The Stationary Navier—Stokes System

Abstract

Let Ώbe a bounded open set of ℝⁿ. Like for the Stokes problem one is looking for a couple

$$ \left( {u,p} \right) $$

((14.1))

representing respectively the velocity of a fluid and its pressure such that

$$ \left\{ \begin{gathered} - \mu \Delta u + \left( {u \cdot \nabla } \right)u + \nabla p = f in \Omega , \hfill \\ div u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} \right. $$

((14.2))

μ is the viscosity of the fluid. Note that here one is taking into account the nonlinear effect. The operator u · ∇ is defined as

$$ u^i \partial _{x_i } $$

((14.3))

with the summation convention in i. We will restrict ourselves to the physical relevant cases - i.e., n = 2 or 3. We refer the reader to [45], [56] for a physical background on the problem (see also [51], [52], [68]). Eliminating the pressure as we did in the preceding chapter we are reduced to find u such that

$$ \left\{ \begin{gathered} u \in \widehat{\mathbb{H}_0^1 }\left( \Omega \right), \hfill \\ \mu \int\limits_\Omega {\nabla u \cdot \nabla vdx} + \int\limits_\Omega {\left( {u \cdot \nabla } \right)u \cdot vdx} = \int\limits_\Omega {f \cdot v \forall v \in \widehat{\mathbb{H}_0^1 }\left( \Omega \right)} , \hfill \\ \end{gathered} \right. $$

((14.4))

where $\widehat{\mathbb{H}_0^1 }$(Ώ) is defined by (13.38). In order to do that we introduce the trilinear form defined as

$$ t\left( {w;u,v} \right) = \int\limits_\Omega {\left( {w \cdot \nabla } \right)u \cdot vdx} = \int\limits_\Omega {w^i \partial _{x_i } u^j v^j dx} $$

((14.5))

(with the summation convention in i, j). One should notice that if

$$ w^i ,\partial _{x_i } u^j ,v^j \in L^2 \left( \Omega \right) $$

it is not clear that the product above is integrable. This will follow however for w,v ∈ ℍ ¹₀ (Ώ) from the Sobolev embedding theorem.