We prove, on one hand, that for a convenient body force with values in the distribution space ( H -1 ( D )) d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V ’ of the divergence free subspace V of ( H 1 0 ( D )) d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.
Applied Mathematics and Optimization – Springer Journals
Published: Oct 1, 2003
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