Geodesic
Regularity of the Solution of the Prandtl Equation
Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 550-568.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $r(x)=1-x^2$.
Keywords: Prandtl equation, weak solution, Fourier integral transformation, integral transformation on the interval. 
@article{MZM_2021_110_4_a5,
     author = {V. \`E. Petrov and T. A. Suslina},
     title = {Regularity of the {Solution} of the {Prandtl} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {550--568},
     publisher = {mathdoc},
     volume = {110},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a5/}
}
TY  - JOUR
AU  - V. È. Petrov
AU  - T. A. Suslina
TI  - Regularity of the Solution of the Prandtl Equation
JO  - Matematičeskie zametki
PY  - 2021
SP  - 550
EP  - 568
VL  - 110
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a5/
LA  - ru
ID  - MZM_2021_110_4_a5
ER  - 
%0 Journal Article
%A V. È. Petrov
%A T. A. Suslina
%T Regularity of the Solution of the Prandtl Equation
%J Matematičeskie zametki
%D 2021
%P 550-568
%V 110
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a5/
%G ru
%F MZM_2021_110_4_a5
V. È. Petrov; T. A. Suslina. Regularity of the Solution of the Prandtl Equation. Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 550-568. http://geodesic.mathdoc.fr/item/MZM_2021_110_4_a5/

[1] V. V. Golubev, Lektsii po teorii kryla, Gostekhizdat, M., 1949

[2] K. Stewartson, “A note on lifting line theory”, Quart. J. Mech. Appl. Math., 13 (1960), 49–56 | DOI | MR

[3] I. P. Krasnov, Raschetnye metody sudovogo magnetizma i elektrotekhniki, Sudostroenie, L., 1986

[4] A. I. Kalandiya, Matematicheskie metody dvumernoi uprugosti, Nauka, M., 1973 | MR

[5] J. I. Diaz, D. Gomez-Castro, J. L. Vazquez, “The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach”, Nonlinear Anal., 177, Part A (2018), 325–360 | DOI | MR

[6] M. M. Fall, “Regularity estimates for nonlocal Schrödinger equations”, Discrete Contin. Dyn. Syst., 39:3 (2019), 1405–1456 | MR

[7] V. E. Petrov, “Integralnoe preobrazovanie na otrezke”, Problemy matem. analiza, 31 (2005), 67–95

[8] V. E. Petrov, “Obobschennoe uravnenie Trikomi, kak uravnenie svertki”, Dokl. AN, 411:2 (2006), 173–177 | MR

[9] I. V. Andronov, V. E. Petrov, “Diffraction by an impedance strip at almost grazing incidence”, IEEE Trans. Antennas and Propagation, 64:8 (2016), 3562–3572 | DOI | MR

[10] V. E. Petrov, “O tochnykh resheniyakh uravnenii Gankelya”, Algebra i analiz, 30:1 (2018), 170–207 | MR

[11] Zh.-L. Lions, E. Madzhenes, Neodnorodnye granichnye zadachi i ikh prilozheniya, T. 1, Mir, M., 1971 | MR | Zbl