Regularity of the Solution of the Prandtl Equation
Tóm tắt
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
$$r(x)=1-x^2$$
.
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