By Luc Tartar

The* advent to Navier-Stokes Equation and Oceanography* corresponds to a graduate direction in arithmetic, taught at Carnegie Mellon collage within the spring of 1999. reviews have been additional to the lecture notes allotted to the scholars, in addition to brief biographical details for all scientists pointed out within the textual content, the aim being to teach that the construction of medical wisdom is a global firm, and who contributed to it, from the place, and while. The aim of the path is to coach a severe viewpoint about the partial differential equations of continuum mechanics, and to teach the necessity for constructing new tailored mathematical tools.

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Charles Louis FEFFERMAN, American mathematician, born in 1949. He received the Fields Medal in 1978. He works at Princeton University, Princeton, NJ. 22 4 Sobolev spaces I say that curl u ∈ L∞ and div u = 0 imply that u ∈ Λ1 , and that means42,43,44 that there exists a constant M such that |u(x+h)+u(x−h)−2u(x)| ≤ M |h| for all x, h, and this implies |u(x + h) − u(x)| ≤ C |h| log(|h|) for |h| small, and OSGOOD’s variant applies. [Taught on Wednesday January 20, 1999. ]45 42 43 44 45 This is the deﬁnition that I heard from Yves MEYER, and I noticed then that it is an interpolation space, for example (C 0,α , C 1,β )θ,∞ with 0 < α, β < 1 and (1 − θ)α + θ β = 1.

HALLEY14,15 had already understood the opposite eﬀects of the Sun and of gravity in 1686: the Sun creates a horizontal variation, heating the waters near the equator much more than near the poles, while gravity likes a vertical variation, drawing the cold waters to the bottom and making the warm waters rise to the surface; the competition between these two eﬀects creates convection. MARSIGLI16 had already studied in 1679 the existence of an undercurrent in the depth of the Bosphorus, created by diﬀerence in salinity between the Mediterranean and the Black Sea (see Lecture 3).

WEISS, Italian-born mathematician, born in 1928. He works at WASHINGTON University, St Louis, MO. George WASHINGTON, American general, 1732–1799. First President of the United States of America, 1789–1797. Charles Louis FEFFERMAN, American mathematician, born in 1949. He received the Fields Medal in 1978. He works at Princeton University, Princeton, NJ. 22 4 Sobolev spaces I say that curl u ∈ L∞ and div u = 0 imply that u ∈ Λ1 , and that means42,43,44 that there exists a constant M such that |u(x+h)+u(x−h)−2u(x)| ≤ M |h| for all x, h, and this implies |u(x + h) − u(x)| ≤ C |h| log(|h|) for |h| small, and OSGOOD’s variant applies.