Existence of global weak solutions to a parabolic $p$-Laplacian problem with convective term

TL;DR

The paper proves the global existence of weak solutions to the parabolic $p$-Laplacian system with a convective term on a bounded domain for all $p\in(1,2)$. It constructs approximations with vanishing viscosity $\nu$ and a mollified convective term $J_\mu(v)\cdot\nabla v$, obtaining local solutions via Bochner pseudo-monotone theory and a duality-based maximum principle. By passing to the limits $\nu\to0^+$ and then $\mu\to0^+$ (with Minty’s trick to identify the nonlinear limits), it establishes a global weak solution that satisfies a maximum principle. This extends previous results to the full subquadratic range $p\in(1,2)$ and provides a robust framework for subquadratic parabolic $p$-Laplacian systems with convective effects, including implications for power-law fluid models.

Abstract

For a given bounded domain $Ω\subset \mathbb R^3$, with $C^2$ boundary, and a given instant of time $T>0$, we prove the existence of a global weak solution on $(0,T)$, which satisfies a maximum principle, to a parabolic $p$-Laplacian system with convective term without divergence constraint for any $p\in (1,2)$.