\tilde{H}}^s(-1,1)$ 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 $r^{-1/2}u,r^{1/2}{u}^{\prime }\in L_2$, where $r(x)=1-x^2$.

17 pages