AT-coercivity approach to the nonlinear Stokes equations

We address the nonlinear Stokes problem with Dirichlet boundary conditions, introducing additional variables into the standard formulation to accommodate solutions with reduced regularity requirements. To ground this analysis, we first review relevant preliminary results, emphasizing the significance of achieving T -coercivity in the context of nonlinear Stokes flows. We then introduce a specially designed operator T , proving its bijectivity and showing that it induces coercivity when applied to the test function space. This result provides a rigorous foundation for solving the quasi- Newtonian Stokes problem with minimal regularity constraint and also sets up the T -coercivity as an alternative to the well-posedness of the nonlinear Stokes problems.