Model for Aqueous Polymer Solutions with Damping Term: Solvability and Vanishing Relaxation Limit

Area of Science:

• Fluid Dynamics
• Nonlinear Partial Differential Equations
• Polymer Physics

Background:

• Investigating the steady-state flow of low-concentrated aqueous polymer solutions is essential for understanding complex fluid behavior.
• The inclusion of a damping term and no-slip boundary conditions presents significant mathematical challenges.

Purpose of the Study:

• To establish the solvability of a steady-state flow model for aqueous polymer solutions with a damping term.
• To define and prove the existence of a full weak solution in the distributions sense.
• To analyze the convergence of solutions to the damped Navier-Stokes system.

Main Methods:

• Utilizing the method of auxiliary viscosity.
• Applying the acute angle theorem for generalized monotone nonlinear operators.
• Employing the Krasnoselskii theorem for superposition operators in Lebesgue spaces.

Main Results:

• Sufficient conditions for the existence of a full weak solution satisfying an energy inequality were derived.
• The existence of a unique solution in the distributions sense was demonstrated.
• Convergence of the solutions to the steady-state damped Navier-Stokes system was proven as relaxation viscosity approaches zero.

Conclusions:

• The study successfully demonstrates the existence of weak solutions for the investigated nonlinear partial differential equations.
• The findings provide a rigorous mathematical foundation for modeling damped polymer fluid flow.
• This work contributes to the theoretical understanding of complex fluids and their dynamics.