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Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space

Chio Chon Kit

Abstract

Based on Dou Huashu's energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to the streamline (i.e., $ u_j \frac{\partial E}{\partial x_j} = 0 $), substituting this critical condition into the NS equations with no-slip boundary conditions leads to the viscous term $ ν\to 0 $. To rigorously analyze the regularity of the solution, Sobolev space $ H_0^1(Ω) $ is introduced for energy estimation. The results show that the velocity field loses $ H^1 $-regularity, and the NS equations degenerate into Euler equations, which admit discontinuous weak solutions. Thus, the position where the mechanical energy gradient is perpendicular to the streamline becomes a weak singularity of the NS equations.

Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space

Abstract

Based on Dou Huashu's energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to the streamline (i.e., ), substituting this critical condition into the NS equations with no-slip boundary conditions leads to the viscous term . To rigorously analyze the regularity of the solution, Sobolev space is introduced for energy estimation. The results show that the velocity field loses -regularity, and the NS equations degenerate into Euler equations, which admit discontinuous weak solutions. Thus, the position where the mechanical energy gradient is perpendicular to the streamline becomes a weak singularity of the NS equations.
Paper Structure (16 sections, 19 equations)

This paper contains 16 sections, 19 equations.

Table of Contents

  1. INTRODUCTION
  2. BASIC DEFINITIONS AND PREMISE CONDITIONS
  3. Total Mechanical Energy
  4. Critical Condition: Mechanical Energy Gradient Perpendicular to Streamline
  5. No-Slip Boundary Condition
  6. Steady, Fully Developed Incompressible NS Equations
  7. SOBOLEV SPACE AND ENERGY ESTIMATION FOUNDATION
  8. Definition of Sobolev Space $H_0^1(\Omega)$
  9. Basic Principle of Energy Estimation
  10. DERIVATION OF VISCOUS TERM $\nu \to 0$
  11. ENERGY ESTIMATION IN SOBOLEV SPACE AND WEAK SINGULARITY DETERMINATION
  12. Energy Estimation and Regularity Analysis
  13. Degeneration of NS Equations and Weak Solutions
  14. Physical Significance of Weak Singularities
  15. CONCLUSION
  16. ...and 1 more sections