Abstract
Existence of weak solutions of the three-dimensional Navier-Stokes problem was first proved by J. Leray in the case of the Cauchy problem, cf. Leray (1934). As is well known, these solutions are important in that they are the only solutions which, so far, are known to exist for all times, without restriction on the data. Unfortunately, however, the question of whether they are classical, in an ordinary sense, is still open, even though partial conclusions regarding regularity are avilable: Leray (1934), Scheffer (1976, 1980), Caffarelli, Kohn, & Nirenberg (1982). Subsequently, E. Hopf (1951), using a different technique, constructed weak solutions for a general initial-boundary value problem. However, hopf’s solution (even for the Cauchy problem) has weaker properties than Leray’s solution. Among others, we refer to the “energy inequality” for the velocity field u(x, t), i.e.
for s = 0, for almost all s > 0 and for all t ≧ s.
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To James Serrin on his 60th birthday
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Galdi, G.P., Maremonti, P. (1989). Monotonic Decreasing and Asymptotic Behavior of the Kinetic Energy for Weak Solutions of the Navier-Stokes Equations in Exterior Domains. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_15
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DOI: https://doi.org/10.1007/978-3-642-83743-2_15
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