Abstract
SupposeU is an open bounded subset of 3-space such that the boundary ofU has Lebesgue measure zero. Then for any initial condition with finite kinetic energy we can find a global (i.e. for all time) weak solutionu to the time dependent Navier-Stokes equations of incompressible fluid flow inU such that the curl ofu is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.
Similar content being viewed by others
References
Almgren, F.J., Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Memoirs of the American Mathematical Society 165. Providence, R.I.: American Mathematical Society 1976
Federer, H.: Geometric measure theory. Berlin, Heidelberg, New York: Springer 1969
Leray, J.: Acta Math.63, 193–248 (1934)
Mandelbrot, B.: Intermittent turbulence and fractal dimension kurtosis and the spectral exponent 5/3+B. In: Turbulence and Navier-Stokes equation. Lecture Notes in Mathematics, Vol. 565. Berlin, Heidelberg, New York: Springer 1976
Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys.55, 97–112 (1977)
Scheffer, V.: The Navier-Stokes equations in space dimension four. Commun. Math. Phys.61, 41–68 (1978)
Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math.66, 535–552 (1976)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton: Princeton University Press 1970
Stein, E.M., Weiss, G.L.: Introduction to fourier analysis on euclidean spaces. Princeton: Princeton University Press 1971
Author information
Authors and Affiliations
Additional information
Communicated by J. Glimm
This research was supported in part by the National Science Foundation Grant MCS-7903361
Rights and permissions
About this article
Cite this article
Scheffer, V. The Navier-Stokes equations on a bounded domain. Commun.Math. Phys. 73, 1–42 (1980). https://doi.org/10.1007/BF01942692
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01942692