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Navier–Stokes Equations

In: Algebraic Approaches to Partial Differential Equations

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  • Xiaoping Xu

    (Academy of Mathematics and System Science, Institute of Mathematics)

Abstract

In this chapter, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent spatial variables and a moving-frame method to solve the three-dimensional Navier–Stokes equations. Seven families of unsteady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blows up on a moving plane, which may be used to study abrupt high-speed rotating flows. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in the study of shock waves. Another family of solutions is partially cylindrical invariant, containing two parameter functions of t, which may be used to describe an incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spatial variables.

Suggested Citation

  • Xiaoping Xu, 2013. "Navier–Stokes Equations," Springer Books, in: Algebraic Approaches to Partial Differential Equations, edition 127, chapter 0, pages 269-316, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-36874-5_9
    DOI: 10.1007/978-3-642-36874-5_9
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