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An Example in Two Dimensions 291
where ν > 0 is a given parameter associated with the kinematic viscosity, and ∆ is
the Laplace operator defined in (13.1.1). The nonlinear relations (13.4.1) and (13.4.2)
are known as momentum equations, while (13.4.3) is the continuity (or incompressibil-
ity) equation. They express the conservation laws of the momentum and the mass
respectively. We note the analogy between the momentum equations and the Burgers
equation, presented in section 10.4 for the case of one space variable. The functions
Vi, 1 ≤ i ≤ 2, require initial and boundary conditions. For example, by imposing
Vi ≡ 0 on ∂Ω×]0,T], 1 ≤ i ≤ 2, we are specifying a no-slip condition at the bound-
ary. Instead, no initial or boundary conditions are needed for the pressure, which is
determined up to an additive constant.
For a theoretical study of the equations (13.4.1)-(13.4.3), the reader is invited to
consult the books of ladyzhenskaya(1969), temam(1985), kreiss and lorenz(1989)
and the numerous references contained therein. On the subjects of numerical approxi-
mation by finite-differences or finite element methods the bibliography is very rich, and
a comprehensive list of references cannot be included here. We would like to remark
that the numerical investigation of the equations relevant to the physics of fluids has
also been a primary source of algorithms and solution techniques in the field of spectral
methods. Due to the abundance of results, we are unable to mention all the contribu-
tors. A distinguished referring point is the book of canuto, hussaini, quarteroni
and zang(1988), which is specifically addressed to computations in fluid dynamics. The
reader will find there material for further extensions.
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