Full Text: PDF
DOI: 10.23952/cot.2023.5
Received July 21, 2022; Accepted October 21, 2022; Published February 4, 2023
Abstract. We consider a quasilinear parabolic equation and its associate stationary problem which correspond to a simplified formulation of a Bingham flow and we mainly study two qualitative properties. The first one concerns with the absence and, respectively, disappearance in finite time, of the movement. We show that there is a suitable balance between the -norm of the forcing datum
and the measure of the spatial domain
(essentially saying that the forcing daum must be small enough) such that the corresponding solution
of the stationary problem is such that
a.e. in
(even if
). Moreover, if
is also the forcing term of the parabolic problem, and if the above mentioned balance is strict, for any
there exists a finite time
such that the unique solution
of the parabolic problem globally stops after
in the sense that
a.e. in
, for any
The second property concerns with the formation of a positively measure “solid region”. We show that if the above balance condition fails (i.e., when the forcing datum is large enough) then the solution
of the stationary problem satisfies that
in
and its “solid region” (defined as the set
) has a positive measure. Similar results are obtained for the symmetric solutions
of the parabolic problem. In addition, the convergence
in
, as
does not take place in any finite time.
How to Cite this Article:
G.R. Cirmi, J.I. Dîaz, Qualitative properties of solutions of some quasilinear equations related to Bingham fluids, Commun. Optim. Theory 2023 (2023) 5.