ISSN:
1432-0673

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics
,
Physics

Notes:
Abstract We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ 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.

Type of Medium:
Electronic Resource

Permalink