On a parabolic p-Laplacian system with a convective term

Abstract: 

In the classical theory of fluid mechanics, Newtonian fluids are characterized by a linear relationship between the stress tensor and the symmetric part of the velocity gradient, leading to the standard Navier-Stokes model. However, many complex materials, such as polymers, gels, and certain biological fluids, exhibit nonlinear rheological behavior better described by power-law models, where the viscosity depends on the magnitude of the shear rate. In this framework, the case p<2 corresponds to shear-thinning fluids, whose effective viscosity decreases as the shear rate increases. These nonlinear models naturally lead to evolutionary systems involving the p-Laplacian operator or its variants, and introduce analytical challenges not present in the Newtonian setting.
In [1] and [2], we study the well-posedness of a parabolic p-Laplacian system with a convective term, derived from the power-law system in the subquadratic case (p<2), by replacing the symmetric gradient with the full gradient and eliminating the pressure term. It should be noted that these modifications make the results less relevant from a Fluid Dynamics perspective, since the corresponding \textit{constitutive law} does not comply with the principle of material invariance. Nevertheless, they are useful to better delimit the expectations for possible results in the fluid dynamics context.
We establish existence and a maximum principle for regular solutions (for p \in (3/2, 2) and weak solutions (for p \in (1, 2)) for an initial datum v_0 (x) \in L^\infty ( Ω ); for regular solutions we analyze the property of extinction in a finite time under suitable smallness assumptions on the initial datum. Moreover, for  v_0 (x) \in L^\infty ( Ω )\cap W^{1,2}_0( Ω ), we are able to prove the uniqueness of regular solutions for p \in (5/3, 2).

The talk is based on two joint works with Francesca Crispo and Michael M. Růžička.

[1] F. Crispo,  A.P. Di Feola, On a parabolic p-Laplacian system with a convective term, Annali di Matematica Pura ed Applicata (1923 -), 204, (2025), no.3, 1119--1146.
[2] A.P. Di Feola, M. Růžička, Existence of global weak solutions to a parabolic p-Laplacian problem with convective term, arXiv:2510.05847, (2025).