Solvability of the heat equation in weighted Sobolev spaces

Wojciech M. Zajączkowski¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland and Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 38 (2011) no. 2, pp. 147-171

Résumé

The existence of solutions to an initial-boundary value problem to the heat equation in a bounded domain in $\Bbb R^3$ is proved. The domain contains an axis and the existence is proved in weighted anisotropic Sobolev spaces with weight equal to a negative power of the distance to the axis. Therefore we prove the existence of solutions which vanish sufficiently fast when approaching the axis. We restrict our considerations to the Dirichlet problem, but the Neumann and the third boundary value problems can be treated in the same way. The proof of the existence is split into the following steps. First by an appropriate extension of initial data the initial-boundary value problem is reduced to an elliptic problem with a fixed $t\in\Bbb R$. Applying the regularizer technique it is considered locally. The most difficult part is to show the existence in weighted spaces near the axis, because the existence in neighbourhoods located at a positive distance from the axis is well known. In a neighbourhood of a point where the axis meets the boundary, the elliptic problem considered is transformed to a problem near an interior point of the axis by an appropriate reflection. Using cutoff functions the problem near the axis is considered in $\Bbb R^3$ with sufficiently fast decreasing functions as $|x|\to\infty$. Then by applying the Fourier–Laplace transform we are able to show an appropriate estimate in weighted spaces and to prove local in space existence. The result of this paper is necessary to show the existence of global regular solutions to the Navier–Stokes equations which are close to axially symmetric solutions.

Détail

DOI : 10.4064/am38-2-2

Keywords: existence solutions initial boundary value problem heat equation bounded domain bbb proved domain contains axis existence proved weighted anisotropic sobolev spaces weight equal negative power distance axis therefore prove existence solutions which vanish sufficiently fast approaching axis restrict considerations dirichlet problem neumann third boundary value problems treated proof existence split following steps first appropriate extension initial initial boundary value problem reduced elliptic problem fixed bbb applying regularizer technique considered locally difficult part existence weighted spaces near axis because existence neighbourhoods located positive distance axis known neighbourhood point where axis meets boundary elliptic problem considered transformed problem near interior point axis appropriate reflection using cutoff functions problem near axis considered bbb sufficiently fast decreasing functions infty applying fourier laplace transform able appropriate estimate weighted spaces prove local space existence result paper necessary existence global regular solutions navier stokes equations which close axially symmetric solutions