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On the existence of the pressure for solutions of the variational Navier-Stokes equations. (English) Zbl 0961.35107

The existence of a velocity-pressure solution of the Navier-Stokes equations in the (real) space domain, whose dimension is greater than or equal to 3, has been proved by many authors for a convenient right-hand side in two steps: first, one proves the existence of a velocity weak solution by solving a weak formulation where the pressure is eliminated and, second, one finds the corresponding pressure by using the de Rham’s theorem or similar arguments. It is known that the existence of the velocity weak solution can be proved even for a more general right-hand side. In this paper it is proved that, in general, the corresponding pressure (field associated with the already determined velocity) doesn’t exist. The main argument for the demonstration is that the functional spaces, to which respectively the right-hand and the left-hand sides belong, can not be imbedded into the same Hausdorff space.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
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