SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HOLDER-alpha DOMAINS

作者：Duran Ricardo G^*; Lopez Garcia Fernando

来源：Mathematical Models and Methods in Applied Sciences, 2010, 20(1): 95-120.

DOI：10.1142/S0218202510004167

• Divergence operator
• Holder-alpha domains
• Stokes equations

If Omega subset of R(n) is a bounded domain, the existence of solutions u is an element of H(0)(1)(Omega)(n) of div u = f for f is an element of L(2)(Omega) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution (u, p) is an element of H(0)(1)(Omega)(n) x L(2)(Omega), where u is the velocity and p the pressure.
It is known that the above-mentioned result holds when Omega is a Lipschitz domain and that it is not valid for arbitrary Holder-alpha domains.
In this paper we prove that if Omega is a planar simply connected Holder-alpha domain, there exist solutions of div u f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal.
For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u, p) is an element of H(0)(1)(Omega)(n) x L(2)(Omega) for some r < 2 depending on the power of the cusp.

• 出版日期2010-1