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Exponential stability of the linearized viscous Saint-Venant equations using a quadratic Lyapunov function

Amaury Hayat, Nathan Lichtlé

Abstract

In this work, we investigate the exponential stability of the viscous Saint-Venant equations by adding to the standard hyperbolic Saint-Venant equations a viscosity term coming from the higher order approximation of the Saint-Venant equations from Navier-Stokes equations. The inclusion of viscosity transforms these equations into more complex second-order partial differential equations, accurately modeling the behavior of real-world fluids that inherently possess viscosity. We construct an explicit quadratic Lyapunov function and demonstrate that it must be diagonal in physical coordinates, revealing that certain quadratic Lyapunov functions effective in non-viscous cases become inadequate when viscosity is introduced. We find explicit sufficient conditions on the parameters of the boundary conditions such that for small viscosities a quadratic Lyapunov function exists. This result ensures the exponential stability of the linearized system around the steady-state solutions in the $L^2$ norm.

Exponential stability of the linearized viscous Saint-Venant equations using a quadratic Lyapunov function

Abstract

In this work, we investigate the exponential stability of the viscous Saint-Venant equations by adding to the standard hyperbolic Saint-Venant equations a viscosity term coming from the higher order approximation of the Saint-Venant equations from Navier-Stokes equations. The inclusion of viscosity transforms these equations into more complex second-order partial differential equations, accurately modeling the behavior of real-world fluids that inherently possess viscosity. We construct an explicit quadratic Lyapunov function and demonstrate that it must be diagonal in physical coordinates, revealing that certain quadratic Lyapunov functions effective in non-viscous cases become inadequate when viscosity is introduced. We find explicit sufficient conditions on the parameters of the boundary conditions such that for small viscosities a quadratic Lyapunov function exists. This result ensures the exponential stability of the linearized system around the steady-state solutions in the norm.
Paper Structure (12 sections, 7 theorems, 134 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the viscous Saint-Venant system and the linearized system
  3. Main results
  4. Exponential stability of the linearized system
  5. Integral term
  6. Boundary term
  7. Numerical Simulations
  8. Conclusion
  9. Proof that $Q$ must be diagonal in physical coordinates
  10. Derivation of the Lyapunov function
  11. Existence of the steady-state and asymptotic estimation of $V^{*}_{x}$ and $V^{*}_{xx}$
  12. Well-posedness of the linear system

Key Result

Proposition 2.1

For any $H_{0}>0$, $V_{0}>0$ satisfying there exists a $\mu^{*}>0$ (depending on $H_{0}$, $V_{0}$) such that for any $\mu\in(0,\mu^{*})$, there is a unique steady-state $(H^{*},V^{*})\in C^{2}([0,L];(0,\infty))^{2}$ to eq:sv_nonlinear_1--eq:sv_nonlinear_2 satisfying eq:subcritical and the last equation of eq:nonlinear_bc together with $H^ where $O(\mu)$ (resp. $O(\mu^{2})$) refers to a function t

Figures (1)

  • Figure 1: Time evolution of the $L^{2}$-norm of the solution $(h,v)$ of \ref{['eq:linear_system']} for various viscosity values, on a linear (left) and logarithmic (right) scale.

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2: Subcritical flow
  • Remark 2.3
  • Proposition 2.1
  • Remark 2.4
  • Definition 3.1: Exponential stability
  • Proposition 3.1
  • Theorem 3.1
  • Remark 3.1: Subcritical condition
  • Remark 3.2: Limitations
  • ...and 9 more