Giuseppa Rita Cirmi, Jésus Ildefonso Dîaz, Qualitative properties of solutions of some quasilinear equations related to Bingham fluids, Vol. 2023 (2023), Article ID 5, pp. 1-20

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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 $L^1$ -norm of the forcing datum $f_{\infty}$ and the measure of the spatial domain $\Omega$ (essentially saying that the forcing daum must be small enough) such that the corresponding solution $u_{\infty }(x)$ of the stationary problem is such that $u_{\infty}\equiv 0$ a.e. in $\Omega$ (even if $f_{\infty }\neq 0$ ). Moreover, if $f_{\infty }$ is also the forcing term of the parabolic problem, and if the above mentioned balance is strict, for any $u_{0}\in L^{\infty }(\Omega )$ there exists a finite time $T_{u_{0},f_{\infty }}>0$ such that the unique solution $u(t,x)$ of the parabolic problem globally stops after $T_{u_{0},f_{\infty }},$ in the sense that $u(t,x)\equiv 0$ a.e. in $\Omega$ , for any $t\geq T_{u_{0},f_{\infty }}.$ 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 $u_{\infty }(x)$ of the stationary problem satisfies that $u_{\infty }\neq 0$ in $\Omega$ and its “solid region” (defined as the set $\mathcal{S(}u_{\infty })=$ $\{x\in \Omega :\nabla u_{\infty }(x)=0\}$ ) has a positive measure. Similar results are obtained for the symmetric solutions $u(t)$ of the parabolic problem. In addition, the convergence $u(t)\rightarrow u_{\infty }$ in $H_{0}^{1}(\Omega )$ , as $t\rightarrow +\infty,$ 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.