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Harmonic analysis tools for solving the incompressible Navier-Stokes equations. (English) Zbl 1222.35139
Friedlander, S. (ed.) et al., Handbook of mathematical fluid dynamics. Vol. III. Amsterdam: Elsevier/North Holland (ISBN 0-444-51556-9/hbk). 161-244 (2004).
From the introduction: Formulated and intensively studied at the beginning of the nineteenth century, the classical partial differential equations of mathematical physics represent the foundation of our knowledge of waves, heat conduction, hydrodynamics and other physical problems. Their study prompted further work by mathematical researchers and, in turn, benefited from the application of new methods in pure mathematics. It is a vast subject, intimately connected to various sciences such as physics, mechanics, chemistry, engineering sciences, with a considerable number of applications to industrial problems.
Although the theory of partial differential equations has undergone a great development in the twentieth century, some fundamental questions remain unresolved. They are essentially concerned with the global existence, regularity and uniqueness of solutions, as well as their asymptotic behavior.
The immediate object of this chapter is to review some improvements achieved in the study of a celebrated nonlinear partial differential system, the incompressible Navier-Stokes equations. The nature of a turbulent motion of a fluid, an ocean for instance, or the creation of a vortex inside it, are two typical problems related to the Navier-Stokes equations, and they are still far from being understood.
From a mathematical viewpoint, one of the most intriguing unresolved questions concerning the Navier-Stokes equations and closely related to turbulence phenomena is the regularity and uniqueness of the solutions to the initial value problem. More precisely, given a smooth datum at time zero, will the solution of the Navier-Stokes equations continue to be smooth and unique for all time? This question was posed in 1934 by Leray and is still without answer, neither in the positive nor in the negative. Smale includes the uniqueness and regularity question for the Navier-Stokes equations as one of the 18 open problems of the twentieth century.
There is no uniqueness proof except for over small time intervals and it has been questioned whether the Navier-Stokes equations really describe general flows. But there is no proof for nonuniqueness either.
Maybe a mathematical ingenuity is the reason for the missing (expected) uniqueness result. Or maybe the methods used so far are not pertinent and the refractory Navier-Stokes equations should be approached with a different strategy.
For the entire collection see [Zbl 1051.76002].

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
42B35 Function spaces arising in harmonic analysis
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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