Showing papers by "Peter Constantin published in 1997"
TL;DR: For the periodic 2D Navier-Stokes equations, this article showed that the set of initial data for which the solution exists for all negative times and has exponential growth is rather rich.
34 citations
TL;DR: In this article, the Littlewood-Paley energy spectrum was introduced and it was shown that k−3 is an upper bound for the energy spectrum in two dimensions and that certain details of the spectrum of the driving forces can be recovered from energy spectrum.
Abstract: We introduce the Littlewood–Paley energy spectrum and prove that k−3 is an upper bound for it in two dimensions. We also show that certain details of the spectrum of the driving forces can be recovered from the energy spectrum.
33 citations
TL;DR: In this paper, the existence and uniqueness of both spatial and space-time statistical solutions of the Navier-Stokes equations on the phase space of vorticity were proved using the methods of Foias and Vishik-Fursikov.
Abstract: Using the methods of Foias [Sem. Math. Univ. Padova 48, 219–343 (1972); 49, 9–123 (1973)] and Vishik–Fursikov [Mathematical Problems of Statistical Hydromechanics (Kluwer, Dordrecht, 1988)], we prove the existence and uniqueness of both spatial and space–time statistical solutions of the Navier–Stokes equations on the phase space of vorticity. Here the initial vorticity is in Yudovich space and the initial measure has finite mean enstrophy. We show under further assumptions on the initial vorticity that the statistical solutions of the Navier–Stokes equations converge weakly and the inviscid limits are the corresponding statistical solutions of the Euler equations.
26 citations
TL;DR: In 1934 J. Leray proved that Navier–Stokes equations—the partial differential equations used to describe viscous incompressible fluids—admit global weak solutions, and did not preclude the possibility that, starting from smooth data, a singularity might form in finite time.
Abstract: In 1934 J. Leray (1) proved that Navier–Stokes equations—the partial differential equations used to describe viscous incompressible fluids—admit global weak solutions. The term global refers to the fact that there are no restrictions, neither in the size of the initial data nor in the length of time these solutions persist. The term weak is technical and refers to the smoothness of these solutions: they are not very smooth. Leray’s work did not preclude the possibility that, starting from smooth data, a singularity might form in finite time. Sixty-three years later we still don’t know. The spontaneous generation and subsequent evolution of singularities in smooth …
1 citations
01 Apr 1997
TL;DR: In this article, the Navier-Stokes equations are used to describe some of the mathematical problems encountered in the study of incompressible fluid turbulence and the equations of motion are described.
Abstract: In this paper I would like to describe some of the mathematical problems encountered in the study of incompressible fluid turbulence. The equations of motion are the Navier-Stokes equations.
$$ \left( {{\partial _t} + u \cdot
abla - v\Delta } \right)u +
abla p = f. $$
01 Jan 1997
TL;DR: In this paper, the dependence of the Nusselt number upon Rayleigh number was studied and it was shown that the exponent 2/7 is obtained if a certain quantitative condition holds; this condition can be interpreted as one of relative horizontal smoothnes of time averaged temperature profiles or as a separation between the thermal and viscous boundary layers.
Abstract: We discuss rigorous results regarding the dependence of the Nusselt number upon Rayleigh number. We find that the exponent 2/7 is obtained if a certain quantitative condition holds; this condition can be interpreted as one of relative horizontal smoothnes of time averaged temperature profiles or as a separation between the thermal and viscous boundary layers.