A variational view on constitutive laws in parabolic problems

TL;DR

This paper develops a variational framework for solving parabolic problems, focusing on $ ext{A}$-quasiconvexity for non-homogeneous operators, and applies it to non-Newtonian Navier--Stokes equations.

Contribution

It introduces a variational approach based on $ ext{A}$-quasiconvexity for anisotropic operators, extending solution concepts to complex non-Newtonian fluid models.

Findings

Established existence results for variational solutions.

Applied theory to non-Newtonian Navier--Stokes equations.

Provided a new perspective on anisotropic parabolic problems.

Abstract

We consider a variational approach to solve parabolic problems by minimising a functional over time and space. To achieve existence results we investigate the notion of $\mathscr{A}$-quasiconvexity for non-homogeneous operators in anisotropic spaces. The abstract theory is then applied to formulate a variational solution concept for the non-Newtonian Navier--Stokes equations.