V. Kokilashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON THE SOLVABILITY OF DIVERGENCE EQUATION IN THE THEORY OF INCOMPRESSIBLE FLUIDS

Let Ω ⊂ R^{n}, n ≥ 2, be a bounded domain with Lipschitz boundary and let x_{0} ∈ Ω.

Let p : Ω → R^{1} be a measurable function satisfying the following conditions:

i) 1 < \underline{p} ≤ p(x) ≤ \overline{p} < ∞; ii) |p(x) − p(y)| ≤ \frac{C}{− ln |x−y|}, |x − y| < \frac{1}{2}.

Put p′(x) = \frac{p(x)}{p(x)−1}.

By L_{ρ}^{p(·)}(Ω) we denote a Banach function space L_{ρ}^{p(·)}, i.e., a space of all measurable functions for which

∥fρ∥_{L^{p(·)}} < ∞,

where ρ(x) = |x − x_{0}|^{α}.

For the definition of the norm in L^{p(·)} see, for example, [1]. Define

L^{∘}_{ρ}^{p(·)} = {f ∈ L_{ρ}^{p(·)}(Ω) : ∫_{Ω} f(x) dx = 0}.

By W_{ρ}^{1,p(·)} we denote the weighted Sobolev space. Theorem. Let p satisfy the conditions (i) and (ii). Assume that \frac{1}{p(x_{0})} <

α < \frac{1}{p′(x_{0})}.

Then for each f ∈ L^{∘}_{ρ}^{p(·)} the divergence equation

div u = f

is solvable in vectorial W_{ρ}^{1,p(·)} space and the estimate

∥∇u∥_{L_{ρ}^{p(·)}} ≤ c∥f∥_{L_{ρ}^{p(·)}}

holds.

References

1. L. Diening and M. Ružićka, Calderon–Zygmund operators in generalized Lebesgue spaces L^{p(·)} and problems related to fluid dynamics. Abert-Ludwig Universität Freiburg, Preprint Nr. 21/2002–04.07.2002.