Non Uniqueness of Power-Law Flows

Area of Science:

• Fluid Dynamics
• Nonlinear Partial Differential Equations
• Mathematical Analysis

Background:

• Power-law fluids are crucial in various industrial applications.
• Understanding the mathematical behavior of these fluids is essential for accurate modeling.
• Existing research has explored solutions but gaps remain for certain power indices.

Purpose of the Study:

• To investigate the existence and uniqueness of solutions for power-law fluid models.
• To analyze the ill-posedness of solutions for different ranges of the power-law index.
• To extend the understanding of non-uniqueness in fluid dynamics.

Main Methods:

• Application of the convex integration technique.
• Analysis of Leray-Hopf solutions.
• Investigation of distributional (non-Leray-Hopf) solutions.

Main Results:

• Established non-uniqueness and existence results for power-law fluids in dimension $$.
• Demonstrated ill-posedness for Leray-Hopf solutions when the power index $$ is below the compactness threshold.
• Showed ill-posedness for distributional solutions for a broader class of indices $$.
• Constructed non-unique solutions for every datum in $$.

Conclusions:

• Convex integration provides a powerful tool for studying power-law fluid models.
• The study reveals significant ill-posedness issues in solutions, particularly for specific power-law fluid behaviors.
• Findings extend previous work and highlight the complexity of fluid dynamics for non-Newtonian fluids.