Regularity of the solution of the Prandtl equation

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to a boundary value problem for the Prandtl equation, introducing new functional spaces and establishing regularity results under specific conditions.

Contribution

It introduces a new scale of function spaces for analyzing the Prandtl equation and proves regularity and solvability results in these spaces.

Findings

Existence and uniqueness of solutions in the spaces ^s(-1,1) for 0 e2 s e2 1.

Regularity results for solutions when f satisfies certain weighted L2 conditions.

Conditions on p(x) ensuring the solvability of the boundary value problem.

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