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10 The Incompressible Navier-Stokes Equations Under Initial and Boundary Conditions

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62312-0
  • Mathematics


Publisher Summary This chapter discusses the linearized equations under boundary conditions. Boundary conditions are chosen in such a way that the theory for mixed parabolic-hyperbolic equations applies for every fixed Ɛ > 0. The boundary contributions appear from integration by parts in the x-variable. There is no restriction to assume the Dirichlet conditions to be homogeneous. If inhomogeneous data are given, one can introduce new variables which satisfy the homogeneous condition, and if one differentiates with respect to y and t, the condition remains homogeneous. Some elementary estimates are proved in the chapter for solutions of Laplace's equation and Poisson's equation in the strip. The linearized system is not of the standard form of an evolution equation as there is no equation for pt. The chapter also reviews the convergence theorem.

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