Abstract
We are concerned with the regularity properties for all times of the equation
which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ɛ>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\). If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\), smoothness of initial data persists indefinitely. If 0≦α<α cr, there exist positive constants ν1(α) and ν2(α), depending on the data, such that global regularity persists for ν>ν1(α), whereas, for 0≦ν<ν2(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α cr, similar results hold for the Navier-Stokes equation.
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Communicated by C. Dafermos
This work was performed while the first author was visiting the University of Nice and the second was at the University of Paris Sud-Orsay
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Bardos, C., Penel, P., Frisch, U. et al. Modified dissipativity for a non-linear evolution equation arising in turbulence. Arch. Rational Mech. Anal. 71, 237–256 (1979). https://doi.org/10.1007/BF00280598
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DOI: https://doi.org/10.1007/BF00280598